Rolf Niedermeier

René Van Bevern

Robert Bredereck

Jiehua Chen

Sepp Hartung

Falk Hüffner

Christian Komusiewicz

André Nichterlein

Manuel Sorge

Ondřej Suchý

Bernard Ries

Florian Sikora

Zsolt Tuza

Pierre-Emmanuel Arduin

Amal Benhamiche

Basile Couëtoux

Edouard Bonnet

Yann Dujardin

Florian Jamain

Sébastien Martin

Renaud Lacour

Amine Louati

Lydia Tlilane

Abdallah Saffi- Dine

Raouia Taktak

Emeric Tourniaire

mes colocs de bureau pour les nombreux bons moments passés ensemble

2 : Preliminaries 2.1 Graph theory: terminology and notations 5 2.2 Decision problems & complexity 9 2.3 Parameterized complexity 10 2.3.1 Fixed-parameter algorithm techniques 12 2.3.2 Kernelization lower bounds 14 2.3.3 Fixed-parameter intractability 14 2.3.4 Parameters hierarchies 16 2.4 Approximation 17 2.4.1 Optimization problem 17 2.4.2 Approximation algorithms 19 2.4.3 Hardness of approximation 20 2.4.4 Approximation preserving reductions 22 25 Chapter 3 : Maximizing the spread of influence Trois ans et neuf mois. C'est exactement le temps qui s'est écoulé entre la signature de mon contrat doctoral et l'écriture de cette phrase. Quatre années de recherche qui ont constituées une expérience incroyablement enrichissante à la fois sur le plan scientifique mais également humain. Cependant, s'il y a bien une chose que j'ai apprise, c'est que la recherche ne se fait pas en solitaire et que de nombreuses personnes se cachent derrière chaque octet de ce fichier, devenu thèse. Je souhaite donc à présent leur dédier ce chapitre. Je voudrais tout d'abord exprimer ma gratitude à mon directeur de thèse Cristina Bazgan. Merci de m'avoir fait confiance quatre ans plus tôt. Merci pour ta rigueur, ta franchise, tes conseils avisés, ton écoute et ta disponibilité qui ont su me guider tout au long de cette aventure. Merci à Michael Fellows et Frances Rosamond pour leur inépuisable énergie à partager leurs connaissances. Je remercie en particulier Michael Fellows pour m'avoir introduit de manière avancée et avec dynamisme au domaine de la complexité paramétrée.

55 Chapter 4 : Finding harmless individuals i Merci à mon frère Julien dont les quatre ans d'avance sur moi dans la voie de la recherche m'ont toujours guidés au fil de mes études. Thanks bro! Merci à Céline Zajakala qui fait mon bonheur au quotidien. Sans toujours le savoir, tu as été un véritable manager durant ces années, à me ramener sur terre quand il le fallait, à supporter les moments difficiles et autres innombrables "attend, je dois terminer un truc pour demain". Je ne crois pas pouvoir trouver les mots pour dire à quel point ta bienveillance et ta bonne humeur à toute épreuve ont été et sont essentielles pour moi.

Enfin, mon dernier remerciement s'adresse à ma mère qui a toujours su prendre les bonnes décisions pour moi et m'a permis de faire les études qui me plaisent. Sans cela, les pages qui suivent n'auraient jamais pu voir le jour (et aussi parce que je me suis toujours assis "à côté de quelqu'un qui a une très bonne moyenne")

Résumé de la thèse

Depuis ces dix dernières années, l'évolution des moyens de communication ainsi que la démocratisation d'Internet ont fait des réseaux sociaux un enjeu crucial à la fois du point de vue social bien sûr, mais également économique [START_REF] Easley | Networks, Crowds, and Markets: Reasoning about a Highly Connected World[END_REF]. En particulier, la propagation (ou diffusion) d'information au sein de telles structures n'a cessé de recevoir un interêt croissant de la part de la commmunauté scientifique, motivé par des applications telles que le marketing viral, la diffusion de rumeur ou même la santé publique. Il existe en effet un lien étroit entre la topologie d'un réseau social et la propagation d'une maladie au sein d'une population [START_REF] Christensen | Disease dynamics in a dynamic social network[END_REF]. Le terme "information" est donc à prendre au sens large puisqu'il peut faire référence à une rumeur, une maladie, un feu, un message publicitaire, etc. Par "propagation", nous sous-entendons un mécanisme qui définit la manière dont l'information se transmet d'un individu à l'autre à travers tout le réseau. Afin de pouvoir étudier formellement ces phénomènes de diffusion, plusieurs modèles théoriques basés sur la théorie des graphes ont été proposés [START_REF] Domingos | Mining the network value of customers[END_REF][START_REF] Richardson | Mining knowledge-sharing sites for viral marketing[END_REF][START_REF] Centola | Complex contagions and the weakness of long ties[END_REF][START_REF] Kempe | Maximizing the spread of influence through a social network[END_REF][START_REF] Chen | Efficient influence maximization in social networks[END_REF][START_REF] Chen | Scalable influence maximization for prevalent viral marketing in large-scale social networks[END_REF][START_REF] Jiang | Simulated annealing based influence maximization in social networks[END_REF][START_REF] Leskovec | Cost-effective outbreak detection in networks[END_REF][START_REF] Leskovec | The dynamics of viral marketing[END_REF]. En effet, étant donné que les réseaux sociaux consistent en un ensemble d'individus inter-connectés selon une relation pré-déterminée (relation d'amitié, de travail, amoureuse, etc.), il est naturel de représenter ces derniers à l'aide d'un graphe (voir Figure 1). On dit alors qu'un sommet du graphe est dans l'état "activé" si l'information lui a été transmise. Selon le contexte, le terme "activé" peut correspondre à un individu infecté par un virus, ou encore à une personne ayant connaissance d'une rumeur.

Figure 1: L'aspect "arborescent" d'un graphe représentant des relations amoureuses dans une école Américaine (image créée par Mark Newman d'après Bearman et al. [START_REF] Bearman | Chains of affection: The structure of adolescent romantic and sexual networks[END_REF]). Les sommets en bleus (gris sombre) et roses (gris clair) correspondent, respectivement, aux garçons et filles. Il y a une arête entre deux individus s'ils ont vécu une relation amoureuse au cours des 18 mois de l'étude.

Une fois qu'un tel graphe a été établi, la prochaine étape consiste à en extraire des informations pertinentes. À ce titre, plusieurs études se sont interessées, entre autre, aux questions suivantes: Comment déterminer les individus les plus influents? Quelle partie de la population est la plus encline à résister à une épidémie? Quelle est la meilleur v vi Résumé de la thèse stratégie de vaccination possible si une maladie vient à se propager? Chacune de ces interrogations peut être abordée comme un problème algorithmique, et tout l'enjeu est alors de savoir si "calculer" les réponses correspondantes est faisable en pratique. Cette thèse se donne donc pour but d'examiner la complexité de résolution de ces problèmes du point de vue algorithmique. Pour ce faire, nous nous proposons d'étudier la complexité exacte et paramétrée ainsi que l'approximation de problèmes d'optimisation combinatoire impliquant un processus de diffusion dans les graphes. Ce travail est structuré autour de trois problématiques décrites dans les paragraphes suivants. Pour chacune d'entre elles, nous définissons le ou les problèmes considérés, nous en faisons l'état de l'art et présentons les résultats obtenus dans ce travail.

Maximiser la diffusion d'information (Chapitre 3).

Maximiser la diffusion d'information dans un réseau social est un problème intervenant dans des contextes très divers comme, par exemple, le viral marketing. Cette technique commerciale consiste à promouvoir un produit auprès de personnes influentes à travers un message persuasif. L'objectif est alors de créer un effet de "bouche à oreille" pour que ce message se diffuse le plus largement possible. L'originalité de cette approche vient du fait que ce sont les clients eux-même qui font la publicité du produit. Le problème d'optimisation qui en découle naturellement consiste en la donnée d'un graphe, une valeur de seuil thr(v) associée à chaque sommet v de ce graphe, ainsi que la règle de propagation suivante: un sommet devient actif s'il possède au moins thr(v) voisins activés. Le processus de propagation se déroule alors en plusieurs étapes et se termine lorsque qu'aucun nouveau sommet ne peut être activé. Étant donné ce modèle de diffusion, l'objectif est alors de trouver et activer un ensemble de sommets de taille minimum de telle sorte que tous les sommets du graphe soient activés à la fin du processus de propagation. Ce problème est connu dans la littérature sous le nom de target set selection 1 et a été introduit par Chen [START_REF] Chen | On the approximability of influence in social networks[END_REF].

Ce problème a été montré NP-hard même dans les graphes bipartis de degré borné avec des seuils au plus égaux à deux [START_REF] Chen | On the approximability of influence in social networks[END_REF]. Il est également difficile à approximer à un ratio O(2 log 1-ε n ) pour tout ε > 0 même pour des graphes de degré borné avec des seuils d'au plus deux [START_REF] Chen | On the approximability of influence in social networks[END_REF]. Ce rapport d'inapproximation reste valide pour des seuils majoritaires (i.e. le seuil de chaque sommet est égal à son degré divisé par deux). Dans le cas des seuils unanimes (i.e. le seuil de chaque sommet est égal à son degré), le problème correspond exactement au problème vertex cover. Par conséquent, il est 2-approximable en temps polynomial et est difficile à approximer à un ratio mieux que 1.36 [START_REF] Dinur | The importance of being biased[END_REF]. Concernant sa compléxité paramétrée, le problème est W [START_REF] Ackerman | Combinatorial model and bounds for target set selection[END_REF]-hard pour le paramètre "taille de la solution", même dans les graphes bipartis avec des seuils majoritaires ou au plus égaux à deux [START_REF] Nichterlein | On tractable cases of target set selection[END_REF]. Cependant, Nichterlein et al. [START_REF] Nichterlein | On tractable cases of target set selection[END_REF] ont donnés plusieurs algorithmes paramétrés lorsque les paramètres sont liés à la structure du graphe. De plus, Ben-Zwi et al. [START_REF] Ben-Zwi | Treewidth governs the complexity of target set selection[END_REF] ont montré que le problème est résoluble en temps polynomial pour des graphes de largeur arborescente bornée.

À la lumière de ce dernier constat, nous proposons, dans un premier temps, d'explorer plus en avant la complexité paramétrée de target set selection à l'aide de paramètres structurels. Nos résultats sont résumés dans la figure 2. Une des conclusions de cette étude est que supposer des seuils constants permet de rendre le problème traitable en pratique. A priori, on pourrait penser que c'est une hypothèse assez restrictive, mais dans de nombreux contextes applicatifs, tel que le viral marketing, supposer de tels seuils 1 Nous gardons la dénomination anglaise pour le nom des problèmes. vii est suffisant. En effet, indépendamment du nombre de mes amis, il pourrait suffir que cinq d'entre eux achètent un certain produit pour que je sois convaincu de son utilité et l'achète à mon tour. Dans un second temps, nous avons envisagé l'approximation paramètrée du problème complémentaire max influence: Étant donné un graphe, une valeur de seuil thr(v) associée à chaque sommet v de ce graphe, et un entier k > 0, l'objectif est de trouver et activer un ensemble de sommets de taille au plus k de telle sorte que le nombre de sommets activés à la fin du processus de propagation soit maximum. Le processus de propagation est le même que celui défini pour target set selection. Ce problème est en fait la version déterministe du problème introduit par Kempe et al. [START_REF] Kempe | Maximizing the spread of influence through a social network[END_REF]. Nos résultats sont résumés dans le tableau 1. Notons qu'il existe deux façons de mesurer la taille de la solution en comptant ou non l'ensemble initialement activé. De ce fait, nous définissons, respectivement, les deux problèmes max closed influence et max open influence. On rappelle qu'un problème d'optimisation est α(n)-approximable en temps fpt par rapport au paramètre k si le problème est α(n)-approximable en temps f (k)•n O (1) où f est une fonction qui ne dépend que de k. Il est intéressant de noter que max open influence, pour le cas unanime, est α(n)-approximable en temps fpt par rapport au paramètre k pour toute fonction croissante α (Corollary 32) alors qu'il est W [START_REF] Aazami | Approximation algorithms and hardness for domination with propagation[END_REF]-hard par rapport à ce même paramètre (Theorem 29) et inapproximable en temps polynomial à un ratio n 1-ε pour tout ε > 0 si NP = ZPP (Theorem 30).

Vertex Cover

Number [START_REF] Nichterlein | On tractable cases of target set selection[END_REF] [90] [START_REF] Nichterlein | On tractable cases of target set selection[END_REF] Feedback Edge Set Number [START_REF] Nichterlein | On tractable cases of target set selection[END_REF] [90] [START_REF] Nichterlein | On tractable cases of target set selection[END_REF] Distance to Clique

Th.18 Th.18

Cluster Vertex Deletion Number

Th. [START_REF] Bousquet | Multicut is FPT[END_REF] Th.16

Clique Cover Number

Th.17

Bandwidth

Th.15Th.15Th.15

Distance to Cograph

Th.13 [START_REF] Ben-Zwi | Treewidth governs the complexity of target set selection[END_REF] Feedback Vertex Set Number [START_REF] Ben-Zwi | Treewidth governs the complexity of target set selection[END_REF] Th.13 [START_REF] Ben-Zwi | Treewidth governs the complexity of target set selection[END_REF] Distance to Interval Th.13 [START_REF] Ben-Zwi | Treewidth governs the complexity of target set selection[END_REF] Pathwidth [START_REF] Ben-Zwi | Treewidth governs the complexity of target set selection[END_REF] Th.13 [START_REF] Ben-Zwi | Treewidth governs the complexity of target set selection[END_REF] Treewidth [START_REF] Ben-Zwi | Treewidth governs the complexity of target set selection[END_REF] Th.13 [START_REF] Ben-Zwi | Treewidth governs the complexity of target set selection[END_REF] Figure 2: Panorama de nos résultats de complexité paramètrée pour target set selection avec des paramètres structurels. Les trois cases en-dessous de chaque paramètre indiquent un résultat pour des seuils (de gauche à droite) bornés par une constante, majoritaire, et sans restriction. Une arête joignant un paramètre k 2 à un autre paramètre k 1 en dessous de k 2 implique l'existence d'une constante c > 0 telle que k 1 ≤ c • k 2 . Le paramètre "Distance to C" où C est une classe de graphe correspond au nombre minimum de sommets à retirer de sorte que le graphe obtenu appartient à C. Pour le paramètre "clique cover number", la case noire indique que le problème est NP-hard pour une valeur constante du paramètre. La couleur violette (gris sombre) indique un résultat de W [START_REF] Aazami | Approximation algorithms and hardness for domination with propagation[END_REF]-hardness alors que la couleur verte (gris clair) signifie que le problème admet un algorithme paramétré. Lorsque la case est blanche, la question est ouverte.

Les résultats présentés dans cette section sont basés sur les papiers suivants: n 1-ε ,∀ε > 0 n 1-ε ,∀ε > 0 n 1-ε ,∀ε > 0 n 1-ε ,∀ε > 0 (Th.27) Maj.

Sup. n n n n Inf.

n 1-ε ,∀ε > 0 n 1-ε ,∀ε > 0 n 1-ε ,∀ε > 0 n

Déterminer un ensemble inoffensif (Chapitre 4).

Dans l'étude précédente, nous avons souligné la difficulté de traitabilité du problème target set selection et avons donc proposé d'autres approches afin d'obtenir des résultats positifs. Au regard de la nature intraitable de ce problème, il est intéressant de se poser la question de la complexité du problème inverse que l'on nomme harmless set: Étant donné un graphe, une valeur de seuil thr(v) associée à chaque sommet v, et un entier k > 0, l'objectif est de trouver un ensemble de sommets de taille au moins k de sorte qu'activer des sommets quelconque dans cet ensemble ne permet d'activer aucun sommet par propagation. Autrement, on souhaite trouver un ensemble S de sommets tel que tout sommet v du graphe a un nombre de voisins dans S inférieur à thr(v). On appelle alors l'ensemble S un ensemble inoffensif. Ce problème pourrait intervenir, par exemple, dans un contexte épidémique. L'objectif serait alors de déterminer la résistance naturelle d'une population face à la propagation d'un virus. Il est intéressant de noter que ce problème peut être mis en relation avec le problème (σ, ρ)-dominating set introduit par Telle [START_REF] Arne | Complexity of domination-type problems in graphs[END_REF]: Étant donné un graphe G = (V, E), deux ensembles d'entiers non-négatifs σ et ρ, et un entier k ≥ 1, l'objectif consiste à trouver un ensemble S ⊆ V de taille au plus k tel que pour tout sommet v ∈ V , le nombre de voisins de v dans S appartient à σ et le nombre de voisins de v n'appartenant pas à S appartient à ρ. On dit alors que S est un ensemble (σ, ρ)-dominant. Il se trouve que si tous les seuils sont égaux alors harmless set est équivalent à (σ, ρ)-dominating set of size k [START_REF] Golovach | Parameterized complexity of generalized domination problems[END_REF] (qui demande un ensemble (σ, ρ)-dominant de taille exactement k) avec σ = ρ = {0, . . . , thr max } où thr max est la valeur du seuil maximum. Puisque ce dernier problème est dans W [START_REF] Aazami | Approximation algorithms and hardness for domination with propagation[END_REF] [61], cela implique que harmless set est également dans W [START_REF] Aazami | Approximation algorithms and hardness for domination with propagation[END_REF] si tous les seuils sont égaux. Á notre connaissance, c'est le seul résultat qui peut être transféré à harmless set.

Dans cette thèse, nous avons étudié la complexité paramétrée (voir le tableau 2) ainsi que l'approximation du problème de maximisation associé max harmless set. Nous avons établi que max harmless set est inapproximable à un ratio n 1-ε pour tout ε > 0 avec des seuils d'au plus deux. Si les seuils sont unanimes, le problème est APX-hard et est 3-approximable en temps linéaire. Nous donnons également un schéma d'approximation polynomiale pour les graphes planaires.

Seuils

Contenir la propagation d'information (Chapitre 5).

Dans cette partie finale, nous avons étudié le problème firefighter. De manière informel, l'objectif est ici de stopper la propagation d'information au sein d'un réseau en ayant la possibilité de "protéger" certains sommets. Un sommet protégé ne pouvant plus être activé. Ce problème a été initialement introduit par Hartnell en 1995 [START_REF] Hartnell | Firefighter! an application of domination[END_REF], et a depuis été très largement étudié [START_REF] Anshelevich | Approximation algorithms for the firefighter problem: Cuts over time and submodularity[END_REF][START_REF] Cai | Firefighting on trees: (1 -1/e)-approximation, fixed parameter tractability and a subexponential algorithm[END_REF][START_REF] Chalermsook | Resource minimization for fire containment[END_REF][START_REF] Finbow | The firefighter problem for graphs of maximum degree three[END_REF][START_REF] Kah | A generalization of the firefighter problem on ZxZ[END_REF][START_REF] Finbow | The firefighter problem: a survey of results, directions and questions[END_REF]. Il est défini de la manière suivante: initialement un sommet particulier d'un graphe est activé. À chaque pas de temps, on applique successivement les deux étapes suivantes: 1) Protéger un sommet non-activé du graphe; 2) Tous les sommets non-protégés et adjacents à un sommet activé sont activés. Le processus se termine lorsque plus aucun nouveau sommet ne peut être activé. Un sommet est alors considéré comme sauvé s'il n'est pas activé. Étant donné un entier k > 0, l'objectif est de trouver une stratégie de protection des sommets permettant de sauver au moins k sommets.

Ce problème a été montré NP-difficile dans les graphes bipartis [START_REF] Macgillivray | On the firefighter problem[END_REF], graphes cubiques [START_REF] King | The firefighter problem for cubic graphs[END_REF], et graphes de disque-unité [START_REF] Fomin | Making life easier for firefighters[END_REF]. Finbow et al. [START_REF] Finbow | The firefighter problem for graphs of maximum degree three[END_REF] ont montré que le problème est NP-hard même dans les arbres de degré au plus trois et résoluble en temps polynomial dans les graphes de degré au plus trois si le sommet initialement activé est de dégré au plus deux. De plus, le problème firefighter est résoluble en temps polynomial dans les caterpillars et les P-arbres [START_REF] Macgillivray | On the firefighter problem[END_REF]. 2 Du point de vue de l'approximation, la version 2 Un P-arbre [START_REF] Macgillivray | On the firefighter problem[END_REF] est un arbre qui ne contient pas la configuration suivante:

niveau i niveau i + 1 niveau i + 2
x Résumé de la thèse maximisation du problème, qui consiste à maximiser le nombre de sommets sauvés, est e e-1 -approximable dans les arbres [START_REF] Cai | Firefighting on trees: (1 -1/e)-approximation, fixed parameter tractability and a subexponential algorithm[END_REF] et non n 1-ε -approximable dans les graphes généraux pour tout ε > 0, si P = NP [START_REF] Anshelevich | Approximation algorithms for the firefighter problem: Cuts over time and submodularity[END_REF]. Pour les arbres où chaque sommet a au plus trois fils, le problème est 1.3997-approximable [START_REF] Iwaikawa | Improved approximation algorithms for firefighter problem on trees[END_REF]. Peu de résultats sont connus concernant la complexité paramétrée du problème. Cai et al. [START_REF] Cai | Firefighting on trees: (1 -1/e)-approximation, fixed parameter tractability and a subexponential algorithm[END_REF] fournissent des algorithmes paramétrés dans le cas des arbres pour chacun des paramètres suivants: le nombre de sommets sauvés, le nombre de sommets brûlés, et le nombre total de sommets protégés. Pour le paramètre "nombre de sommets sauvés", les auteurs donnent un noyau polynomial.

Dans cette thèse, nous considérons la version plus générale du problème où b ≥ 1 (appelé budget) sommets peuvent être protégés à chaque pas de temps. Nous étudions également le dual noté dual firefighter dont l'objectif est de sauver au moins nk b sommets où n est la taille du graphe et k b est un entier positif. Pour terminer, nous considérons le problème bounded firefighter qui est défini de manière similaire au problème firefighter excepté que l'on est autorisé à protéger un total d'au plus k p ≥ 1 sommets, où k p est un entier donné en entrée du problème. Nous montrons que le problème firefighter est NP-complet dans les arbres de degré au plus b + 3 ainsi que dans les arbres de pathwidth au plus trois. Cependant, nous montrons que le problème est résoluble en temps polynomial pour la classe des graphes dont le degré maximum et le pathwidth sont bornés. Nous fournissons également un algorithme polynomial pour résoudre le problème (et la version pondérée correspondante) pour une sous classe d'arbres de pathwidth deux, les k-caterpillars. Nous établissons des bornes de compléxité paramétrée inférieures et supérieures pour les problèmes firefighter, dual firefighter, et bounded firefighter dans les graphes généraux par rapport aux paramètres standards (see Figure 3). Nous donnons également des algorithmes paramétrés dans les arbres qui améliorent les résultats obtenus par Cai et al. [START_REF] Cai | Firefighting on trees: (1 -1/e)-approximation, fixed parameter tractability and a subexponential algorithm[END_REF]. Nous répondons également à plusieurs questions ouvertes de [START_REF] Cai | Firefighting on trees: (1 -1/e)-approximation, fixed parameter tractability and a subexponential algorithm[END_REF]. De plus, nous établissons plusieurs algorithmes paramétrés par rapport à des paramètres liés à la structure du graphe (see Figure 4). Pour terminer, nous observons que la version minimisation du problème firefighter est inapproximable à un ratio n 1-ε même dans les arbres pour tout ε > 0 et tout budget b ≥ 1 si P =NP. Nous répondons de manière négative à une question ouverte de Finbow and MacGillivray [START_REF] Finbow | The firefighter problem: a survey of results, directions and questions[END_REF].

Figure 3: Résumé de nos résultats de compléxité paramétrée incluant la paramétrisation standard. À chaque colonne est associé un problème et son paramètre standard. À chaque ligne, excepté la première, correspond également un paramètre. L'intersection d'une ligne et d'une colonne donne un résultat de complexité paramétrée par rapport au paramètre combiné de la ligne et de la colonne.

Les résultats présentés dans cette section sont basés sur les papiers suivants:

O

ver the last past decade, the evolution of communication technology together with the democratization of Internet have made social networks a major economic and social stake [START_REF] Easley | Networks, Crowds, and Markets: Reasoning about a Highly Connected World[END_REF]. In particular, the propagation (or diffusion) of information through these networks has gained a lot of interest driven by applications such as viral marketing, rumor spreading or even public health (for instance, there is a close connection between the topology of social networks and the propagation of a disease through a population [START_REF] Christensen | Disease dynamics in a dynamic social network[END_REF]). The term "propagation of information" should be considered in a broad sense here as it may appear in various contexts. In order to better understand its meaning, let us consider the following real-world examples. In 1996, the email service Hotmail added the following simple message to the footer of every mail sent out by users "Get your free email at Hotmail". As a result, the number of subscribers grew by 12 million in 18 months. In this case, the propagated information was a textual advertisement message that goes (propagates) from one user to many others. As a matter of fact, this marketing technique is called "viral advertisement" due to its analogy with the spread of viruses or computer viruses. Another example is from a study of Christakis and Fowler [START_REF] Christakis | The Spread of Obesity in a Large Social Network over 32 Years[END_REF]. In their work, the authors analyzed the obesity propagation during 32 years through a social network having more than 12000 persons. They found that having obese friends increase by 57% the chance of developing obesity. Moreover, they also observed that obesity may influence up to three degrees of separation i.e. if the friend of a friend of my friend is obese then my obesity risk is increased. Here the propagation is clearly of psychological nature. To formally capture these diffusion phenomenons, several theoretical models have emerged recently [START_REF] Domingos | Mining the network value of customers[END_REF][START_REF] Richardson | Mining knowledge-sharing sites for viral marketing[END_REF][START_REF] Centola | Complex contagions and the weakness of long ties[END_REF][START_REF] Kempe | Maximizing the spread of influence through a social network[END_REF][START_REF] Chen | Efficient influence maximization in social networks[END_REF][START_REF] Chen | Scalable influence maximization for prevalent viral marketing in large-scale social networks[END_REF][START_REF] Jiang | Simulated annealing based influence maximization in social networks[END_REF][START_REF] Leskovec | Cost-effective outbreak detection in networks[END_REF][START_REF] Leskovec | The dynamics of viral marketing[END_REF]. As social networks consist of individuals who are linked together by a pre-determined relationship, they are often represented as graphs. A graph is a mathematical object that comes up with a collection of vertices together with a collection of edges that connect pairs of vertices (see Figure 1.1).

Given a graph that represents a social network, a natural next step is to determine what kind of knowledge one can learn from it. For instance, several studies investigate the following questions among others: who are the most influencer individuals? what part of a population is the most resistant to an epidemic outbreak? in the case of a spread of a disease, what is the best vaccination strategy? Each of these interrogations can be regarded as a particular problem to solve, and the question that naturally follows is whether it is an easy task or not. This leads us to the main objective of this thesis which is to address the complexity of these problems from the computer science point of view. More specifically, the goal is to design efficient algorithms that compute the answer of the previous questions. By "efficient" we mean that the running time of these algorithms on a computer is required to be reasonable i.e. they should not run more than a few days. If such an algorithm exists, we say that the problem is practically solvable. As a matter of fact, all problems in this work are combinatorial problems. At its most general form, a combinatorial problem is a problem for which the goal is to find an optimal solution among a large finite set of solutions. Consider for example a well known combinatorial problem "Traveling Salesman Problem (TSP)" [START_REF] David | The Traveling Salesman Problem: A Computational Study[END_REF]: Given a list of cities and distances between each pair of cities, TSP consists of finding a shortest possible route that visits each city exactly once and returns to the origin city. One might think that it suffices to exhaustively Figure 1.1: The "tree-like" aspect of a graph representing romantic relationships in an American high school (image drawn by Mark Newman from Bearman et al. [START_REF] Bearman | Chains of affection: The structure of adolescent romantic and sexual networks[END_REF]). Blue (dark gray) and pink (light gray) vertices correspond to male and female, respectively. An edge joins two individuals if they were romantically involved during the 18 months in which the study was conducted. check every route to eventually find the optimal one. However, a typical combinatorial problem has a number of possible solutions which is so large that the best solution could not be obtain within a reasonable amount of time using this naive technique. This is what we call the combinatorial explosion. Considering the traveling salesman problem with only 20 cities, such exhaustive algorithm may take centuries before we get the optimal solution, even with the world-fastest computer. To be convinced, let us consider a really fast computer able to compare two possible routes and say which one is the shortest in less than 0.000000001 second or, equivalently, one nanosecond i.e. faster than the time taken for light to travel one meter. It turns out that for 20 cities, the number of possible routes is about 2.4 × 10 18 . Since the computer needs to compare every route with each other, this would take 2.4 × 10 18 nanoseconds to find the shortest route which corresponds roughly to 770 years! This is definitely not a reasonable running time to solve the problem. The central question is then to determine whether there exists a better approach i.e. for a given problem, is there an algorithm that could possibly solve it in reasonable time? This leads us to the concept of computational "hardness" of a problem. This notion is rigorously defined in computational complexity theory but, for the sake of clarity, we only say here that a problem is hard if its solution requires significant resources (essentially time and memory) to be computed. Computer scientists have grouped problems into classes based on how long they take to be solved. The class NP gathers problems for which an answer can be verified in a reasonable amount of time. Some problems of NP can in fact be solved quickly. Those problems are said to be in P, which stands for polynomial time. However, there are other problems in NP which have never been solved in polynomial time and widely believed of not being so. Those latter problems are said NP-hard (like the traveling salesman problem). In fact, problems in NP and P are required to be decision problems, i.e. a problem with only two possible solutions "yes" or "no". However, this is not restrictive since every combinatorial optimization problem has a computationally equivalent decision problem. For example, computing an optimal solution for the traveling salesman problem is at least as hard as to solve its decision version: given an integer k, is there a route of length at most k that visits each city exactly once and returns to the origin city?

In this thesis, we consider combinatorial optimization problems related to the diffusion of information in social networks. We then classify them either by proving that they are polynomial-time solvable or by proving that they are in fact NP-hard. In the last case, we try to rely on other type of methods such as approximation or parameterized algorithms to solve them. For an approximation algorithm, the basic idea is to release the constraint of optimality while the running time is required to be reasonable, and the computed solution must be guaranteed to be "close" to the optimum. A parameterized algorithm tries to get rid of the easy-to-solve parts of a problem so that it only remains what makes the problem difficult. The idea is to cope with a smaller equivalent version of the problem, thus making algorithms running faster on it. We can even combine the two approaches to get a parameterized approximation algorithm.

Let us consider a viral marketing problem as an illustration of the problems studied in this thesis. The aim of viral marketing is to advertise a product to the most influential customers who are most likely to produce a "word-of-mouth" effect through their social network. The advantage of this technique is that the customers perform themselves the advertisement of a product, saving thus a lot of money for the company. The Figure 1.2 depicts the partial social network of a customer called "Bob" by the use of an undirected graph. Each vertex corresponds to a customer and an edge between two vertices indicates that these two individuals know each other. The number inside each vertex is called "threshold". A customer with threshold t is convinced of a product's usefulness and buys it if at least t of its friends have one. In this thesis, we speak in a more general sense and thus say that a vertex of a graph is "active" instead of "convinced". Depending of the application context, the term "active" could also mean "infected" or "burned" as well. In this example, Bob would buy an xphone if three of his friends have it. Suppose now that the company can offer a very limited number of xphone, say two in this example. The question is now to find who are the two customers to give the product. Of course, the stake here is to choose the two most influencer ones. In fact, one optimal solution would be to choose Edward and Carol. In this little example, the optimal solution is very easy to find and an exhaustive search approach would have solved the problem quickly. However, realworld social networks might involve over millions of individuals with complex relationship, thus making exhaustive algorithms useless. This thesis is structured around three different problematics described hereafter and further developed in the subsequent chapters. Firstly, we recall basic theoretical background in Chapter 2. In Chapter 3, we consider two problems with the objective of max-imizing the spread of influence in social networks. The first one is the target set selection problem. Given a social network represented as a graph and a threshold value thr(v) associated to each vertex v, the task is to find and activate a vertex subset of minimum size such that all the vertices become active at the end of the propagation process defined as follows. A vertex v becomes active if at least thr(v) of its neighbors are active. The propagation process proceeds in several steps and stops when no further vertex becomes active. The next investigated problem is maximum influence which is defined similarly except that the input has an extra integer k (it corresponds to some "budget") and we ask to activate at most k vertices such that the total number of activated vertices at the end of the propagation process is maximized (this is exactly the problem in the above viral marketing example). In Chapter 4, we turn our attention to a converse objective: Find the largest set of vertices such that if any vertices get activated in it then no new vertex can be activated by the application of the propagation rule. One motivation for this problem arises from the context of preventing the spread of dangerous ideas or epidemics. It also corresponds to the objective of finding a population resistant to an epidemic outbreak. In Chapter 5, we now consider the problem of containing a malicious agent (fire, virus, . . . ) which has already started to propagate through a network. Initially, the outbreak starts at a single vertex of a graph. At each time step, we have to choose one vertex which will be protected. Then the outbreak spreads to all unprotected neighbors of the "infected" vertices. The process ends when the outbreak can no longer spread, and then all vertices that are not infected are considered as saved. The objective consists of choosing, at each time step, a vertex which will be protected such that a maximum number of vertices in the graph is saved at the end of the process. Finally, the Chapter 6 provides conclusions and future research directions.

≥ 2 Alice ≥ 2 F rank ≥ 2 Gordon ≥ 2 Dave Edward ≥ 3 Bob Carol ≥ 2 Alice ≥ 2 F rank ≥ 2 Gordon ≥ 2 Dave Edward ≥ 3 Bob Carol

Chapter 2 Preliminaries

2.1

Graph theory: terminology and notations T the purpose of this chapter is to give the basic backgrounds on classical and parameterized complexity, approximation as well as some basic graph theory concepts and notations used throughout this thesis. For more details about parameterized complexity theory, the reader is referred to the books of Downey and Fellows [START_REF] Downey | Parameterized Complexity[END_REF], Niedermeier [START_REF] Niedermeier | Invitation to Fixed-Parameter Algorithms[END_REF], and Flum and Grohe [START_REF] Flum | Parameterized Complexity Theory[END_REF]. Concerning the approximation theory, we recommend the following books of Ausiello, Crescenzi, Gambosi, Kann, Marchetti-Spaccamela, and Protasi [START_REF] Ausiello | Complexity and Approximation: Combinatorial Optimization Problems and Their Approximability Properties[END_REF], Hochbaum [START_REF] Hochbaum | Approximation algorithms for NP-hard problems[END_REF], Vazirani [START_REF] Vazirani | Approximation Algorithms[END_REF], and Williamson and Shmoys [START_REF] Williamson | The Design of Approximation Algorithms[END_REF].

Graph theory: terminology and notations

We denote a graph by an ordered pair G = (V, E) where V is the set of vertices and E ⊆ V × V the set of edges. We now define basic graph terminology.

A vertex v ∈ V is adjacent to another vertex u ∈ V if there is an edge uv ∈ E connecting them. The neighbors (or neighborhood) of a vertex v is the set of vertices adjacent to v. The degree of a vertex is the number of its neighbors.

Two vertices are said twins if they have the same neighborhood. They are called true twins if they are moreover neighbors, false twins otherwise.

A path is either a single vertex or a graph where every vertex has degree one or two, and exactly two vertices have degree one (called endpoints). The length of a path is the number of edges.

The diameter of a graph is the longest shortest path between any two vertices.

A graph is connected if there is a path joining every pair of vertices.

A cycle is a connected graph where every vertex has degree two.

A subgraph H = (V ′ , E ′ ) of G is a graph where V ′ ⊆ V and E ′ ⊆ E. A subgraph H = (V ′ , E ′ ) of G is said to be induced by V ′ if, for any pair of vertices u, v ∈ V ′ , we have uv ∈ E ′ if and only if uv ∈ E.
A linear layout of G is a bijection π : V → {1, . . . , n}. For convenience, we express π by the list L = (v 1 , . . . , v n ) where π(v i ) = i. Given a linear layout L, we denote the distance between two vertices in L by d L (v i , v j ) = |i -j|.

The cutwidth cw(G) of G is the minimum integer k such that the vertices of G can be arranged in a linear layout L = (v 1 , . . . , v n ) in such a way that, for every i = 1, . . . , n -1, there are at most k edges with one endpoint in {v 1 , . . . , v i } and the other in {v i+1 , . . . , v n }

The bandwidth bw(G) of G is the minimum integer k such that the vertices of G can be arranged in a linear layout L = (v 1 , . . . , v n ) in such a way that, for every edge

v i v j of G we have d L (v i , v j ) ≤ k.
If the graph G is a directed graph (or digraph for short), then we need to adjust some previously established terminologies. In a directed graph every edge uv is called an arc and is directed either from u to v, denoted (u, v), or from v to u, denoted (v, u). We will denote by A rather than by E the set of arcs in G. The in-neighbors (resp. outneighbors) of a vertex v is the set of vertices

N + (v) = {u ∈ V : (u, v) ∈ A} (resp. N -(v) = {u ∈ V : (v, u) ∈ A}).
Unless otherwise specified, all graphs in this thesis are undirected, finite (bounded number of vertices) and simple (at most one edge connects two vertices).

Notations. Let G = (V, E) be a graph, we will use the following standard notations.

• N G (v) : open neighborhood of a vertex v i.e. N G v) = {u ∈ V : uv ∈ E}. • N G [v] : close neighborhood of a vertex v i.e. N G [v] = N G (v) ∪ {v}. • N G (S) : open neighborhood of a set S ⊆ V i.e. N G (S) = u∈S N G (u). • N G [S] : close neighborhood of a set S ⊆ V i.e. N G [S] = N G (S) ∪ S.
• dist G (u, v) : distance between vertices u and v i.e. minimum length of a path with endpoints u, v ∈ V .

• N i G (v) : set of vertices which are at distance at most i from vertex v (called

i th neigh- borhood of v) i.e. N i G (v) = {u ∈ V : dist G (v, u) ≤ i}. Thus N 1 G (v) = N G (v).
• G[S] : the subgraph induced by a set S ⊆ V .

• deg G (v) : degree of vertex v i.e. deg G (v) = |N G (v)|.
• ∆(G) : maximum degree of G.

• tw(G) : treewidth of G (see Definition 2).

• ltw r (G) : local treewidth of G with respect to r ∈ N (see Definition 4).

• pw(G) : pathwidth of G (see Definition 6).

• cw(G) : cutwidth of G.

• bw(G) : bandwidth of G.

• V (H) : the set of vertices of a graph H.

• E(H) : the set of edges of a graph H.

We may skip the subscript or the argument if the graph G is clear from the context.

Graph classes. We briefly review the graph classes encountered throughout this work.

A tree is a connected graph without cycles. Let T be a tree and let r be a vertex of T designated as root. We say that T is a rooted tree. We define the level i of T to be the set of vertices that are at distance exactly i from r. A leaf is a vertex of degree one. An ancestor (resp. descendant) of a vertex v in T is any vertex on the path from r to v (resp. from v to a leaf which does not contain any ancestors of v). The height of T is the length of a longest path from r to a leaf. A child of a vertex v in T is an adjacent descendant of v. The tree T is said to be complete if every non-leaf vertex has exactly the same number of children. The tree T is said to be full if it is complete and all leaves are at the same distance from the root.

A t-ary tree is a rooted tree in which every vertex other than the leaves has t children.

A graph is bipartite if the vertices can be partitioned into two sets such that any two vertices in the same set are not adjacent.

A graph is regular if all the vertices have the same degree. A ∆-regular graph is a regular graph where vertices have degree ∆.

A graph is complete (or is a clique) when every vertex is adjacent to every other vertex.

A graph is planar if it can be drawn in the plane without any edges crossing.

A caterpillar is a tree such that the vertices with degree at least two induce a path. In other words, a caterpillar consists of a path P such that all edges have at least one endpoint in P .

A k-caterpillar , for some integer k ≥ 1, is a caterpillar in which every pending edge, i.e. every edge uv with exactly one endpoint, say u, in P , may be replaced by a path of length at most k. This path is then called a leg of the k-caterpillar at vertex u.

A star is a tree consisting of one vertex, called the center of the star, adjacent to all the other vertices.

A k-star , for some integer k ≥ 1, is a tree obtained from a star in which every edge may be replaced by a path of length at most k. Notice that a k-star is a special case of a k-caterpillar.

A cograph is a graph that does not contain an induced P 4 , that is, a path on four vertices.

An interval graph is a graph G = (V, E) for which there exists a set of real intervals

{I v : v ∈ V } such that I v ∩ I u = ∅ if and only if uv ∈ E.
(Local) Treewidth. Informally speaking, the treewidth of a graph is a number that reflects how "close" the graph is to being a tree. This notion was first introduced by Halin [START_REF] Halin | s-functions for graphs[END_REF] and further developed by Robertson and Seymour [START_REF] Robertson | Graph minors. III. Planar tree-width[END_REF]. As a matter of fact, treewidth can be defined in several equivalent ways. Here we use the notion of tree decomposition of a graph to define it.

Definition 1: Tree decomposition

A tree decomposition of a graph G = (V, E) is a pair T = (T, H) where T is a tree with vertex set X and H = {H x : x ∈ X} is a family of subsets of V, such that the following conditions are met 1. x∈X H x = V.

2.

For each uv ∈ E there is an x ∈ X with u, v ∈ H x .

3. For each v ∈ V , the set of nodes {x ∈ X : v ∈ H x } induces a subtree of T .

In the above definition, the third condition is equivalent to assuming that if v ∈ H x ′ and v ∈ H x ′′ then v ∈ H x for all nodes x of the unique path from x ′ to x ′′ in T . The width of a tree decomposition T is max x∈X |H x | -1. We are now ready to give the definition of treewidth.

Definition 2: Treewidth

The treewidth tw(G) of a graph G is the minimum width over all possible tree decompositions of G.

The "-1" in the definition is included for the convenience that trees have treewidth 1 (rather than 2). It is worth noting that finding the treewidth of a graph is NP-hard [START_REF] Arnborg | Complexity of finding embeddings in a k-tree[END_REF] but can be found in linear time for graphs of bounded treewidth [START_REF] Bodlaender | A linear-time algorithm for finding tree-decompositions of small treewidth[END_REF]. Concerning the appearance of substructures, one can see that the subtree T x of T rooted at node x represents the subgraph G x induced by precisely those vertices of G which occur in at least one H y where y runs over the nodes of T x .

In an algorithmic perspective, we rather use a more refined decomposition called nice tree decomposition.

Definition 3: Nice tree decomposition

A nice tree decomposition T = (T, H) of a graph G is a tree decomposition satisfying the following conditions 1. Each node of T has at most two children.

2. For each node x with two children y, z, we have

H y = H z = H x (x is called join node).
3. If a node x has just one child y, then

H x ⊂ H y (x is called forget node) or H y ⊂ H x (x is called insert node) and ||H x | -|H y || = 1.
It is not hard to show that any tree decomposition T = (T, H) of a graph can be transformed in linear time into a nice tree decomposition T ′ = (T ′ , H ′ ) of same width, with |T ′ | ≤ c • |T | for some constant c > 0 and with H x = ∅ for all H x ∈ H.

Eppstein [START_REF] Eppstein | Diameter and treewidth in minor-closed graph families[END_REF] generalized the notion of treewidth by introducing the notion of bounded local treewidth. Informally speaking, a graph has bounded local treewidth if, for any vertex v, the treewidth of the induced subgraph by the r th neighborhood of v is bounded by a function that solely depends on r > 0.

Definition 4: Local treewidth

Given an integer r > 0, the local treewidth of a graph G = (V, E) is the number ltw r (G) defined as follows

ltw r (G) = max v∈V {tw(G[N r (v)])}
We then say that a graph G has bounded local treewidth if there exists a function f : N → N such that ltw r (G) ≤ f (r) for all integer r > 0. Here are some class of graphs of bounded local treewidth.

Every graph G of bounded treewidth has bounded local treewidth since ltw r (G) ≤ tw(G) for all r > 0.

Every graph of maximum degree ∆ has local treewidth bounded by ∆(∆ -1) r-1 for all r > 0.

Finally, a planar graph has local treewidth bounded by 3r -1 for all r > 0 [START_REF] Hans | A partial k-arboretum of graphs with bounded treewidth[END_REF].

Pathwidth. As treewidth measure the "tree-likeness" of a graph, the pathwidth reflects how "close" the graph is to being a path. Because of this analogy, we directly give the relevant definitions.

Definition 5: Path decomposition

A path decomposition of a graph G = (V, E) is a pair P = (P, H) where P is a path with vertex set X and H = {H x : x ∈ X} is a family of subsets of V, such that the following conditions are met 1. x∈X H x = V.

2. For each uv ∈ E there is an x ∈ X with u, v ∈ H x .

3. For each v ∈ V , the set of nodes {x ∈ X : v ∈ H x } induces a path.

Definition 6: Pathwidth

The pathwidth pw(G) of a graph G is the minimum width over all possible path decompositions of G.

We also have the notion of nice path decomposition but we skip its definition here since we do not make use of it.

Decision problems & complexity

Before we give the definition of a decision problem, we first recall some basics from the theory of formal languages. An alphabet, denoted by Σ, is a set of symbols. A string (or word) is a finite sequence of symbols from a given alphabet. The length of a string x, denoted |x|, is the number of symbols in the string. The set of all strings over an alphabet Σ is denoted by Σ * . A language L is a subset of Σ * .

Definition 7: Decision problem

A decision problem is a language L ⊆ Σ * over a binary alphabet Σ = {0, 1}.

In fact, when we introduce a new decision problem, we will make use of the following more informal and standard way to define it.

Problem Name

Input: Some inputs Question: A yes-no question that solely depends upon the inputs. We do not discuss here how to encode such definition into a regular decision problem i.e. into a language L ⊆ Σ * , and we assume, throughout this thesis, that we use only reasonable encoding to do so (see Garey and Johnson [60,Chapter 2.1]). More generally, the specification of any kind of problem (decision problem, parameterized problem (see Definition 11), and optimization problem (see Definition 20)) as well as integers, graphs, formulas, etc. is assumed to be encoded in binary in the usual way. Therefore, we will address these objects directly instead of working with their formal string representations.

An instance of a decision problem is a concrete utterance of the problem represented as a string x ∈ Σ * . The size of an instance is then the size of the corresponding string. Given an instance x ∈ Σ * of a decision problem L ⊆ Σ * , we say that x is a yes-instance if x ∈ L and a no-instance otherwise.

P & NP classes. The subsequent definitions make use of the concept of Turing machine, see for instance Arora and Barak [START_REF] Arora | Computational Complexity: A Modern Approach[END_REF] for more details.

Definition 8: NP class

The class NP contains every decision problem L ⊆ Σ * for which the question "Does x belongs to L?" where x ∈ Σ * can be decided by a non-deterministic Turing machine that runs in polynomial time i.e. , the number of steps performed by the machine is upper bounded by a polynomial expression in |x|.

Let L 1 , L 2 ⊆ Σ * be two decision problems. We say that L 1 polynomial-time reduces to L 2 if there exists an algorithm that takes as input an instance x 1 ∈ Σ * and outputs in polynomial time a new instance

x 2 ∈ Σ * such that x 1 ∈ L 1 if and only if x 2 ∈ L 2 .
A decision problem L is NP-hard if every problem of NP polynomial-time reduces to L. If a decision problem is NP-hard and is in NP then it is NP-complete.

Definition 9: P class

The class P contains every decision problem L ⊆ Σ * for which the question "Does x belongs to L?" where x ∈ Σ * can be decided by an algorithm that runs in polynomial time i.e. , the number of steps performed by the algorithm is upper bounded by a polynomial expression in |x|.

Asymptotic notation. In order to express the running time of an algorithm, we use the following standard notation.

Definition 10: Big O notation

Let g be a real function. We denote by O(g(n)) the set of all real functions f for which there exist a constant c > 0 and a value n 0 such that f (n) ≤ c • g(n) for all n > n 0 .

We might use the above notation in a more involved way. For example, we denote by n O(1) some function f of the form f (n) = n d where d ∈ O(1).

Parameterized complexity

The parameterized complexity is a framework which provides a new way to express the computational complexity of decision problems. For example, consider the well known NP-complete Vertex Cover problem: given an graph G = (V, E) and an integer k, determine whether there is a subset S ⊆ V , |S| ≤ k, such that every edge is covered by S i.e., for all uv ∈ E we have u ∈ S or v ∈ S. Since the problem is NP-hard it is unlikely that there is an algorithm that solves any instance of the problem in polynomial time. However, one can solve an instance using the following exponential-time algorithm: for each subset S ⊆ V of vertices, pick the one that is a vertex cover with size at most k. This trivial procedure has running time O(n k ) = O(2 k log n ) where n = |V |. Of course this algorithm is not satisfying i.e. we would like to have an algorithm that does something smarter than just trying all possibilities. One way to achieve this goal is to design a socalled parameterized algorithm. The general idea of this approach is to shift the exponential blowup from the input size n to a parameter of the input, i.e. we would like to get an algorithm with running time f (k) • n O (1) where f is a typically exponential function that solely depends on the parameter k (see Figure 2.1). As an illustration, let us consider the Vertex Cover problem with the solution size k as the parameter. Observe that since each edge uv has to be covered, either u or v (or both) must be in the solution. Using this remark, we can solve the problem by the following recursive procedure. Let uv ∈ E be any edge. The graph G has a vertex cover of size k if either G \ v or G \ u has a vertex cover of size k -1. It is not hard to see that there are at most k recursive calls implying a running time of O(2 k • n). In this case, for "small" enough value of k, we can decide an instance efficiently, no matter how large the input graph is. Indeed the running time grows linearly with the graph size. We say that Vertex Cover parameterized by k is fixed-parameter tractable. As a matter of fact, the best known algorithm for solving Vertex Cover takes time 1.2738 k • n O(1) [START_REF] Chen | Improved upper bounds for vertex cover[END_REF]. A parameterized problem is then a decision problem L ⊆ Σ * where each instance is associated with some integer value k ∈ N. We also say that L is parameterized by k. For example, a lot of decision problems are obtained from an optimization version, thus a natural and well studied parameter candidate is the solution size k also called standard parameterization. However, we would like to emphasize that the parameter can be something totally different and less explicit e.g. structural parameterization for graph problems (see Figure 2.2). Finally, it is worth noting that there exists a general definition of a parameterized problem that allows for more complicated parameters. For example, in the following problem the parameter could be the graph H.

Graph Minor Testing

Input: Two graphs H and G Question: Does G contains H as a minor? However, the Definition 11 is sufficient for our purpose since all parameters considered in this work are integers.

Definition 12: FPT class

The class FPT contains every parameterized problem Q ⊆ Σ * × N for which the question "Does (x, k) belongs to Q?" can be decided by an algorithm that runs in f (k) • |x| O(1) time (or fpt-time) where (x, k) ∈ Σ * × N and f is a function depending solely on k.

A problem in FPT is called fixed-parameter tractable. We also define the class XP as follows.

Definition 13: XP

The class XP contains every parameterized problem Q ⊆ Σ * ×N for which the question "Does (x, k) belongs to Q?" can be decided by an algorithm that runs in |x| g(k) time where (x, k) ∈ Σ * × N and g is a function depending solely on k.

Fixed-parameter algorithm techniques

In this section, we will review some basic tools used in this thesis for proving fixedparameter tractability.

Kernelization. One of the main tool to devise parameterized algorithms is the kernelization technique. This can be regarded as a pre-processing over the instance of a parameterized problem. It consists of getting rid of the "easy-to-solve" part of the instance so that it only remains the kernel of the problem i.e. the "hard" part to solve. It turns out that a parameterized problem is in FPT if and only if there exists such pre-processing. Let us now define more formally this notion.

Definition 14: Kernelization

Let Q ⊆ Σ * × N be a parameterized problem. A kernelization is an algorithm that takes as input a pair (x, k) ∈ Σ * × N and outputs in (|x| + k) O (1) time, a new pair

(x ′ , k ′ ) ∈ Σ * × N with k ′ ≤ k such that 1. (x, k) ∈ Q ⇔ (x ′ , k ′ ) ∈ Q 2. |x ′ | ≤ g(k) and k ′ ≤ g(k)
for some computable function g

The instance (x ′ , k ′ ) is called a kernel of size g(k). If g is a polynomial then we have a polynomial kernel.

The following well-known theorem shows the equivalence between a problem being fixed-parameter tractable and admitting a kernel.

Theorem 1 A parameterized problem Q is fixed parameter tractable if and only if Q admits a kernelization.

Proof. Let (x, k) ∈ Σ * × N. Assume that Q admits a kernelization. We first apply the kernelization on (x, k) to get, in

(|x| + k) O(1) time, a new equivalent instance (x ′ , k ′ ) ∈ Σ * × N such that k ′ ≤ g(k) and |x ′ | ≤ g(k)
for some computable function g. Next, it suffices to apply any algorithm on the kernel to solve the former instance. The overall running time is then h(g(k)) + (|x| + k) O (1) for some computable function h.

Conversely, suppose that Q is fixed parameter tractable and can be solved in time f (k) • |x| c for some constant c > 0. We have to distinguish between the following two cases

• if |x| ≤ f (k) then the instance itself is a kernel.

• if |x| > f (k) then simply run the parameterized algorithm to solve (x, k). This is done in polynomial time since f (k) • |x| c ≤ |x| c+1 . Depending of the answer, we return a trivial no-instance or yes-instance. This completes the proof.

In practice, we use so-called "reduction rules" to iteratively reduce the size of an instance and finally obtain the desired kernel. A reduction rule is simply an algorithm that computes in polynomial time a new equivalent instance with reduced size. As an illustrative example, consider the following reduction rule for Vertex Cover.

Reduction rule 2 Let (G, k) be an instance of Vertex Cover. If there exists a vertex v with deg(v) > k then remove v from G and decrease k by one.

After applying iteratively the above reduction rule, one gets a new equivalent instance (G ′ , k ′ ) where G ′ has size n ′ . Indeed, observe that if a vertex v has degree larger than k then it must be in the solution, otherwise we would have taken all its neighbors to cover all edges. Thus, we can safely remove v from G and decrease k by one to reflect the fact that v is part of any vertex cover. Now, suppose that there is a vertex cover S of size k ′ for G ′ . Thus, we must have n ′ ≤ k 2 since S covers every edge and the maximum degree of G ′ in k. The kernelization is then defined as follows. If n ′ > k 2 then return any trivial no-instance, otherwise we have n ′ ≤ k 2 giving us a polynomial kernel.

Monadic Second Order Logic (MSOL). The monadic second order logic is another powerful tool to prove fixed parameter tractability with respect to parameter treewidth (see Definition 2). This logic is an extension of the first order logic which is, in turns, an extension of the propositional logic. The key point here is that whenever a graph problem can be expressed as a MSO-formula, we can derive the result that it is in FPT with respect to parameter treewidth. The language of MSOL for graphs includes the logical connectives ∨, ∧, ¬, ↔, →, variables for vertices, edges, sets of vertices, and sets of edges, the quantifiers ∀, ∃ that can be applied to these variables, and the following four binary relations:

1. u ∈ U , where u is a vertex variable and U is a vertex set variable.

2. d ∈ D, where d is an edge variable and D is an edge set variable.

3. adj(u, v), where u, v are vertex variables, and the interpretation is that u and v are adjacent.

4. Equality, =, of variables representing vertices, edges, sets of vertices, and sets of edges.

Given a formula φ, we denote by G |= φ the fact that the graph G satisfies the property φ -we also say that G is a model of φ. For example the following statement,

G |= ∀x(x ∈ V → (∃y(y ∈ V ∧ y ∈ P ∧ adj(x, y))) ∨ x ∈ P )
holds if and only if the set P is a dominating set in the graph G.

For our purpose, the central result of this theory is as follows.

Theorem 3 (Courcelle [START_REF] Courcelle | The monadic second-order logic of graphs I. Recognizable sets of Finite Graphs[END_REF]) Let G be a graph and φ a MSO formula of length ℓ.

The problem of deciding whether G |= φ is fixed parameter tractable with respect to the combined parameter tw(G) and ℓ.

Kernelization lower bounds

As previously hinted, any fixed parameter problem admits a kernel of size depending of the running time of the parameterized algorithm. This arises the natural question whether there always exists a "small" kernel i.e. a kernel of polynomial size. We observed that it is the case for the Vertex Cover problem. Unfortunately, it is not always possible to obtain such kernel for all problems in FPT. Bodlaender et al. [START_REF] Hans | On problems without polynomial kernels[END_REF] introduced a technique for proving that a parameterized problem does not admit a polynomial kernel unless a well-established complexity theory assumption unexpectedly collapses. This technique is based on the previous results of Bodlaender et al. [START_REF] Hans | On problems without polynomial kernels[END_REF] and Fortnow and Santhanam [START_REF] Fortnow | Infeasibility of instance compression and succinct PCPs for NP[END_REF].

We first recall here the crucial definitions.

Definition 15: Polynomial equivalence relation [START_REF] Hans | Cross-composition: A new technique for kernelization lower bounds[END_REF] An equivalence relation R on Σ * is called a polynomial equivalence relation if the following conditions are met 1. There is an algorithm that given two strings x, y ∈ Σ * decides whether R(x, y) in (|x| + |y|) O(1) time. Definition 16: Cross-composition [START_REF] Hans | Cross-composition: A new technique for kernelization lower bounds[END_REF] Let L ⊆ Σ * be a decision problem and Q ⊆ Σ * × N be a parameterized problem. We say that L cross-composes into Q if there is a polynomial equivalence relation R and an algorithm which, given t strings x 1 , x 2 , . . . x t belonging to the same equivalence class of R, computes an instance (x

* , k * ) ∈ Σ * × N in time polynomial in t i=1 |x i | such that 1. (x * , k * ) ∈ Q ⇔ x i ∈ L for some 1 ≤ i ≤ t.
2. k * is bounded polynomially in max t i=1 |x i | + log t. The central result for proving kernel size lower bound is the following. Theorem 4 ([21], Theorem 9) Let L ⊆ Σ * be a decision problem, and let Q ⊆ Σ * × N be a parameterized problem. If L is NP-hard and cross-composes into Q then Q that has no polynomial kernel unless NP ⊆ coNP/poly.

Fixed-parameter intractability

Parameterized complexity theory also provides methods for proving the inherent intractability of a parameterized problem. To this end, we need to introduce the notion of parameterized reduction (or fpt-reduction) defined as follows.

Definition 17: Parameterized reduction

Let Q 1 , Q 2 ⊆ Σ * × N be two parameterized problems. We say that Q 1 fpt-reduces to Q 2 if there exists an algorithm that takes as input an instance (x 1 , k 1 ) ∈ Σ * × N and outputs in f

(k 1 ) • |x 1 | O(1) time a new instance (x 2 , k 2 ) ∈ Σ * × N such that 1. (x 1 , k 1 ) ∈ Q 1 ⇔ (x 2 , k 2 ) ∈ Q 2 2. k 2 ≤ g(k 1
) for some computable functions f and g.

Before introducing the different classes of complexity, we need to define the concept of boolean circuit.

Definition 18: Boolean circuit

A boolean circuit C = (V, A) is a directed acyclic graph whose vertices V are called gates. The gates of in-degree 0 are called inputs. There is exactly one gate of outdegree 0 called output. Every gate that is neither an input nor an output is labeled by an element of {OR, AN D, N OT }. A gate with label N OT has in-degree exactly one.

We need to define some additional terminology. Let C = (V, A) be a boolean circuit. A gate with in-degree bounded by some fixed constant is said small and large otherwise. The weft of C is the maximum number of large gates on a path from an input to the output. The depth is the maximum number of all gates on a path from an input to the output. A truth assignment for C is a function τ : V → {true, f alse} that associates the value true or f alse to each input gates. The (Hamming) weight of a truth assignment is the number of input gates set to true. Given an assignment τ for C, the value ν(g) of a gate g is recursively defined as follows

ν(g) =                  τ (g) if g is an input h∈N -(g) ν(h) if g is labeled by OR h∈N -(g) ν(h) if g is labeled by AND ¬ν(h) if g is labeled by NOT with N -(g) = {h}
A truth assignment satisfies C if the value of the output gate is true. We can now introduce the following central problem that is used to define the hierarchies of parameterized complexity classes.

Weft-t Circuit Satisfiability

Input: A boolean circuit C with constant depth and weft at most t and an integer k. Question: Is there a truth assignment of weight k that satisfies C?

Definition 19: W[t] class A parameterized problem Q ⊆ Σ * ×N belongs to W[t] , for fixed t > 0, if Q fpt-reduces to Weft-t Circuit Satisfiability parameterized by k.
The inclusion relationship of the above classes are as follows.

FPT ⊆ W[1] ⊆ W[2] . . . ⊆ XP
Informally speaking, a problem in W[t] is considered "harder" than those lying in W[t-1] where t > 1. We say that a parameterized problem is W[t]-hard if every problem of W[t] fpt-reduces to it. If the problem moreover belongs to W[t] then it is W[t]-complete. There is a good reason to believe that W[t]-hard problems are unlikely to be in FPT. It is worth pointing out that, in practice, we only consider W [START_REF] Aazami | Approximation algorithms and hardness for domination with propagation[END_REF] and W [START_REF] Ackerman | Combinatorial model and bounds for target set selection[END_REF] as the basic classes of parameterized intractability.

Cesati [START_REF] Cesati | The Turing way to parameterized complexity[END_REF] introduced a more "classical" way to prove that a given parameterized problem belongs to the class W [START_REF] Aazami | Approximation algorithms and hardness for domination with propagation[END_REF] (or W [START_REF] Ackerman | Combinatorial model and bounds for target set selection[END_REF]) by the use of Turing machine. For that purpose, we need to introduce the following two problems.

Short Nondeterministic Turing Machine

Input: A nondeterministic Turing machine M , a string x on the input alphabet of M , and an integer k. Question: Is there a computation of M on input x that reaches a final accepting state in at most k steps? Short Multi-tape Nondeterministic Turing Machine Input: A multi-tape nondeterministic Turing machine M , a string x on the input alphabet of M , and an integer k. Question: Is there a computation of M on input x that reaches a final accepting state in at most k steps?

We have now the following result due to Cesati [START_REF] Cesati | The Turing way to parameterized complexity[END_REF].

Theorem 5 A parameterized problem is in W [START_REF] Aazami | Approximation algorithms and hardness for domination with propagation[END_REF] (resp. W [START_REF] Ackerman | Combinatorial model and bounds for target set selection[END_REF]) if it can be fpt-reduced to the Short Nondeterministic Turing Machine (resp. Short Multi-tape Nondeterministic Turing Machine) problem parameterized by k.

Observe that the previous theorem plays the same role as the non-deterministic Turing machine does for the class NP.

Parameters hierarchies

We conclude this section on parameterized complexity with the following useful property.

Theorem 6 Let Q 1 , Q 2 ⊆ Σ * ×N be two parameterized problems. If the parameters k 1 of Q 1 and k 2 of Q 2 always satisfy the inequality k 1 ≤ c • k 2 for
some constant c > 0 then the following assertions are true.

1. If Q 2 is W[t]-hard (resp. NP-hard for constant values of k 2 ) for some integer t > 0 then Q 1 is W[t]-hard (resp. NP-hard for constant values of k 1 ). 2. If Q 1 is in FPT (resp. XP) then Q 2 is in FPT (resp. XP).
As an application of the above theorem, consider the Figure 2.2. If a parameterized problem is in FPT (resp. W[t]-hard, NP-hard for constant value of k) with respect to some structural parameter k then, using Theorem 6, it is in FPT (resp. W[t]-hard, NPhard for constant value of k) with respect to each parameter which is an ancestor (resp. descendant) of k. A parameter k 1 is an ancestor (resp. descendant) of a parameter k 2 if there exists a directed path from k 1 to k 2 (resp. from k 2 to k 1 ).

Approximation

Another approach for coping with NP-hard problems is to use the approximation framework. Indeed, computing an optimal solution might not be mandatory and could be a really hard task even with time efficient algorithm. That is where the algorithms with performance guarantee come in. In such algorithms, the basic idea is to "release" the constraint of optimality while the running time is required to be reasonable. Moreover, the computed solution must be not "too far" from the optimum. In what follows we define more formally these notions.

Optimization problem

So far, we have dealt with decision problems. In the context of approximation algorithms, we rather consider optimization problems. While in a decision problem the solution is either "yes" or "no", an optimization problem asks to find a solution that maximizes (or minimizes) an objective function.

Definition 20: Optimization problem & NPO class

An optimization problem O is a 4-tuple (D, sol, cost, goal) where 1. D ⊆ Σ * is the set of instances recognizable in polynomial time.

2. For each instance x ∈ D, sol(x) ∈ P(Σ * ) is the set of feasible solutions of x, where P(X) is the set of all subsets of a set X. 4. It can be decided in polynomial time whether y ∈ sol(x) holds for given x and y.

5. Given an instance x and a feasible solution y, cost(x, y) is a polynomial-time computable positive integer.

The class that contains all optimization problems is denoted by NPO .

A minimization (resp. maximization) problem is an optimization problem with goal = min (resp. goal = max).

As a matter of fact, the above definition stands for an NP optimization problem which is a particular optimization problem. However, this thesis only deals with optimization problems that are NP optimization problems, allowing us the use of this shortcut.

Given an instance x of some optimization problem, the goal is to find an optimal solution, that is, a feasible solution y ∈ sol(x) such that cost(x, y) = goal

y ′ ∈sol(x) {cost(x, y ′ )}
The cost of an optimum solution for an instance x is denoted by opt(x).

As an illustration, the Max Clique problem is to find a complete subgraph of a given graph of maximum size (see Appendix A). Formally, this problem is defined as follows.1

• D = {G = (V, E) : G is a graph}. • Let G be any graph, sol(G) = {H = (V ′ , E ′ ) : H is a complete subgraph of G}
• Let G be any graph and H a complete subgraph of G of size n, cost(G, H) = n.

• goal = max

In this thesis, when a new optimization problem will be introduced, we will make use of the following more convenient definition.

Problem Name

Input: Some inputs Output: Objective to achieve. An optimization problem is polynomial time solvable if there exists an algorithm that computes, for every instance, an optimal solution within a running time polynomial in the instance size.

Definition 21: PO class

The class PO contains all problems of NPO that are polynomial-time solvable.

The link between optimization problems and decision problems is the following. If O = (D, sol, cost, goal) is a maximization problem then we can define its decision version as follows (for minimization problems the definition is analogous).

O k

Input: An instance x ∈ D and an integer k. Question: Is opt(x) ≥ k?

In fact the decision version of a optimization problem is in NP. In general, we are interested in optimization problems for which the decision version is NP-hard. By definition, an optimization problem is NP-hard if its decision version is NP-hard.

Approximation algorithms

Before defining what an approximation algorithm is, we need to determine how to measure the quality of a solution. For that purpose, we introduce the performance ratio.

Definition 22: Performance ratio & error

Given an instance x of an optimization problem O = (D, sol, cost, goal), the performance ratio r(x, y) of a solution y ∈ sol(x) is

r(x, y) = max cost(x, y) opt(x) , opt(x) cost(x, y)
The error of y, denoted ε(x, y), is defined by ε(x, y) = r(x, y) -1.

Notice that the performance ratio is always > 1 and it gets closer to 1 as the solution gets closer to the optimum.

Definition 23: α-approximation algorithm

Let O = (D, sol, cost, goal) be an optimization problem and α : N →]1, +∞[ be a function. An α-approximation algorithm is an algorithm that takes as input any instance x ∈ D of size n = |x| and returns a solution y ∈ sol(x) such that r(x, y) ≤ α(n).

A problem admitting an α-approximation algorithm is said to be α-approximable. Notice that we did not impose any specific running time in the above definition. A polynomial-time α-approximation algorithm is an α-approximation algorithm with running time polynomial in the instance size. Let F be a set of functions, we denote by F -APX the class of optimization problems for which there exists such algorithm with α ∈ F . We now review the different types of approximation classes encountered throughout this thesis. ,O(n c ) for some fixed constant c > 0).

Definition 25: PTAS class

An optimization problem admits a polynomial-time approximation scheme (ptas for short) if for any fixed constant ε > 0, there exists a polynomial-time (1 + ε)approximation algorithm for that problem. The class PTAS contains all optimization problems that admit a ptas.

Generally, the running time of a ptas is exponential in 1/ε, like n O(1/ε) if the size of the instance is n. We have the following inclusions

PO ⊆ PTAS ⊆ APX ⊆ log-APX ⊆ poly-APX ⊆ NPO
We conclude this section by introducing the concept of parameterized approximation. We first extend the definition of an optimization problem as follows.

Definition 26: Parameterized optimization problem

A parameterized optimization problem is an optimization problem where each instance is associated with a nonnegative integer called parameter. We denote such optimization problem by O p = (D p , sol, cost, goal) where D p ⊆ Σ * × N.

We are now ready to give the following definition.

Definition 27: fpt-time α-approximation algorithm

Let O p = (D p , sol, cost, goal) be a parameterized optimization problem and α : N →]1, +∞[ a function. An fpt-time α-approximation algorithm is an algorithm that takes as input any instance (x, k) ∈ D p of size n = |x| and returns a solution y ∈ sol(x) such that r(x, y) 1) time for some computable function f that solely depends on k.

≤ α(n) in f (k) • n O(
We establish in [START_REF] Bazgan | Parameterized approximability of influence in social networks[END_REF] the following simple but useful lemma.

Lemma 7 Let O p be a parameterized optimization problem. If there is an fpt-time α 1approximation algorithm for O p and some strictly increasing function α 1 depending solely on the parameter then there is also an fpt-time α 2 -approximation algorithm for O p and any strictly increasing function α 2 depending solely on the instance size.

Proof.

Let α -1 1 and α -1 2 be the inverse functions of α 1 and α 2 , respectively. Let (x, k) ∈ D p be an instance of a parameterized maximization problem O p = (D p , sol, cost, goal) (the proof is analogous for minimization problems). We distinguish the following two cases.

Case 1: k ≤ α -1 1 (α 2 (|x|)). In this case, we apply the α 1 -approximation algorithm and directly get a solution y ∈ sol(x) such that r(x, y) 1) for some computable function f . Case 2: k > α -1 1 (α 2 (|x|)). We then have |x| < α -1 2 (α 1 (k)) and thus we can solve x by exhaustively checking every solution y in sol(x) and return the one with the largest cost(x, y) value. Since |y| ≤ |x| O(1) by the definition of an optimization problem, we know that there are at most 1) different solutions in the set sol(x). It follows that the running time in this case is 1) ) = f (k) for some computable function f . This completes the proof.

≤ α 1 (k) ≤ α 1 (α -1 1 (α 2 (|x|))) = α 2 (|x|) in time f (k) • |x| O(
|Σ| |x| O(1) ≤ |Σ| α -1 2 (α 1 (k)) O(
O(|Σ| α -1 2 (α 1 (k)) O(
As an illustration of this lemma, if a problem admits a polynomial-time k-approximation then we can approximate this problem within any arbitrarily small ratio depending on the instance size in fpt-time e.g. log(log(. . . log(|x|)).

Hardness of approximation

The main tool used in this thesis for showing directly the inapproximability of an optimization problem is the gap-introducing reduction.

Definition 28: Gap-introducing reduction

Let L ⊆ Σ * be a decision problem and O = (D, sol, cost, goal) be a maximization (resp. minimization) problem. We say that there is a gap-introducing reduction from L to O if there exist two functions α : N →]1, +∞[ and f : D → Q together with an algorithm that takes as input an instance x ∈ Σ * and outputs in polynomial time a new instance x ′ ∈ D such that the following conditions are met 1. if x is a yes-instance then opt(x

′ ) ≥ f (x ′ ) (resp. opt(x ′ ) ≤ f (x ′ )). 2. if x is a no-instance then opt(x ′ ) < f (x ′ ) α(|x ′ |) (resp. opt(x ′ ) > α(|x ′ |) • f (x ′ )).
The function α is called the gap or hardness factor. We have the following result.

Theorem 8 If there is a gap-introducing reduction from a NP-hard decision problem to an optimization problem with gap α, then there is no polynomial-time αapproximation algorithm for that optimization problem unless P = NP.

For proving that a given parameterized optimization problem is unlikely to have a fpttime approximation algorithm, we need a slightly different version of the above reduction.

Definition 29: Parameterized gap-introducing reduction

Let Q ⊆ Σ * × N be a parameterized problem and O p = (D p , sol, cost, goal) be a parameterized maximization (resp. minimization) problem. We say that there is a parameterized gap-introducing reduction from Q to O p if there exist two functions α : N →]1, +∞[ and f : Σ * → Q together with an algorithm that takes as input an instance (x, k) ∈ Σ * × N and outputs, in g(k) • |x| O (1) time for some computable function g, a new instance (x ′ , k ′ ) ∈ D p such that the following conditions are met 1. k ′ ≤ h(k) for some computable function h that depends solely on k.

2. if (x, k) ∈ Q then opt(x ′ ) ≥ f (x ′ ) (resp. opt(x ′ ) ≤ f (x ′ )). 3. if (x, k) ∈ Q then opt(x ′ ) < f (x ′ ) α(|x ′ |) (resp. opt(x ′ ) > α(|x ′ |) • f (x ′ )).
We then have a similar result as the one above.

Theorem 9 If there is a parameterized gap-introducing reduction from a W[t]-hard parameterized problem, t > 0, to a parameterized optimization problem with gap α, then there is no fpt-time α-approximation algorithm for that optimization problem unless

FPT = W[t]. Proof. Let Q ⊆ Σ * × N be a W[t]
-hard parameterized problem and O p = (D p , sol, cost, goal) a parameterized maximization problem (the proof is analogous for the minimization case). Suppose that there is a parameterized gap-introducing reduction from Q to O p with gap α and a fpt-time α-approximation algorithm A for O p . We show that we can solve every instance (x, k) of Q in fpt-time with respect to

k implying FPT = W[t].
Since there is a parameterized gap-reduction from Q to O p , there must exist an algorithm that maps the instance (x, k) into an instance (x ′ , k ′ ) ∈ D p in fpt-time with respect to k. Now we run the approximation algorithm A onto (x ′ , k ′ ) to get a solution y ∈ sol(x ′ ) such that opt(x ′ ) cost(x ′ ,y) ≤ α(|x ′ |) in fpt-time with respect to k ′ and thus in fpt-time with respect to k (since k ′ ≤ h(k) for some function h).

By the definition of the reduction, if (x, k) is a yes-instance then opt(x ′ ) ≥ f (x ′ ) and then

f (x ′ ) α(|x ′ |) ≤ cost(x ′ , y). Conversely, if (x, k) is a no-instance then opt(x ′ ) < f (x ′ ) α(|x ′ |) and thus cost(x ′ , y) < f (x ′ ) α(|x ′ |) (since cost(x ′ , y) ≤ opt(x ′ ))
. It follows that by comparing the values cost(x ′ , y) and f (x ′ ) α(|x ′ |) , we are able to distinguish between yesand no-instances of Q in fpt-time with respect to k. This completes the proof.

Approximation preserving reductions

For proving that a problem is at least as hard to approximate as another, we need to introduce the notion of approximation preserving reduction. This kind of reduction is more involved than the one to prove NP-hardness for decision problems. This is essentially because we have to deal with quantitative solutions and not simply a "yes" or "no" solution. Although the literature references many different approximation reductions [START_REF] Ausiello | Complexity and Approximation: Combinatorial Optimization Problems and Their Approximability Properties[END_REF], we only present the L-reduction and E-reduction in this thesis. We start with the L-reduction ("linear reduction") introduced by Papadimitriou and Yannakakis [START_REF] Papadimitriou | Optimization, approximation, and complexity classes[END_REF].

Definition 30: L-reduction

Let O 1 = (D 1 , sol 1 , cost 1 , goal 1 ) and O 2 = (D 2 , sol 2 , cost 2 , goal 2 ) be two optimization problems. We say that O 1 is L-reducible to O 2 if there are two constants α, β > 0 and two polynomial time computable functions f , g such that 1. f maps each instance

x 1 ∈ D 1 into an instance x 2 ∈ D 2 such that opt 2 (x 2 ) ≤ α • opt 1 (x 1 ). 2. g maps each solution y 2 ∈ sol 2 (x 2 ) into a solution y 1 ∈ sol 1 (x 1 ) such that | cost 1 (x 1 , y 1 ) -opt 1 (x 1 )| ≤ β • | cost 2 (x 2 , y 2 ) -opt 2 (x 2 )|.
An optimization problem is APX-hard if every problem of APX L-reduces to that problem. An APX-hard optimization problem is unlikely to be in PTAS, just as NP-hard decision problems are unlikely to be in P. Furthermore, if a problem O 1 is L-reducible to a problem O 2 then the following holds:

If O 1 is APX-hard then O 2 is also APX-hard and if O 2 ∈ PTAS then O 1 ∈ PTAS.
Unfortunately, this property is no longer true for a class beyond APX. For instance, let ε ∈ (0, 1) be any fixed constant, if a problem O 1 is n 1-ε -APX-hard and L-reduces to another problem O 2 then we cannot deduce that O 2 is also n 1-ε -APX-hard. To get rid of this problem, we need to use the E-reduction ("error-preserving reduction") introduced by Khanna et al. [START_REF] Khanna | On syntactic versus computational views of approximability[END_REF].

Definition 31: E-reduction Let O 1 = (D 1 , sol 1 , cost 1 , goal 1 ) and O 2 = (D 2 , sol 2 , cost 2 , goal 2 ) be two optimiza- tion problems. We say that O 1 is E-reducible to O 2 if there exist polynomial time computable functions f , g and a constant β such that 1. f maps each instance x 1 ∈ D 1 into an instance x 2 ∈ D 2 such that opt 1 (x 1 )
and opt 2 (x 2 ) are related by a polynomial factor, i.e. there exists a polynomial p such that opt

2 (x 2 ) ≤ p(|x 1 |) • opt 1 (x 1 ). 2. g maps each solution y 2 ∈ sol 2 (x 2 ) into a solution y 1 ∈ sol 1 (x 1 ) such that ε(x 1 , y 1 ) ≤ βε(x 2 , y 2 ).
An important property of an E-reduction is that it can be applied uniformly to all levels of approximability. In other words, if a problem O 1 is E-reducible to a problem Chapter 3 Maximizing the spread of influence T his chapter essentially addresses the computational complexity of the problem of finding the most influential individuals in a network. Formally, given a graph and a threshold value thr(v) associated to each vertex v, the task is to find and activate a vertex subset of minimum size such that all of the vertices become active at the end of the propagtion propagation process defined as follows. A vertex becomes active if at least thr(v) neighbors of v are active. The propagation process proceeds in several steps and stops when no further vertex becomes active.

In the graph below, one could select the gray vertices into a target set to activate the entire graph within three steps. The numbers inside the vertices denote the thresholds. In this chapter, we investigate the parameterized complexity and parameterized approximability of this problem and its complementary version i.e. given an integer k, find and activate a vertex subset of size at most k that maximizes the total number of activated vertices at the end of the propagation process.

The content of this chapter is based on the following papers.

Introduction

Optimization problems that involve a diffusion process in a graph are widely studied [START_REF] Peleg | Local majorities, coalitions and monopolies in graphs: a review[END_REF][START_REF] Kempe | Maximizing the spread of influence through a social network[END_REF][START_REF] Chen | On the approximability of influence in social networks[END_REF][START_REF] Aazami | Approximation algorithms and hardness for domination with propagation[END_REF][START_REF] Dreyer | Irreversible k-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion[END_REF][START_REF] Chang | Spreading messages[END_REF][START_REF] Ben-Zwi | Treewidth governs the complexity of target set selection[END_REF][START_REF] Thirumala Reddy | Variants of spreading messages[END_REF]. Such problems share the common property that, according to a specified propagation rule, a chosen subset of vertices activates all or a fixed fraction of the vertices, where initially all but the chosen vertices are inactive. Such optimization problems model the spread of influence or information in social networks via word-of-mouth recommendations, of diseases in populations, or of faults in distributed computing [START_REF] Peleg | Local majorities, coalitions and monopolies in graphs: a review[END_REF][START_REF] Kempe | Maximizing the spread of influence through a social network[END_REF][START_REF] Dreyer | Irreversible k-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion[END_REF]. One representative problem that appears in this context is the influence maximization problem introduced by Kempe et al. [START_REF] Kempe | Maximizing the spread of influence through a social network[END_REF]. Given a directed graph, the task is to choose a vertex subset of size at most a fixed number such that the number of activated vertices at the end of the propagation process is maximized. The authors show that the problem is polynomial-time ( e e-1 + ε)-approximable for any ε > 0 under some stochastic propagation models, but NP-hard to approximate within a ratio of n 1-ε for any ε > 0 for general propagation rules. In this thesis, we rather use the following deterministic propagation model. We are given a graph, a threshold value thr(v) associated to each vertex v, and the following propagation rule: a vertex becomes active if at least thr(v) neighbors of v are active. The propagation process proceeds in several steps and stops when no further vertex becomes active. A subset S of vertices is called a target set if all the vertices of the graph get active when the vertices of S are initially activated. Given this model and an integer k > 0, determining the existence of a target set of size at most k is known as the Target Set Selection problem [START_REF] Chen | On the approximability of influence in social networks[END_REF]. The minimization version is denoted by Min Target Set Selection and consists of finding a target set of minimum size.

Chen [START_REF] Chen | On the approximability of influence in social networks[END_REF] showed hardness of approximating this last problem within a ratio O(2 log 1-ε n ) for any ε > 0, even for constant degree graphs with thresholds at most two and for general graphs when the threshold of each vertex is half its degree (later called majority thresholds). When the threshold of each vertex equals its degree (later called unanimity thresholds), the problem turns out to be polynomial-time equivalent to the vertex cover problem [START_REF] Chen | On the approximability of influence in social networks[END_REF] and, thus, admits a 2-approximation, is hard to approximate within a ratio better than 1.36 [START_REF] Dinur | The importance of being biased[END_REF], and the decision version is fixed-parameter tractable with respect to the parameter k. Ben-Zwi et al. [START_REF] Ben-Zwi | Treewidth governs the complexity of target set selection[END_REF] obtained a W[1]-hardness result with respect to the parameter "treewidth" of the input graph. However, Target Set Selection is polynomial-time solvable on graphs of bounded treewidth with the degree of the polynomial depending on the treewidth. They also proved fixed-parameter tractability for the same parameter once the threshold values are bounded by any constant. Recently, further parameterized complexity studies for the structural graph parameters "diameter", "cluster editing number", "vertex cover number", and "feedback edge set number" have been undertaken [START_REF] Nichterlein | On tractable cases of target set selection[END_REF]. Moreover, polynomial-time algorithms for Target Set Selection restricted to special graph classes including chordal graphs and block-cactus graphs have been developed [START_REF] Centeno | Irreversible conversion of graphs[END_REF][START_REF] Chiang | Some results on the target set selection problem[END_REF][START_REF] Thirumala Reddy | Variants of spreading messages[END_REF]. Finally, there are numerous combinatorial studies concerning the sizes of optimal target sets (upper and lower bounds) mostly with respect to special graph classes [START_REF] Ackerman | Combinatorial model and bounds for target set selection[END_REF][START_REF] Spence | Dynamic monopolies and feedback vertex sets in hexagonal grids[END_REF][START_REF] Spence | Modeling the spread of fault in majority-based network systems: Dynamic monopolies in triangular grids[END_REF][START_REF] Chang | Spreading messages[END_REF][START_REF] Chiang | Some results on the target set selection problem[END_REF][START_REF] Dreyer | Irreversible k-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion[END_REF][START_REF] Reichman | New bounds for contagious sets[END_REF]. The role of the threshold values and threshold functions has been studied in the past. For instance, Dreyer and Roberts [START_REF] Dreyer | Irreversible k-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion[END_REF] showed NPhardness for Target Set Selection when all vertices have the same threshold t, t ≥ 3. Later, Chen [START_REF] Chen | On the approximability of influence in social networks[END_REF] extended the previous result to t = 2. Centeno et al. [START_REF] Centeno | Irreversible conversion of graphs[END_REF] and Chiang et al. [START_REF] Chiang | Some results on the target set selection problem[END_REF] exploited threshold values being upper-bounded by two to develop polynomial-time algorithms for Target Set Selection on chordal graphs.

From the previous works, it can be seen that Target Set Selection is computationally hard from both a parameterized and an approximation point of view. In light of this fact, we tackle the complexity of this problem using two different approaches. First, we investigate the parameterized complexity of the problem using parameters related to the structure of the graph. Indeed, the structural parameters of some social networks might be small and/or have useful properties that help in solving the problem. For instance networks modeling romantic relationship (see [52, Chap. 2, Fig. 2.7] and Figure 1.1) are sparse and thus parameters related to the sparseness of the input graph are of particular interest. Moreover, we consider for each parameter three types of thresholds: unanimity, majority and thresholds bounded by a constant. One might think that assuming thresholds to be constant bounded is quite a restrictive hypothesis. However in several applications, such as viral marketing, it is conceivable that constant thresholds suffice to model the underlying application scenarios. For instance, independent of my total number of friends it may suffice that at least five of my friends in a social network buy a certain product to convince me about the product's usefulness. Overall, our results are pictorially summarized in Figure 3.1. Second, we combine approximation with parameterized complexity theory toward the goal of getting positive approximation results. More specifically, we turn our attention to the complementary version of Target Set Selection called Influence. This problem is defined similarly except that the input is extended with an integer ℓ > 0 and we ask the existence of a subset of vertices of size k to activate such that at least ℓ vertices are activated at the end of the propagation process. We then consider the parameterized approximability of the maximization version where the objective is to find a subset of vertices of size k to activate that maximizes the total number of activated vertices at the end of the propagation process. This is actually the deterministic version of the problem by Kempe et al. [START_REF] Kempe | Maximizing the spread of influence through a social network[END_REF]. In that problem, there are two possibilities of measuring the value of a solution: counting the vertices activated by the propagation process including or excluding the initially chosen vertices. This leads to two versions of the problem denoted by Max Closed Influence and Max Open Influence, respectively. Observe that whether or not counting the chosen vertices might change the approximation factor. In this chapter, we consider both cases and our approximability results are summarized in Table 3.1.

Max Open Influence

Max Closed Influence thr Bounds poly-time fpt-time poly-time fpt-time Cons.

Upper

n n n n Lower n 1-ε ,∀ε > 0 n 1-ε ,∀ε > 0 n 1-ε ,∀ε > 0 n 1-ε ,∀ε > 0 (Th.27) Maj. Upper n n n n Lower n 1-ε ,∀ε > 0 n 1-ε ,∀ε > 0 n 1-ε ,∀ε > 0 n 1-ε ,∀ε > 0 (Th.26) Una. Upper 2 k (Th.31) α(n), ∀α (Th.32) 2 k α(n), ∀r Lower n 1-ε ,∀ε > 0 (Th.30) ? 1 + ε (Th.36) ? Table 3.1:

Cluster Vertex Deletion Number

Th. [START_REF] Bousquet | Multicut is FPT[END_REF] Th.16

Clique Cover Number

Th.17

Bandwidth

Th.15Th.15Th.15

Distance to Cograph

Th.13 [START_REF] Ben-Zwi | Treewidth governs the complexity of target set selection[END_REF] Feedback Vertex Set Number [START_REF] Ben-Zwi | Treewidth governs the complexity of target set selection[END_REF] Th.13 [START_REF] Ben-Zwi | Treewidth governs the complexity of target set selection[END_REF] Distance to Interval

Th.13 [START_REF] Ben-Zwi | Treewidth governs the complexity of target set selection[END_REF] Pathwidth [START_REF] Ben-Zwi | Treewidth governs the complexity of target set selection[END_REF] Th.13 [START_REF] Ben-Zwi | Treewidth governs the complexity of target set selection[END_REF] Treewidth [START_REF] Ben-Zwi | Treewidth governs the complexity of target set selection[END_REF] Th.13 [START_REF] Ben-Zwi | Treewidth governs the complexity of target set selection[END_REF] This chapter is organized as follows. The Section 3.2 is dedicated to the problem definitions as well as some preliminaries. In the Sections 3.3 and 3.4, we study the parameterized complexity of Target Set Selection with respect to parameters related to sparse and dense structures, respectively. In Section 3.5, we turn our attention to the parameterized approximability of Max Open Influence and Max Closed Influence. We also consider the complexity of the decision version Influence. Conclusion and open problems are given in Section 3.6.

As the present work was done in close collaboration with another PhD student, I will specify my main contributions to this chapter. First, I extended the W[1]-reduction proof for Target Set Selection and general thresholds to the majority case (Theorem 13). Secondly, I proved the parameterized inapproximability of Max Open Influence and Max Closed Influence (Theorem 26 and Theorem 27), as well as the polynomial-time inapproximability result for the unanimity case (Theorem 30). Furthermore, I also stated the general result of Lemma 7.

Problem definitions and preliminaries

Before we define formally the investigated problems, we need to define some notions. Let G = (V, E) be a graph and a threshold function thr : V → N. In this work, we consider majority thresholds i.e. thr(v) = ⌈ deg(v)

2 ⌉ for each v ∈ V , unanimity thresholds i.e. thr(v) = deg(v) for each v ∈ V , and constant thresholds i.e. thr(v) ≤ c for each v ∈ V and some constant c > 1.

We now define the propagation process involved in the studied problems. Initially, all vertices of G are not activated and we select a subset S ⊆ V . The propagation unfolds in discrete steps. At time step 0, only the vertices in S are activated. At time step t + 1, a vertex v is activated if and only if the number of its activated neighbors at time t is at least thr(v). We apply the rule iteratively until no more activations are possible. Formally, we define the set of vertices that are activated by S at step i as A i G,thr (S) with

A 0 G,thr (S) = S and A i+1 G,thr (S) = A i G,thr (S) ∪ {v ∈ V : |N (v) ∩ A i G,thr (S)| ≥ thr(v)}
We denote by r(S) = max{i : Let G = (V, E) be a graph and a threshold function thr : V → N, and a set S ⊆ V . We say that S is a target set for (G, thr) if σ G,thr [S] = V .

A i-1 G,thr = A i G,
We omit the subscript (G, thr) if the graph and the threshold function are clear from the context. We denote the maximum threshold of an instance (G, thr) by thr max (G, thr) = max{thr(v) : v ∈ V (G)}. We can now formally define the studied problems.

Target Set Selection

Input: A graph G = (V, E), a threshold function thr : V → N and an integer k. Question: Is there a target set S ⊆ V for (G, thr) such that |S| ≤ k?

We also consider the complementary problem of Target Set Selection defined as follows.

Influence

Input: A graph G = (V, E), a threshold function thr : V → N, and two integers k and ℓ. Question: Is there a subset S ⊆ V , |S| ≤ k, such that |σ(S)| ≥ ℓ?

In this chapter, we investigate the approximability of the corresponding optimization problem. As previously hinting, the fact that we count or not the initially activated vertices in the solution might change the approximation factor. Thus we need to define two maximization versions of Influence. Max Open Influence Input: A graph G = (V, E), a threshold function thr : V → N, and an integer k.

Output: A subset S ⊆ V , |S| ≤ k, such that |σ(S)| is maximum.

Max Closed Influence

Input: A graph G = (V, E), a threshold function thr : V → N, and an integer k.

Output: A subset S ⊆ V , |S| ≤ k, such that |σ[S]| is maximum.
We will sometimes make use of the following relation between Max Open Influence and Max Closed Influence.

Lemma 10 If there is an α-approximation algorithm for Max Open Influence then there is an α-approximation algorithm for Max Closed Influence.

Proof. Let A be an α-approximation algorithm for Max Open Influence. Let I = (G, thr, k) be an instance of Max Closed Influence with |V | = n and optimum value opt(I). When we apply A on I, we obtain a solution S such that |σ(S)| ≥

opt(I)-k α(n) and thus |σ[S]| = k + |σ(S)| ≥ opt(I) α(n) .
We use the following two data reduction rules throughout this work. If the threshold of a vertex exceeds its degree, it cannot be activated by its neighbors and, hence, the vertex is part of any target set. Moreover, we consider vertices with threshold 0 as already active.

Reduction rule 11 ([90, Reduction

Rule 1]) Let G = (V, E) and v ∈ V . If thr(v) > deg(v)
, then delete v, decrease the threshold of all its neighbors by one and decrease k by one. If thr(v) = 0, then delete v and decrease the thresholds of all its neighbors by one.

In an instance that is reduced with respect to Reduction rule 11, every degree-one vertex has threshold one. Thus, considering an arbitrary degree-one vertex, we do not select it into the target set as choosing its neighbor is at least as good. This is formalized in the next data reduction rule.

v ∈ V (G) with thr(v) = deg(v) = 1. Then, delete v from G.
Parameter Identification. In this paragraph, we further justify the choice of the structural parameters in this study. Fixed-parameter algorithms are efficient in practice if the considered parameter is small. As previously hinted, target set selection having many applications on social networks [START_REF] Easley | Networks, Crowds, and Markets: Reasoning about a Highly Connected World[END_REF], it is natural to extract small parameters from typical properties of social networks. A widely accepted property of social networks is the socalled "small-world phenomenom", roughly stating that the diameter of social networks is usually small. Unfortunately, the diameter of the input graph turns out not to be a suitable parameter as the problem is NP-hard for constant diameter values [START_REF] Nichterlein | On tractable cases of target set selection[END_REF]. When the network models friendships, we expect the network to be made up of multiple cliques (or dense substructures) that overlap. This motivates considering the number of cliques needed to cover all vertices [START_REF] Karp | Reducibility among combinatorial problems[END_REF] (the "clique cover number") or the number of vertices to remove to obtain a clique (the "distance to clique"). As the latter parameter is somewhat restrictive, we also considered the number of vertices to delete in order to obtain a collection of disjoint cliques (the "cluster vertex deletion number"). Recently, the cluster vertex deletion number was also used to parameterize problems related to coloring and hamiltonicity [START_REF] Doucha | Cluster vertex deletion: A parameterization between vertex cover and clique-width[END_REF]. In some applications, we deal with very sparse social networks, for instance networks modeling romantic relationship [52, Chap. 2, Fig. 2.7]. In these cases, parameters related to the sparseness of the input graph are interesting. Among them, we consider the number of vertices to remove to obtain an edgeless graph ("vertex cover number"), the number of edges or vertices to remove to obtain a forest ("feedback edge set number" and "feedback vertex set number") as well as some graph width parameters (treewidth, pathwidth, and bandwidth).

Basic reductions I & II

In the following, we present several "basic reductions" that are used as starting point for several reductions in this chapter.

Basic reduction I. This fpt-reduction, which we will refer to as basic reduction I, was introduced by Ben-Zwi et al. [START_REF] Ben-Zwi | Treewidth governs the complexity of target set selection[END_REF] and is from the

W[1]-hard problem Multicolored Clique (see Appendix A). Let (G, col, k) be an instance of Multicolored Clique. An equivalent instance (G ′ , thr, k ′ ) of Target Set Selection is constructed as follows.
For each color c ∈ {1, . . . , k}, create a vertex-selection gadget X c consisting of a star whose leaves one-to-one correspond to vertices with color c in G. For each pair of distinct colors c 1 , c 2 ∈ {1, . . . , k}, let E {c 1 ,c 2 } ⊆ E be the set of all edges that connect vertices of color c 1 with vertices of color c 2 and create the following edge-selection gadget X {c 1 ,c 2 } . The edge-selection gadget X {c 1 ,c 2 } consists of a star whose leaves one-to-one correspond to edges in E {c 1 ,c 2 } . The center vertex of any star is called guard.

The second type of gadgets is a validation gadget. They use the arbitrary bijection low : V → {1, . . . , n} and the bijection high : V → {n, . . . , 2n -1} defined as high(v) = 2nlow(v) for each v ∈ V . For each {c 1 , c 2 } with c 1 , c 2 ∈ {1, . . . , k}, add two validation gadgets V c 1 ,c 2 and V c 2 ,c 1 each consisting of two vertices. Now, for each uv ∈ E {c 1 ,c 2 } such that col(v) = c 1 , connect the first validation gadget V c 1 ,c 2 as follows: Let v ′ be the vertex in X c 1 corresponding to v. First, add low(v) vertices and connect them to v ′ and to the first vertex of V c 1 ,c 2 . Next, add high(v) vertices and connect them to v ′ and to the second vertex of V c 1 ,c 2 . Analogously, with e uv denoting the vertex in X {c 1 ,c 2 } corresponding to the edge uv, add high(v) vertices and connect them to e uv and to the first vertex of V c 1 ,c 2 . Then, add low(v) vertices and connect them to e uv and to the second vertex of V c 1 ,c 2 . The second validation gadget V c 2 ,c 1 is analogously connected to the vertex of X c 2 that corresponds to u and to e uv in X {c 1 ,c 2 } .

We call all the vertices adjacent to vertices of a validation gadget connection vertices. The thresholds are set as follows: Guard vertices and connection vertices have threshold one, the two vertices in each validation gadget have threshold 2n, and the remaining vertices in the selection gadgets have a threshold equal to their degree. Finally, k ′ = k+ k 2 . This completes the reduction.

As to the correctness: If the instance (G, col, k) is a yes-instance of Multicolored Clique then the vertices chosen to be in the target set of G ′ refer to the multicolored clique in G: For each vertex in the clique, the corresponding vertex in the vertex-selection gadget is in the target set. Furthermore, for each edge in the clique the corresponding vertex in the edge-selection gadget is in the target set. This target set activates the whole graph. In the reverse direction the validation gadgets play a central role: Each validation gadget connects a vertex-selection gadget with and edge-selection gadget. The vertices in the validation gadget only become activated if a vertex in the vertex-selection gadget and a vertex in the edge-selection gadget are in the target set such that the corresponding vertex and edge in G are incident. Basically this ensures that one has to choose vertices in G ′ into the target set that refer to a multicolored clique in G. We refer the reader to Ben-Zwi et al. [START_REF] Ben-Zwi | Treewidth governs the complexity of target set selection[END_REF] for more details.

Basic reduction II. In the following, we introduce a fpt-reduction from the W[2]-hard problem Dominating Set (see Appendix A), denoted as basic reduction II. Given an instance (G = (V, E), k) of Dominating Set we construct the instance (G ′ = (V ′ , E ′ ), thr, k) of Target Set Selection as follows. For each vertex v ∈ V , we add two vertices v t and v b (t and b respectively standing for top and bottom) to V ′ as well as the edge v t v b to E ′ . For each edge uv ∈ E, add the edges

u t v b and u b v t in G ′ . Finally, set thr(v t ) = deg G ′ (v t )
and thr(v b ) = 1 for every top vertex v t and every bottom vertex v b , respectively. This completes the reduction (see Figure 3.

2).

As to the correctness: For the forward direction, suppose there exists a dominating set S ⊆ V in G of size k. Consider the solution S ′ ⊆ V ′ containing the corresponding top vertices. After the first step, all bottom vertices are activated since they have thresholds one and S is a dominating set. Finally, after the second step, all top vertices are activated too. For the reverse direction, suppose there is a subset

S ′ ⊆ V ′ of size k in G ′ such that σ[S ′ ] = V ′ .
We can assume without loss of generality that S ′ contains no bottom vertex. Since all bottom vertices are activated we have that {v i :

v t i ∈ S ′ } is a dominating set in G. v 5 v 4 v 3 v 2 v 1 v t 1 v b 1 v t 2 v b 2 v t 3 v b 3 v t 4 v b 4 v t 5 v b 5 Figure 3.2: Sample construction of the bipartite graph G ′ (right) from a graph G (left) of Dominating Set. All vertices v t i , 1 ≤ i ≤ 5 have thresholds deg G ′ (v t i ) while all vertices v b i , 1 ≤ i ≤ 5 have thresholds 1.

Parameters related to sparse structures

In this section, we consider parameters that measure the sparseness of the input graph. Since trees are the most sparse connected graphs and Target Set Selection is polynomial-time solvable on trees [START_REF] Chen | On the approximability of influence in social networks[END_REF], parameters measuring the distance to trees are most interesting. Canonical candidates for this are the treewidth, the pathwidth, and the feedback vertex set number of the input graph. Notably, if the maximum threshold thr max is bounded by a constant, then a fixed-parameter algorithm of Ben-Zwi et al. [START_REF] Ben-Zwi | Treewidth governs the complexity of target set selection[END_REF] for the parameter treewidth tw can solve Target Set Selection in thr 1) time, implying fixed-parameter tractability for the three parameters mentioned above. Furthermore, Ben-Zwi et al. [START_REF] Ben-Zwi | Treewidth governs the complexity of target set selection[END_REF] proved W [START_REF] Aazami | Approximation algorithms and hardness for domination with propagation[END_REF]-hardness for Target Set Selection with respect to the parameter treewidth when the thresholds are unbounded. We extend this result by showing W[1]-hardness for treewidth when the thresholds respect the majority condition. The proof even shows hardness for the combined parameter feedback vertex set, pathwidth, distance to cographs, and distance to interval graphs. Finally, we show that Target Set Selection is fixed-parameter tractable when parameterized by the bandwidth. This result even holds for general thresholds.

O(tw) max •n O(

Extending the basic reduction

In the following, we show that the basic reduction I (see Section 3.2.1) can be extended to prove hardness for the combination of various sparseness-related parameters.

Theorem 13 Target Set Selection with majority threshold is W[1]-hard with respect to the combination of the following parameters: feedback vertex set, distance to cograph, distance to interval graph, and pathwidth.

Proof. We modify the basic reduction I to get the new, equivalent instance (G ′′ , thr ′ , k ′′ ) as follows. For each vertex v in a validation gadget, add deg G ′ (v)-4n vertices adjacent to v. Moreover, for each guard vertex v add deg G ′ (v) -2 neighbors. Let X be the set of vertices added so far. Insert a new vertex u adjacent to all vertices in X and add |X|+2(k ′ +2) vertices to the neighborhood of u. To complete the modification of the graph, for every vertex v in a selection gadget, attach deg

G ′ (v) neighbors. Finally, set thr ′ (v) = ⌈deg G ′′ (v)/2⌉ for all v ∈ V (G ′′ ) and k ′′ = k ′ + 1.
We claim that (G ′′ , thr ′ , k ′′ ) is a yes-instance if and only if (G ′ , thr, k ′ ) is a yesinstance.

"⇒": Suppose that there is a solution S ′ for (G ′′ , thr ′ , k ′′ ). First, observe that u ∈ S ′ since, otherwise, u would not become active. Indeed, even if all the vertices in G ′′ plus k ′′ degree-one neighbors of u are activated, the vertex u will not be activated since its threshold is |X| + k ′′ + 1. Since u is in all solutions, we may consider the equivalent instance where u is removed together with all its neighbors (they all have threshold one and thus get activated by u). Moreover, for each removed vertex v, we have to decrease the threshold of the vertices in N (v) by one. This operation leaves a graph with many degree-one vertices of threshold one. Applying Reduction rule 12, we arrive at the instance (G ′ , thr, k ′ ). By the correctness of Reduction rule 12, it follows that S ′ is also a solution for (G ′ , thr, k ′ ).

"⇐": Conversely, let S be a solution for (G ′ , thr, k ′ ). Since activating u and exhaustively applying Reduction rule 12 on (G ′′ , thr ′ , k ′′ ) results in (G ′ , thr, k ′ ), it is clear that S ∪ {u} is a target set for (G ′′ , thr ′ ) of size k ′ + 1.

To complete the proof of the theorem, it is enough to observe that if we remove the vertex u, all guard vertices, and the validation gadgets (that is, k 2 + k + 1 vertices), then we get stars and isolated vertices.

Bandwidth

Another possible measure for sparseness is the bandwidth of the input graph. Here, our result is of more positive nature: we show that Target Set Selection is fixed-parameter tractable with respect to the bandwidth, even for general threshold functions, by using an algorithm of Ben-Zwi et al. [START_REF] Ben-Zwi | Treewidth governs the complexity of target set selection[END_REF].

Theorem 14 Target Set Selection is fixed-parameter tractable with respect to the combined parameter "treewidth" and "maximum degree" of the input graph.

Proof.

Let (G = (V, E), thr, k) be an instance of Target Set Selection. [START_REF] Ben-Zwi | Treewidth governs the complexity of target set selection[END_REF] gave a (thr max ) O(tw) •n-time algorithm for solving Target Set Selection, where tw is the treewidth of the input graph and thr max is the maximum threshold value. It follows that this algorithm applied to

First, exhaustively apply Reduction rule 11 to get a new equivalent instance (G

′ = (V ′ , E ′ ), thr ′ , k ′ ). Observe that thr ′ (v) ≤ deg G ′ (v) for all v ∈ V ′ . Moreover, Ben-Zwi et al.
G ′ runs in time ∆(G ′ ) O(bw(G ′ )) • n since tw(G ′ ) ≤ 2 bw(G ′ ).
By the definition of bandwidth we can deduce that ∆(G) ≤ 2 bw(G) for any graph G which implies the following corollary.

Corollary 15 Target Set Selection is fixed-parameter tractable with respect to the parameter "bandwidth" of the input graph.

Parameters related to dense structures

In contrast to the previous section, we now consider Target Set Selection with respect to parameters related to the denseness of the input graph. Since cliques are the most dense graphs and Target Set Selection is polynomial-time solvable on cliques [START_REF] Nichterlein | On tractable cases of target set selection[END_REF], parameters measuring the distance to cliques are most interesting. In particular, we consider the vertex deletion distance to clique and to a collection of disjoint cliques (also called cluster vertex deletion number or "cvd number" for short), and the clique cover number.

Starting with the case of unrestricted thresholds in Section 3.4.1, we show that Target Set Selection parameterized by the size of a minimum cluster vertex deletion (cvd) set is W[1]-hard. Furthermore, we show NP-hardness when restricting to instances with clique cover number two. Then, in Section 3.4.2, we study restricted threshold functions. For constant or majority thresholds, Target Set Selection parameterized by the distance to a clique is fixed-parameter tractable. Furthermore, we show an exponential-size kernel with respect to the combined parameter maximum threshold value and cvd number, implying fixed-parameter tractability with respect to the cvd number on inputs with thresholds bounded by a constant.

Unrestricted thresholds

Here, we research the general Target Set Selection setting without constraints on the thresholds of the input. As hinted in the introduction, the next two theorems state that these variants are parameterized intractable with respect to the employed denseness measures.

Theorem 16 Target Set Selection is W[1]-hard with respect to the parameter "cvd number".

Proof. We modify the basic reduction I (see Section 3.2.1) as follows. Make all connection vertices that have a common vertex in a (vertex-or edge-) selection gadget to a clique. The thresholds of the connection vertices are modified as follows: In each maximal clique of the connection vertices, an arbitrary ordering of the vertices is fixed. Then the first vertex has threshold one, the second has threshold two, and the i th vertex has threshold i. See Figure 3.3 for a scheme of the reduction.

Note that each connection vertex is contained in exactly one maximal clique.

Notice also that the following holds. Let C be such a maximal clique, V 1 (resp. V 1 and V 2 ) denote the validation gadget adjacent to C, and v the vertex adjacent to C in the vertex-selection gadget (resp. the edge-selection gadget). Then all connection vertices in C are activated if and only if either v is activated or all the vertices in V 1 (resp. V 1 and V 2 ) are activated. We now prove that (G, col, k) is a yes-instance of Multicolored Clique if and only if (G ′ , thr, k 2 + k) is a yes-instance of Target Set Selection. "⇒": Suppose that (G, col, k) has a multicolored clique C ⊆ V of size k. Then the set S = {v ∈ C} ∪ {e uv : u, v ∈ C} is a target set for (G ′ , thr, k ′ ). Indeed, in the first step of the propagation process every guard vertex is activated since they are all adjacent to a vertex in S. After 4n steps, all the connection vertices adjacent to a vertex in S get activated. During the next step, all 4 k 2 vertices in validation pairs will be activated since C is a multicolored clique of size k. From now on, it is not hard to see that the entire graph will be activated.

"⇐": Conversely, assume that (G ′ , thr, k ′ ) has a target set S ⊆ V ′ of size k. First, we may assume that S does not contain any guard vertex since they all have threshold one. Moreover, one has to pick up in the target set at least one vertex in each selection gadget to activate the guard vertex of the latter. Indeed, recall that every neighbor of a guard vertex has a threshold equal to its degree and the guard vertex is not in the target set. Thus, every target set contains at least one vertex in each selection gadget. Furthermore, since k ′ = k 2 +k we conclude that there is exactly one vertex from each selection gadget in a minimal target set. Suppose now that we select two vertices u ∈ X c 1 and v ∈ X c 2 together with an edge-vertex e u ′ v ′ ∈ X {c 1 ,c 2 } for some c 1 , c 2 ∈ {1, . . . , k} such that e u ′ v ′ is not incident to both u and v. Without loss of generality, we may assume that u = u ′ . Then at least one vertex in the validation gadgets V c 1 ,c 2 and V c 2 ,c 1 will not be activated. To see this, recall that for all w ∈ V ′ it holds that high(w) + low(w) = 2n and, since u = u ′ , we have either low(u) + high(u ′ ) < 2n or high(u) + low(u ′ ) < 2n. This implies, as previously discussed, that some connection vertices will not be activated, a contradiction.

Theorem 17 Target Set Selection is NP-hard and W[2]-hard with respect to the parameter k even on graphs with clique cover number two.

Proof. We present a parameterized reduction from the W[2]-hard problem Hitting Set (see Appendix A). Given an instance (F, U, k) of Hitting Set consisting of a set family F = {F 1 , . . . , F m } over a universe U = {u 1 , . . . , u n } and an integer k ≥ 0, we construct an instance (G, thr, k) of Target Set Selection as follows.

We start with the construction of the graph G. A set of vertices V U contains a vertex for every element u ∈ U , that is, V U = {v u : u ∈ U }. Analogously, a second set W F contains a vertex for every subset, that is, W F = {w F : F ∈ F}. The vertices in V U are called element vertices and the vertices in W F are called subset vertices.

There is an edge between an element vertex v u and a subset vertex w F if and only

if u ∈ F . Next, add a new vertex x ∈ (V U ∪ W F ) to G and connect x to all vertices in W F . Then, make V 1 = V U ∪{x} a clique. Add |F|-1 sets of vertices V B 1 , . . . , V B |F |-1
to the graph, each set containing α = |U |+2 vertices and let

V B = |F |-1 i=1 V B i . Finally, make V 2 = W F ∪ V B 1 ∪ . . . ∪ V B |F |-
F i ) = (i-1)α+i, for every element vertex v u ∈ V U , set thr(v u ) = |{F ∈ F : u ∈ F }|+k+1, and for each vertex v ∈ V B i , 1 ≤ i ≤ |F|-1, set thr(v) = (i-1)α+i. Finally, complete the construction by setting thr(x) = |W F | + k
Since V 1 and V 2 are cliques, the constructed graph G is a diameter-two graph whose vertices can be covered by two cliques, see Figure 3.4.

For the correctness it remains to show that (F, U, k) is a yes-instance of Hitting Set if and only if (G, thr, k) is a yes-instance of Target Set Selection.

"⇒": If (F, U, k) is a yes-instance, then there exists a hitting set U ′ of size k for F. We show that S = {v u : u ∈ U ′ } is a target set for G of size k . Since U ′ is a hitting set, every vertex in W F has at least one neighbor in S. Thus, all vertices in V 1 become active in 2|W F | -1 steps: In the first step w F 1 is activated since thr(w F 1 ) = 1. Then in the second step, all vertices in V B 1 are activated since all these vertices also have threshold one and w F 1 is active. For 2 ≤ i ≤ |W F |, in the (2i-1) th step the vertex w F i is activated and in the next step all vertices in V B i : The neighbors of w F i that are active in step 2i -2 are: All vertices in V B 1 ∪ . . . ∪ V B i-1 , the vertices w F 1 , . . . , w F i-1 and at least one vertex in S. Since the threshold is thr(w F i ) = (i -1)α + i, the vertex w F i is activated. Then, there are (i -1)α + i active vertices in V 2 and, hence, all vertices in V B i are activated in the 2i th step. After all vertices in V 2 are active, x is activated. Finally, in the last step all vertices in V U \ S are activated since for every vertex in V U \ S all neighbors in W F and x have been activated. "⇐": If (G, thr, k) is a yes-instance of Target Set Selection, then there is a target set S of size at most k. We first show that S ⊆ V U .

Assume towards a contradiction that there is a vertex in

S \ V U . Then, |S ∩ V U | ≤ k -1. Let ℓ denote the first step in which a vertex in V U \ S is activated, that is, ℓ = min{j : A j G,thr (S) ∩ (V U \ S) = ∅}. Moreover, let v u ∈ A ℓ G,thr (S) ∩ (V U \ S). Note that, by definition of ℓ, it holds that |A ℓ-1 G,thr (S) ∩ V 1 | ≤ k. Hence |N G (v u ) ∩ A ℓ-1 G,thr (S)| =|N G (v u ) ∩ A ℓ-1 G,thr (S) ∩ W F | + |N G (v u ) ∩ A ℓ-1 G,thr (S) ∩ V 1 | ≤|{F ∈ F : u ∈ F }| + k <|{F ∈ F : u ∈ F }| + k + 1 = thr(v u ) a contradiction. Therefore, S ⊆ V U .
Finally, we show that U ′ = {u : v u ∈ S} is a hitting set for F. To this end, we show that every subset vertex w F i has a neighbor in S and, hence, is hit by U ′ . Assume towards a contradiction that there exists a vertex w

F i ∈ W F with N G (w F i ) ∩ S = ∅. Let X i = ( i≤j≤|F |-1 V B j ) ∪ ( i≤j≤|F | {w F i }). Let ℓ denote the first step in which a vertex in X i is activated, that is, ℓ = min{j : A j G,thr (S) ∩ X i = ∅}. Hence |A ℓ-1 G,thr (S) ∩ W F | = |A ℓ-1 G,thr (S)∩(W F \X i )| ≤ i-1 and |A ℓ-1 G,thr (S)∩V B | = |A ℓ-1 G,thr (S)∩(V B \X i )| ≤ | 1≤j<i V B j | = (i -1)α. Let v ∈ X i ∩ V B , then we have |A ℓ-1 G,thr (S) ∩ N G (v)| = |A ℓ-1 G,thr (S) ∩ V B | + |A ℓ-1 G,thr (S) ∩ W F | ≤ (i -1)α + i -1 < (i -1)α + i ≤ thr(v) and, thus, A ℓ G,thr (S) ∩ X i ∩ V B = ∅. Now consider v ∈ X i ∩ (W F \ {w F i }). Observe that |A ℓ-1 G,thr ∩ V 1 | = |S| = k. Thus, we have |A ℓ-1 G,thr (S) ∩ N G (v)| ≤ |A ℓ-1 G,thr (S) ∩ V B | + |A ℓ-1 G,thr (S) ∩ W F | + |A ℓ-1 G,thr (S) ∩ V 1 | ≤ (i -1)α + i -1 + k < iα + i + 1 ≤ thr(v) and, thus, A ℓ G,thr (S) ∩ X i ∩ (W F \ {w F i }) = ∅. Finally, consider v = w F i |A ℓ-1 G,thr (S) ∩ N G (v)| ≤ |A ℓ-1 G,thr (S) ∩ V B | + |A ℓ-1 G,thr (S) ∩ W F | ≤ (i -1)α + i -1 < (i -1)α + i = thr(v)
and, thus,

w F i / ∈ A ℓ G,thr (S). Altogether we have A ℓ G,thr (S) ∩ X i ∩ V B = ∅, A ℓ G,thr (S) ∩ X i ∩ (W F \ {w F i }) = ∅, and w F i / ∈ A ℓ G,thr (S) and, hence, A ℓ G,thr (S) ∩ X i = ∅, a contradiction. v u i • • • w F 3 2α + 3 w F 4 3α + 4 w F 5 4α + 5 V U W F w F 1 1 • • • w F 2 α + 2 . . . 1 V B 1 . . . α + 2 V B 2 . . . 2α + 3 V B 3 . . . 3α + 4 V B 4 . . . 4α + 5 V B 5 x thr(x) = |W F | + k V 2 V 1 Figure 3.4:
A schematic picture of the constructed graph. Each of the two vertex sets V 1 (upper box) and V 2 (lower box) form a clique. The number below a vertex denotes its threshold. The only way to activate all vertices with a target set of size k is to choose k vertices in V U such that these k vertices activate all vertices in W F .

Restricted thresholds

In the spirit of researching the influence of bounded thresholds on Target Set Selection, we consider the parameters distance to clique and cluster vertex deletion number (cvd number). Recall that we showed W[1]-hardness for the parameter cvd number (for unbounded thresholds) in the previous paragraph. By presenting an exponential-size kernel, we show that the problem becomes tractable with respect to this parameter if the maximum threshold is a constant. First, we show that Target Set Selection with majority thresholds or constant thresholds is fixed-parameter tractable with respect to the parameter distance to clique. We can even show fixed-parameter tractability for less restrictive threshold functions. To this end, let P(V ) be the set of all subsets of V .

Theorem 18 Target Set Selection on graphs with vertex set V is fixedparameter tractable with respect to the parameter "distance to clique", denoted by ℓ, when the threshold function thr fulfills the restriction

thr(v) > g(ℓ) ⇒ thr(v) = f (N (v))
for all vertices v ∈ V and arbitrary functions f : P(V ) → N and g : N → N.

Proof. We prove the theorem by giving a fixed-parameter algorithm computing a target set of minimum size for (G, thr). To this end, we introduce some notations. Let X ⊂ V , |X| = ℓ, denote a set of vertices such that G[V \ X] is a clique. We define a non-standard "twins" equivalence relation ≡ by

u ≡ v ⇐⇒ (N [u] = N [v]) ∧ (thr(u) = thr(v)) ∧ (u ∈ X ⇐⇒ v ∈ X)
Since the thresholds and neighborhoods of all vertices in an equivalence class Z are equal, we can denote this threshold and this neighborhood by thr(Z) and N [Z], respectively. Let Z 1 , Z 2 , . . . , Z s be a list of all nonempty equivalence classes of ≡.

Since G[V \ X] is a clique, we know that for all u, v ∈ V \ X : N [u] = N [v] if and only if N G[X] [u] = N G[X] [v]. Due to the condition thr(v) > g(ℓ) ⇒ thr(v) = f (N (v))
, for each subset X ′ ⊆ X, there are at most g(ℓ) + 1 equivalence classes disjoint from X whose neighborhood in X is exactly X ′ . Hence, s ≤ 2 ℓ (g(ℓ) + 1) + ℓ.

Let S be a minimum-size target set for (G, thr). With S, we can define r i as the number of the first activation step in which all vertices of Z i are active. More formally, r i = min{j : Z i ⊆ A j G,thr (S)}. Let r = max{r i : 1 ≤ i ≤ s}. In the following, we upper-bound r by s. We do this by showing that for each 1 ≤ j ≤ r, there is an 1 ≤ i ≤ s such that r i = j. Assume this was false, that is, there is some activation step j such that none of the equivalence classes gets activated in step j. Since j ≤ r, there is some vertex v that gets activated in step j.

Let Z i denote the equivalence class of v. Since j ≥ 1, we know that |N (v) ∩ A j-1 G,thr (S)| ≥ thr(v). Since for each vertex u ∈ Z i , thr(u) = thr(v) and N (u) = N (v), we conclude that Z i ⊆ A j
G,thr (S), contradicting the assumption that r i = j. Now we describe our algorithm. In the first phase, we guess the correct values of r i for all 1 ≤ i ≤ s. There are at most r s ≤ s s possibilities to do so.

In the second phase of the algorithm, we use an ILP formulation to solve the problem. Each variable x i in the ILP represents the number of vertices in the equivalence class i that are in the target set S. We use constraints to model the activation process: For each equivalence class Z i , the number of active neighbors in step r i exceeds thr(Z i ). Two types of active neighbors are considered. First, the vertices in N [Z i ] ∩ S. Second, the vertices in all equivalence classes Z j ⊆ N [Z i ] that are active in step i, that is, r j < r i . More formally, Minimize:

s i=1 x i subject to: ∀1 ≤ i ≤ s, thr(Z i ) ≤ Z j ⊆N[Z i ] r j ≥r i x j + Z j ⊆N[Z i ] r j <r i |Z j | ∀1 ≤ i ≤ s, x i ∈ {0, 1}
By the discussion above, a solution to this ILP corresponds to a minimum-size target set for (G, thr). Since the ILP formulation has s variables, a result by Lenstra [START_REF] Lenstra | Integer programming with a fixed number of variables[END_REF] implies that solving it is fixed-parameter tractable with respect to s. Since at most s s such ILPs have to be solved and s ≤ 2 ℓ (g(ℓ) + 1) + ℓ, fixed-parameter tractability with respect to ℓ follows.

Clearly, Theorem 18 is a pure complexity classification result. Since the majority thresholds and constant thresholds both satisfy the restrictions required in Theorem 18, the next corollary immediately follows.

Corollary 19 Target Set Selection with majority thresholds or constant thresholds is fixed-parameter tractable with respect to the parameter distance to clique.

Next, we show fixed-parameter tractability for Target Set Selection with respect to the parameter "cvd number" and constant thresholds. In the following, we assume that an optimal cvd set X of the input graph is given. If this is not the case, then one might instead use a simple polynomial-time 3-approximation algorithm. 1 Either way, we abbreviate ℓ = |X|.

In this section we use the notion of "critical cliques". Here, a clique K in a graph is critical if all its vertices have the same closed neighborhood and K is maximal with respect to this property.

First, we present a data reduction rule allowing us to bound the number of vertices with the same open or closed neighborhood by the maximum threshold thr max .

Reduction rule 20 Let I = (G = (V, E), thr, k) be an instance of Target Set Selection that is reduced with respect to Reduction rule 11 and let v 1 , v 2 , . . . , v thr max +1 ∈ V be vertices such that either

N (v 1 ) = N (v 2 ) = . . . = N (v thrmax +1 ) or N [v 1 ] = N [v 2 ] = . . . = N [v thrmax +1 ]
Furthermore, let v 1 be the vertex with the highest threshold, that is, for all 1 ≤ i ≤ thr max +1 it holds that thr(v 1 ) ≥ thr(v i ). Then delete v 1 .

Lemma 21 Reduction rule 20 is correct and can be applied exhaustively in O(n + m) time.

Proof. For the running time, note that computing the critical cliques of a graph can be done in linear time [START_REF] Mcconnell | Linear-time modular decomposition and efficient transitive orientation of comparability graphs[END_REF]. Thus, we first compute the critical cliques of the graph in linear time. Then we iterate over the critical cliques and if one of them has size thr max +r, r > 0, then we delete the r vertices of this critical clique having the largest thresholds. This can clearly be done in linear time. Notice that a maximal set of vertices with the same open neighborhood form a critical clique in the complement graph. Hence, in a second step, we repeat the procedure with the complement graph. Then the graph is reduced with respect to Reduction rule 20. Furthermore observe that Reduction rule 11 is not applicable after Reduction rule 20 was applied.

To show the correctness, we prove that the instance (G ′ = (V ′ , E ′ ), thr, k) that is produced by Reduction rule 20 is a yes-instance if and only if the input instance I is a yes-instance.

"⇒:" Since (G ′ , thr, k) is a yes-instance, there exists a target set S ⊆ V ′ , |S| ≤ k, that activates all vertices in G ′ . Hence, S activates all vertices of V \ {v 1 } in G. Since (G, thr, k) is reduced with respect to Reduction rule 11, the vertex v 1 is activated by its neighbors. Thus, S is also a target set for (G, thr, k).

"⇐:" Since (G, thr, k) is a yes-instance, there exists a target set S ⊆ V , |S| ≤ k, activating all vertices in G. Let W = {v 1 , v 2 , . . . , v thrmax +1 } be the vertices considered in the reduction rule. First observe that we can assume W \ S = ∅, (that is, not all vertices of W are in the target set) since otherwise, S ′ = S \ {v 1 } is also a target set: In the first activation step all vertices in N (v 1 ) become active and, since (G, thr, k) is reduced with respect to Reduction rule 11, it follows that v 1 is active after the second step. Thus, S ′ is also a target set. Now consider the case that v 1 / ∈ S. Since for all v i ∈ W it holds that thr(v 1 ) ≥ thr(v i ) and v 1 is activated by its neighbors, it is clear that all vertices in W \ {v 1 } are active once v 1 is active. Since |W | > thr max this implies that all vertices in N G (v 1 ) become active in G ′ and, thus, S is a target set for G ′ .

Finally, consider the case that v 1 ∈ S. Let w ∈ W \ S be the vertex with the highest threshold, that is, for all v i ∈ W \ S it holds that thr(w) ≥ thr(v i ). Observe that S ′ = (S \{v 1 })∪{w} is a target set for G ′ : Since S activates all vertices in W it is clear that S ′ activates all vertices in W \ {v 1 }. This implies that all vertices in N (v 1 ) are activated by S ′ in G ′ since |W \ {v 1 }| = thr max and all vertices in W have the same neighborhood. Thus, S ′ activates all vertices in G ′ .

In the following we assume that the input graph G is reduced with respect to Reduction rule 20. Thus, G[V \ X] consists of disjoint cliques, each of size at most 2 ℓ thr max . Hence, in order to get the desired kernel it remains to bound the number of cliques in G[V \ X]. To this end, we introduce the following notation:

Definition 35: Clusters equivalence Let I = (G = (V, E), thr, k) be an instance of Target Set Selection, let X ⊆ V be a cvd set, and let S ⊆ V . Let C 1 , C 2 ⊆ V be two clusters in G[V \ X]. We call C 1 and C 2 equivalent with respect to X, denoted by C 1 ≡ X C 2 , if there exists a bijection f : C 1 → C 2 such that for every v ∈ C 1 it holds that thr(v) = thr(f (v)) and N (v) ∩ X = N (f (v)) ∩ X.
Furthermore, we call C 1 and C 2 equivalent with respect to X and S, denoted by

C 1 ≡ S X C 2 , if the bijection f additionally fulfills v ∈ S ⇐⇒ f (v) ∈ S for all v ∈ C 1 .
Note that ≡ X is an equivalence relation on the clusters in G[V \ X] with at most (thr max +1) 2 ℓ thrmax equivalence classes. To see this, observe that each equivalence class is uniquely determined by 2 ℓ (possibly empty) sequences of thresholds. One for each subset of X. Since G is reduced with respect to Reduction rule 20, each such sequence contains between 0 and thr max thresholds. Since each threshold is at most thr max , the number of equivalence classes is at most

thrmax i=0 thr i max 2 ℓ ≤ (thr max +1) thrmax 2 ℓ = (thr max +1) 2 ℓ thrmax
In the following, our goal is to bound the number of cliques in one equivalence class in a function depending only on thr max and ℓ. Note that once we achieve this goal, we have a kernel with respect to the parameter "cvd number". The next lemma is a first step towards this goal.

Lemma 22 Let I = (G = (V, E), thr, k) be an instance of Target Set Selection, let X ⊆ V be a cvd set for G, and let S ⊆ V , |S| ≤ k, be a target set for G. Furthermore let C 1 , C 2 , . . . , C thrmax +1 ⊆ V be clusters in G[V \ X] that are pairwise equivalent with respect to X and S. Then, S \ C 1 is a target set for G[V \ C 1 ]. Proof. Let S ′ = S \ C 1 and G ′ = G[V \ C 1 ]
. We prove the lemma by contradiction: Assume that S ′ is not a target set for G ′ . Let Y ⊆ V \ C 1 be the set of vertices that are activated in G in some step i but are not activated in G ′ in the step i. Formally,

Y = {v ∈ V \ C 1 : ∃i ≥ 1 s.t. v ∈ A i G,thr (
S) and v / ∈ A i G ′ ,thr (S ′ )}. Since S ′ is not a target set for G ′ , the set Y is not empty. In particular, Y contains all vertices in G ′ that are not activated by S ′ . Let v ∈ Y be the vertex that is activated first in G i.e. for all u ∈ Y it holds that u

∈ A i G,thr (S) ⇒ v ∈ A i G,thr (S), 1 ≤ i. Since v ∈ Y and Y ⊆ V \ C 1 , it holds that v / ∈ S. Let i ≥ 1 be the step in which v becomes active in G, that is, v ∈ A i G,thr (S) \ A i-1 G,thr (S). Thus, |N G (v) ∩ A i-1 G,thr (S)| ≥ thr(v). Since v is in G ′ not activated by S ′ , it follows that |N G ′ (v) ∩ A i-1 G ′ ,thr (S ′ )| < thr(v). From the selection of v it follows that Y ∩ A i-1 G,thr (S) = ∅. Thus, A i-1 G,thr (S) \ A i-1 G ′ ,thr (S ′ ) ⊆ C 1 . Since N G (v)\N G ′ (v) ⊆ C 1 , it follows that N G (v)∩A i-1 G,thr (S)∩C 1 = ∅ and v ∈ X. Let u ∈ N G (v) ∩ A i-1
G,thr (S) ∩ C 1 . Note that C 1 and C j , 1 < j ≤ thr max +1, are equivalent with respect to X and S and, hence, there is a bijection f j as described in Definition 35. Thus, it is easy to see that u

∈ A i-1 G,thr (S) ⇒ f j (u) ∈ A i-1
G,thr (S). Moreover, since u ∈ N G (v) it follows that f j (u) ∈ N G (v) and, thus,

f j (u) ∈ N G ′ (v). Hence, f j (u) ∈ N G ′ (v) ∩ A i-1 G ′ ,thr (S ′ ) for all 2 ≤ j ≤ thr max +1 and thus |N G ′ (v) ∩ A i-1 G ′ ,thr (S ′ )| ≥ thr max . Hence, thr(v) > |N G ′ (v) ∩ A i-1 G ′ ,thr (S ′ )| ≥ thr max , a contradiction.
Since we do not know the target set S for G, two problems have to be solved in order to convert this lemma into a data reduction rule: The first problem is to find out by how much we have to decrease k, or, equivalently, how to compute |S ∩ C 1 | in polynomial time? The second problem is that we do not know the target set S. As we show in the following, the key in overcoming these two problems is to increase the number of equivalent clusters C j in the assumption of the lemma.

To this end, we first compute a lower and upper bound on the size of the target set for G. Let G X be the graph that results from activating all vertices in X and applying Reduction rule 11 exhaustively. Let C

X 1 , C X 2 , . . . , C X ζ denote the maximal cliques of G X . Clearly, for each clique C X of G X there is a cluster C in G[V \ X] such that C X ⊆ C.
Let S X ⊆ V be an optimal solution for G X . Note that S X can be computed in linear time [START_REF] Nichterlein | On tractable cases of target set selection[END_REF]. By construction of G X it is clear that |S X | is a lower bound for the size of any target set for G. Furthermore, S X ∪ X is a target set for G. Hence, if k < |S X | we can immediately answer no and if k ≥ |S X | + |X| = |S X | + ℓ we can answer yes. Thus, we assume in the following that |S X | ≤ k < |S X | + ℓ. Besides these general bounds on the target set size we can also derive bounds for the number of vertices in a target set for each cluster

C in G[V \ X]: If there is a (uniquely determined) clique C X in G X such that C X ⊆ C, then set min(C) = |S X ∩ C X |.
In case there is no such clique in G X , set min(C) = 0. Finally, set max(C) = min{thr max , min(C) + ℓ}. Clearly, min(C) and max(C) are lower resp. upper bounds on the number of vertices of C that are in an optimal target set for G. Note that if two clusters C 1 and C 2 in G[V \X] are equivalent with respect to X, then min(C 1 ) = min(C 2 ). Furthermore, having ℓ + 1 clusters C 1 , . . . , C ℓ+1 in G[V \ X] that are equivalent with respect to X, we can conclude that for any optimal target set S there is a cluster C i , 1 ≤ i ≤ ℓ+1, having exactly min(C 1 ) vertices in the target set, since otherwise, the solution S X ∪ X for G contains fewer vertices than S. Likewise, if there are ℓ + r clusters C 1 , . . . , C ℓ+r that are equivalent with respect to X, then it is clear that for any optimal target set S at least r of these clusters contain exactly min(C 1 ) vertices of S. Hence, increasing the number of equivalent clusters to at least ℓ + thr max +1 solves the first problem.

We overcome the second problem by relaxing the condition "equivalent with respect to X and S" for the clusters C 1 , . . . , C thrmax ⊆ V to "equivalent with respect to X" and increase the number of equivalent clusters: We can assume that, out of each cluster C, at most max(C) ≤ thr max vertices are in a target set. Thus, there are at most thr 2 ℓ max possibilities for choosing thr max vertices from a cluster to be in a target set: Choose at most thr max vertices with the highest threshold from each of the at most 2 ℓ critical cliques of the cluster. Having a set of vertices with the same closed neighborhood and the task is to choose s of them to be in a target set, it is best to choose the s vertices with the highest thresholds [90, Observation 7]. Thus, when increasing the number of clusters that have to be equivalent with respect to X to ℓ + thr 2 ℓ max (thr max +1) we can conclude with the pigeonhole principle that there are clusters C i 1 , . . . , C i thrmax +1 that are equivalent with respect to X and S for any target set S and each cluster C i j contains min(C i j ) vertices of S. Hence, applying Lemma 22 to this set we arrive at the following reduction rule.

Reduction rule 23 Let I = (G = (V, E), thr, k) be an instance of Target Set Selection that is reduced with respect to Reduction rule 11 and let X ⊆ V be a cvd set of size ℓ.

Let C 1 , C 2 , . . . , C α ⊂ V be disjoint clusters in G[V \X] such that α = ℓ+thr 2 ℓ
max (thr max +1) and for each pair C i , C j , 1 ≤ i, j ≤ α, it holds that C i ≡ X C j . Then delete C 1 and reduce k by min(C 1 ).

The correctness of the data reduction rule follows from Lemma 22 and the above discussion. As to the running time, note that Reduction rule 23 can be exhaustively applied in O(n 2 ) time. Since we require that the cvd set X is given, we can compute the clusters in G[V \X] in linear time. Then, we sort the vertices in these clusters by neighborhood and threshold. This can be done in O(n log(n)) time. After this sorting the check whether two clusters are equivalent with respect to X can be done in linear time: Simply iterate over the sorted vertices and check whether the current vertices in both clusters have the same neighborhood and threshold. Overall, iterating over all clusters in G[V \ X], determining the equivalent clusters, and deleting the respective clusters can be done in O(n 2 ) time.

With these data reduction rules we now arrive at the following theorem.

Theorem 24 Target Set Selection admits a kernel with thr O(2 ℓ thrmax) max ℓ vertices, where ℓ is the cluster vertex deletion number and thr max is the maximum threshold.

Proof. Let I = (G = (V, E), thr, k) be an instance of Target Set Selection that is reduced with respect to Reduction rules 11, 20, and 23. Furthermore let X ⊆ V be a cvd set and let ℓ = |X|.

Since I is reduced with respect to Reduction rule 20, the clusters in G[V \ X] have size at most 2 ℓ thr max . Hence, there are at most (thr max +1) 2 ℓ thrmax clusters in G[V \ X] that are all pairwise not equivalent with respect to X. Furthermore, since I is reduced with respect to Reduction rule 23, each equivalence class of ≡ X contains at most ℓ + thr 2 ℓ max (thr max +1) clusters. Thus, the number of clusters in G[V \ X] is bounded by (ℓ+thr 2 ℓ max (thr max +1))(thr max +1) 2 ℓ thrmax , each of these clusters contains at most 2 ℓ thr max vertices. Overall this gives thr Clearly this kernel implies that Target Set Selection is fixed-parameter tractable with respect to the combined parameter thr max and ℓ. This yields the following result for Target Set Selection with constant thresholds.

O(2 ℓ thrmax) max ℓ vertices in G[V \ X]
Corollary 25 Target Set Selection with constant thresholds is fixed-parameter tractable with respect to the parameter "cvd number".

Complementary problem

In this section, we turn our attention to the complementary problem named Influence. First, we consider the parameterized approximability of the optimization version. As explained in the preliminaries, there are two versions of it denoted by Max Closed Influence and Max Open Influence. Notably, we prove that both versions are strongly parameterized inapproximable with respect to k even for constant and majority thresholds. However, in the unanimity case, we observe that they admit a fpt-time approximation algorithm with respect to k whereas their decision version is W[1]-hard and Max Open Influence cannot be approximated within n 1-ε in polynomial time for all ε > 0 unless NP = ZPP. Second, we prove that Influence is fixed-parameter tractable with respect to the combined parameter k and maximum degree of the input graph.

Inapproximability results

Here, we consider the parameterized approximability of both Max Closed Influence and Max Open Influence with respect to the parameter k. We show that these problems are W[2]-hard to approximate within n 1-ε for any ε > 0 for majority thresholds and thresholds at most two, respectively.

Theorem 26 For any ε > 0, Max Closed Influence and Max Open Influence with majority thresholds cannot be approximated within n 1-ε in fpt-time with respect to the parameter k even on bipartite graphs, unless FPT = W [START_REF] Ackerman | Combinatorial model and bounds for target set selection[END_REF].

Proof. By Lemma 10, it suffices to show the result for Max Closed Influence. We construct a parameterized gap-introducing reduction from the W[2]-hard problem Dominating Set (see Appendix A) to Max Closed Influence with majority. In this reduction, we will make use of the ℓ-edge gadget, for some integer ℓ. An ℓ-edge between two vertices u and v consists of ℓ vertices of threshold one adjacent to both u and v.

Given an instance I = (G = (V, E), k) of Dominating Set with n = |V |, m = |E|, we define an instance I ′ = (G ′ , k + 1) of Max Closed Influence with majority thresholds as follows. We start with the graph obtained from I as described in the basic reduction II (see Section 3.2.1), then we modify it to get G ′ as follows. Replace every edge v t v b by an (k + 2)-edge between v t and v b . Moreover, for a given constant β = 8 ε -5, let L = ⌈n β ⌉ and we add nL more vertices x 1 1 , . . . , x 1 n , . . . , x L 1 , . . . , x L n . For i = 1, . . . , n, vertex x 1 i is adjacent to all the bottom vertices. Moreover, for any j = 2, . . . , L, each x j i is adjacent to x j-1 k , for any i, k ∈ {1, . . . , n}. We also add a vertex w and an n+(k+2)(deg G (v)-1)-edge between w and v b , for any bottom vertex v b . For i = 1, . . . , n, vertex x 1 i is adjacent to w. For i = 1, . . . , n add n pending-vertices (i.e. degree one vertices) adjacent to x L i . For any vertex v t add (deg G (v) + 1)(k + 2) pending-vertices adjacent to v t . Add also n + n 2 + (k + 2)(2mn) pending-vertices adjacent to w. All vertices of the graph G ′ have the majority thresholds (see also Figure 3.5). Let n ′ be the order of G ′ , notice that we have n ′ ≤ n 4 + nL ≤ n β+5 .

We claim that if I is a yes-instance then opt(I ′ ) ≥ n β+1 ; otherwise opt(I ′ ) < n β+1

(n ′ ) 1-ε . Suppose that there exists a dominating set S ⊆ V in G of size at most k. Consider the solution S ′ for I ′ containing the corresponding top vertices and vertex w. After and some u t ∈ V and n + (k + 2)(deg G (v) -1) neighbors activated belonging to an n + (k + 2)(deg G (v) -1)-edge between v b and w. Thus, every vertex x 1 i gets active after the third step, and generally after the jth step, j = 4, . . . , L + 2 the vertices x j-2 i are activated, and at the (L + 3)th step all pending-vertices adjacent to x L i are activated. Therefore, the size of an optimal solution is at least nL ≥ n β+1 .

v 5 v 4 v 3 v 2 v 1 w v t 1 v b 1 v t 2 v b 2 v t 3 v b 3 v t 4 v b 4 v t 5 v b 5 x 1 1 x 1 2 x 1 3 x 1 4 x 1 5 x 2 1 x 2 2 x 2 3 x 2
Suppose that there is no dominating set in G of size k. Without loss of generality, we may assume that no pending-vertices are in a solution of I ′ since they all have threshold one. If w does not take part of a solution in I ′ , then no vertex x 1 i could be activated and in this case opt(I ′ ) is less than n ′ -nL ≤ n 4 . Consider now the solutions of I ′ of size k + 1 that contain w. Observe that if a top-vertex v t gets active through bottom-vertices then v t can not activate any other bottom-vertices. Indeed, as a contradiction, suppose that v t is adjacent to a non-activated bottom-vertex. It follows that v t could not have been activated because of its threshold and that no pendingvertices are part of the solution, a contradiction. Notice also that it is not possible to activate a bottom vertex by selecting some x 1 i vertices since of their threshold. Moreover, since there is no dominating set of size k, any subset of k top vertices cannot activate all bottom vertices, therefore no vertex x k i , i = 1, . . . , n, k = 1, . . . , L can be activated. Hence, less than n ′ -nL vertices can be activated in G ′ and the size of an optimal solution is at most n 4 ≤ n β+1 (n ′ ) 1-ε . We then have a gap of (n ′ ) 1-ε , and the result follows from Theorem 9.

Theorem 27 For any ε > 0, Max Closed Influence and Max Open Influence with thresholds at most two cannot be approximated within n 1-ε in fpt-time with respect to the parameter k even on bipartite graphs, unless FPT = W [START_REF] Ackerman | Combinatorial model and bounds for target set selection[END_REF].

Proof. By Lemma 10, it suffices to prove the result for Max Closed Influence. We construct a parameterized gap-introducing reduction from the W[2]-hard problem Dominating Set (see Appendix A) to Max Closed Influence with thresholds at most two. In this reduction, we will make use of the directed edge gadget. A directed edge from a vertex u to another vertex v consists of a 4-cycle {a, b, c, d} such that a and u as well as c and v are adjacent. Moreover thr(a) = thr(b) = thr(d) = 1 and thr(c) = 2. The idea is that the vertices in the directed edge gadget become active if u is activated but not if v is activated. Hence, the activation process may go from u to v via the gadget but not in the reverse direction. In the rest of the proof, we may assume that no vertices from {a, b, c, d} are part of a solution of Max Closed Influence. Indeed, it is always as good to take the vertex u instead. We will also make use of a directed tree with leaves x 1 , . . . , x n and root r defined as follows: introduce n -1 new vertices y 2 , . . . , y n and insert directed edges from x 1 to y 2 , from x 2 to y 2 , from y i to y i+1 , for i = 2, . . . , n -1, from x i to y i , for i = 3, . . . , n, from y n to r. Moreover thr(y i ) = 2, i = 2, . . . , n and thr(r) = 1. The idea is that the vertices in the directed tree become active if all vertices x 1 , . . . , x n are activated but not if r is activated. So, we may assume that no vertex from y 2 , . . . , y n is part of a solution of Max Closed Influence.

Given an instance I = (G = (V, E), k) of Dominating Set with n = |V |, we define an instance I ′ = (G ′ , thr, k) of Max Closed Influence as follows. We start with the graph and thresholds obtained from I as described in the basic reduction II (see Section 3.2.1), then we modify them to get G ′ and the function thr as follows. Set the thresholds of top-vertices to two. Replace every edge between a top vertex v t and a bottom vertex v b by a directed edge from v t to v b . Let L = ⌈n β ⌉ where β = 4 ε -3. For j = 1, . . . , L, add vertices p j 1 , . . . , p j n and a directed tree between leaves v b i , i = 1, . . . , n and root p 1 ℓ , for ℓ = 1, . . . , n. Moreover for j = 1, . . . , L -1 add directed trees between leaves p j 1 , . . . , p j n and root p j+1 ℓ , for ℓ = 1, . . . , n. This completes the construction (see Figure 3.6). Let n ′ be the order of G ′ , notice that we have

n ′ = 2n + n 2 n β + 4(2n -1)n β+1 < n β+3 .
We claim that if I is a yes-instance then opt(I ′ ) > n β+2 ; otherwise opt(I ′ ) <

n β+2 (n ′ ) 1-ε .
Suppose that there exists a dominating set S ⊆ V in G of size at most k. Consider the solution S ′ for I ′ containing the corresponding top vertices. Since S is a dominating set in G, after the fourth step, all the bottom vertices are activated. It follows that at the end of the activation process all the vertices of the graph G ′ are activated except the top vertices outside S ′ and the vertices of some directed edges of the basic gadget. The optimum solution is opt( A gray arrow from u to v indicates a directed tree where u is one of the leafs and v is the root.

I ′ ) > n ′ -5n 2 > n β+2 . v 5 v 4 v 3 v 2 v 1 v t 1 v b 1 v t 2 v b 2 v t 3 v b 3 v t
Suppose that there is no dominating set in G of size k. Consider a solution S ′ for I ′ of size k. Without loss of generality, we may assume that no p j i vertices or bottom vertices are contained in S ′ since they all have threshold one. For the reason previously mentioned, we know that no vertices from the directed edge gadgets and no vertices from the directed trees are in S ′ . It follows that S ′ only contains top-vertices. Since there is no dominating set of size k in G then at least one bottom-vertex is not activated. Moreover, because of the directed edges the activated bottom-vertices cannot activate new top-vertices. Thus at least vertex of each directed tree with roots p 1 i , i = 1, . . . , n cannot be activated implying that no p j i vertices can be activated. This leads to a solution of size at most 5n 2 < n 3 ≤ n β+2

(n ′ ) 1-ε . We then have a gap of (n ′ ) 1-ε , and the result follows from Theorem 9. Using Lemma 7, Theorem 26, and Theorem 27 we can deduce the following corollary.

Corollary 28 For any strictly increasing function α, Max Closed Influence and Max Open Influence with thresholds at most two or majority thresholds cannot be approximated within α(k) in fpt-time with respect to the parameter k unless FPT = W[2].

The unanimity case

For the unanimity thresholds case, we will give some results on general graphs before focusing on bounded degree graphs and regular graphs.

General graphs. We first show that, in the unanimity case, Influence parameterized by the combined parameter solution size k and number of newly activated vertices ℓ is W[1]-hard, and Max Open Influence is not approximable within n 1-ε for any ε > 0 in polynomial time, unless NP = ZPP. However, if we are allowed to use fpt-time then Max Open Influence with unanimity is α(n)-approximable in fpt-time with respect to the parameter k for any strictly increasing function α.

Theorem 29 Influence with unanimity thresholds is W[1]-hard with respect to the combined parameter solution size k and number of newly activated vertices ℓ even for bipartite graphs.

Proof.

We construct a fpt-reduction from the W[1]-hard problem Clique (see Appendix A) to Influence. Given an instance (G = (V, E), k) of Clique, we construct an instance (G ′ = (V ′ , E ′ ), thr, k, ℓ) of Influence as follows. For each vertex v ∈ V add a copy v ′ to V ′ . For each edge uv ∈ E, add k + 1 edge-vertices e 1 uv , . . . , e k+1 uv adjacent to both u ′ and v ′ . Set ℓ = (k + 1) k 2 and thr(u

) = deg G ′ (u) for all u ∈ V ′ .
We claim that there is a clique of size k in G if and only if there exists a subset S ⊆ V ′ of size k such that |σ(S)| ≥ ℓ.

"⇒": Assume that there is a clique C ⊆ V of size k in G. One can easily verify that the set

S = {v ′ ∈ V ′ : v ∈ C} activates |σ(S)| ≥ (k + 1) k 2 = ℓ edge-vertices in G ′ since C is clique.
"⇐": Suppose that there exists a subset S ⊆ V ′ of size k such that |σ(S)| ≥ ℓ. We may assume without loss of generality that no edge-vertices belong to S. Indeed, each edge-vertex is adjacent to only vertices with threshold at least k + 1. Thus choosing some edge-vertices to S cannot activate any new vertices in G ′ . Since the solution S activates at least (k + 1) k 2 edge-vertices, this implies that S is a clique in G.

Theorem 30 For any ε > 0, Max Open Influence with unanimity thresholds cannot be approximated within n 1-ε in polynomial time, unless NP = ZPP.

Proof.

We will show how to transform any approximation algorithm for Max Open Influence into another one with the same ratio for Max Independent Set (see Appendix A). Consider an instance I = (G, k) of Max Open Influence with unanimity thresholds. One can note and easily check that the following holds. Given a solution S ⊆ V of I, σ(S) is obtained in only one step of the diffusion process and is an independent set. Therefore there exists an integer k * ∈ {1, . . . , n} such that σ(OP T (I k * )) is the maximum independent set in G, where OP T (I k * ) is the optimal solution of the instance

I k * = (G, k * ) of Max Open Influence (and thus |σ(OP T (I k * ))| = opt(I k * )).
Suppose that Max Open Influence has a polynomial-time α(n)-approximation algorithm A, we then have |σ(S A )| ≥ opt(I k * ) α(n) , where S A is the solution given by A when applied on the instance I k * . Furthermore, it follows from the previous observation that σ(S A ) is an independent set in G and then an α(n)-approximate solution of Max Independent Set on the instance I k * . Now, it suffices to apply the approximation algorithm A for each k = 1, . . . , n and return the approximate solution S max that has the largest value. Given this solution, we have |σ(S max )| ≥ |σ(S A )|. Hence, we get a polynomial-time α(n)-approximation algorithm for Max Independent Set problem. Since Max Independent Set cannot be approximated within n 1-ε for any ε > 0 unless NP = ZPP [START_REF] Håstad | Clique is hard to approximate within n 1-ǫ[END_REF], the result follows.

In the following, we provide the fpt-time approximation algorithm for Max Open Influence and Max Closed Influence. As a first step toward this goal, we first give the following polynomial-time approximation algorithm.

Theorem 31 Max Open Influence and Max Closed Influence with unanimity thresholds are 2 k -approximable in polynomial time.

Proof. By Lemma 10, it suffices to show the result for Max Open Influence. Let I = (G = (V, E), k) be an instance of Max Open Influence with unanimity thresholds. The polynomial-time algorithm consists in the following two steps: (i) Find F ⊆ V , the largest set of false twins (see Section 2.1) such that deg(v) ≤ k, ∀v ∈ F , and (ii) Return N (F ). The first step can be done for example by searching for the largest set of identical lines with at most k ones in the adjacency matrix of the graph. Since F is a false-twins set with vertices of degree at most k, the size of the neighborhood of F is also bounded by k. Consider the activation of the set N (F ). After one step, this will activate |σ(N (F ))| ≥ |F | vertices, since all the neighborhood of the vertices in F are activated.

To complete the proof, observe that for any solution of size at most k, there are at most 2 k different "false-twins sets". Therefore, any optimal solution could activate at most 2 k • |F | vertices, providing the claimed approximation ratio.

Using Lemma 7 and Theorem 31 we directly get the following.

Corollary 32 For any strictly increasing function α, Max Open Influence and Max Closed Influence with unanimity thresholds are α(n)-approximable in fpttime with respect to the parameter k.

For example, Max Open Influence is log(n)-approximable in fpt-time with respect to the parameter k.

Finding dense subgraphs. In the following we show that Max Open Influence with unanimity thresholds is at least as difficult to approximate as the Densest k-Subgraph problem (see Appendix A). In particular, any positive approximation result for Max Open Influence with unanimity would directly transfers to Densest k-Subgraph.

Theorem 33 For any strictly increasing function r, if Max Open Influence with unanimity thresholds is α(n)-approximable in fpt-time with respect to the parameter k then Densest k-Subgraph is α(n)-approximable in fpt-time with respect to the parameter k.

Proof. We give an E-reduction from Densest k-Subgraph to Max Open Influence. Consider an instance I of Densest k-Subgraph formed by a graph G = (V, E). We construct an instance I ′ of Max Open Influence with unanimity thresholds consisting of graph G ′ = (V ′ , E ′ ) as follows. For each vertex v ∈ V add a copy v ′ to V ′ ; for each edge uv ∈ E add a vertex e uv to V ′ , moreover add k + 1 vertices x 1 , . . . , x k+1 . For any edge uv ∈ E add edges u ′ e uv , e uv v ′ to E ′ , and add an edge between x i and v ′ for any 1 ≤ i ≤ k + 1.

Let S ⊆ V , |S| = k be an optimum solution for I that is opt(I) is the number of edges induced by S. The set S ′ = {v ′ : v ∈ S} is such that |σ(S ′ )| = opt(I) since no x vertex will be activated. Thus opt(I ′ ) ≥ opt(I).

Given any solution S ′ ⊆ V ′ of size k, we can consider that S ′ contains only vertices of type v ′ such that v ∈ V . Thus the set S = {v : v ′ ∈ S ′ } has value cost(I, S) = cost(I ′ , S ′ ). Moreover if S ′ is optimal, then opt(I) ≥ opt(I ′ ) and thus opt(I) = opt(I ′ ). Therefore, we have ε(I, S) = ε(I ′ , S ′ ).

Using Theorem 33 and Corollary 32, we have the following corollary, independently established in [START_REF] Bourgeois | Exact and approximation algorithms for densest k-subgraph[END_REF].

Corollary 34 For any strictly increasing function α, Densest k-Subgraph is α(n)-approximable in fpt-time with respect to the parameter k.

Bounded degree graphs. We show in the following that Max Open Influence and thus Max Closed Influence are constant approximable in polynomial time on bounded degree graphs with unanimity thresholds. Moreover, Max Closed Influence and then Max Open Influence have no polynomial-time approximation scheme even on 3-regular graphs if P = NP. Moreover, we show that Influence parameterized by k is in FPT.

Lemma 35 Max Open Influence and Max Closed Influence with unanimity thresholds on bounded degree graphs are constant approximable in linear time.

Proof. By Lemma 10, it suffices to show the result for Max Open Influence. Indeed on graphs of degree bounded by ∆, the optimum is bounded by k • ∆ and we can construct in polynomial time a solution S of value at least ⌊ k ∆ ⌋ by considering iteratively vertices with disjoint neighborhoods and putting their neighbors in S.

Theorem 36 Max Open Influence and Max Closed Influence with unanimity thresholds have no polynomial-time approximation scheme even on 3-regular graphs for k = θ(n), unless P = NP.

Proof. By Lemma 10, it suffices to show the result for Max Closed Influence. We show that if Max Closed Influence with unanimity thresholds has a polynomialtime approximation scheme A ε ′ , ε ′ ∈ (0, 1), on 3-regular graphs when k = θ(n), then Min Vertex Cover has also a polynomial-time approximation scheme on 3-regular graphs. Consider an instance I of Min Vertex Cover consisting of a 3-regular graph G = (V, E) of order n. Clearly, a minimum vertex cover has a value opt(I) satisfying n 2 ≤ opt(I) < n. For any ε ∈ (0, 1), we apply the polynomial-time approximation scheme A ε ′ that establishes an (1 + ε ′ )-approximation for Max Closed Influence on graph G for each k between n 2 and n and ε ′ = ε 2-ε . By applying A ε ′ on G for k between n 2 and n, we obtain a solution S k ⊂ V of size k such that S k ∪ σ(S k ) is an (1 + ε ′ )-approximation. The set V \ σ(S k ) is a vertex cover in G of size denoted by val k . We show in the following that the best solution obtained in this way is an (1+ε)approximation for Min Vertex Cover on G. Indeed the best solution obtained in this way has a value val * ≤ val ℓ , where val ℓ is the value of the solution obtained for ℓ = opt(I). Thus val ℓ = |V \ σ(S ℓ )|. Since |S ℓ ∪ σ(S ℓ )| is an (1 + ε ′ )-approximation and the optimum solution activates all vertices, we have

|S ℓ ∪ σ(S ℓ )| ≥ n 1+ε ′ and |V \ (S ℓ ∪ σ(S ℓ ))| ≤ n ε ′ 1+ε ′ . Thus val * ≤ val ℓ ≤ ℓ + n ε ′ 1+ε ′ ≤ ℓ(1 + 2ε ′ 1+ ε ′ ) = ℓ(1 + ε).
The theorem follows from the fact that Min Vertex Cover has no polynomial-time approximation scheme on 3-regular graphs, unless P = NP [START_REF] Alimonti | Some APX-completeness results for cubic graphs[END_REF].

In Theorem 29 we showed that Influence with unanimity thresholds parameterized by the combined parameter k and ℓ is W[1]-hard. In the following we give several fixedparameter tractability results for Influence parameterized by k on regular graphs and bounded degree graphs with unanimity thresholds. First we show that using results of Cai et al. [START_REF] Cai | Random separation: A new method for solving fixed-cardinality optimization problems[END_REF] we can obtain fixed-parameter tractable algorithms. Then we establish an explicit and more efficient combinatorial algorithm.

Theorem 37 Influence with unanimity thresholds can be solved in 2 O(k∆ 3 ) n 2 log n time where ∆ denotes the maximum degree and in 2 O(k 2 log k) n log n time for regular graphs.

Proof. For graphs of maximum degree ∆, we simply apply the result from [26, Theorem 4] with i = 3.

Let G be a ∆-regular graph. When ∆ > k, any k vertices of the graph form a solution since no vertex outside the set becomes active. Hence, we assume in the following that ∆ ≤ k. Since G is regular, it follows that any subset S, |S| = k can activate at most k vertices. Hence, the graph G[S ∪ σ(S)] contains at most 2k vertices and, thus, ℓ ≤ k. Furthermore, since we consider unanimity thresholds, every vertex v ∈ σ(S) has exactly ∆ neighbors in S and, thus, Phase 2: Check whether H is a subgraph of G.

Phase 1 is realized by simply iterating over all possible graphs H with k + ℓ vertices. A simple upper bound on the number of different graphs with k + ℓ vertices is 2 ( k+ℓ 2 ) ≤ 2 4k 2 . Hence, in Phase 1 the algorithm tries at most O(2 4k 2 ) possibilities. Note that Phase 2 can be done in 2 O(∆k log k) n log n using a result from [START_REF] Cai | Random separation: A new method for solving fixed-cardinality optimization problems[END_REF]Theorem 1]. Altogether this gives a running time of O(2 4k 2 2 O(∆k log k) n log n). Since ∆ ≤ k, this gives 2 O(k 2 log k) n log n. The correctness of the algorithm follows from the exhaustive search.

While the previous results use general frameworks to solve the problem, we now give a direct combinatorial algorithm for Influence with unanimity thresholds on bounded degree graphs. For this algorithm we need the following definition and lemma.

Definition 36: Realizing vertex

Let (α, β) be a pair of positive integers, G = (V, E) a graph with unanimity thresholds. We call a vertex v a realizing vertex for the pair (α, β) if there exists a vertex subset

V ′ ⊆ N 2α-1 [v] of size |V ′ | ≤ α such that |σ(V ′ )| ≥ β and σ[V ′ ] is connected. Furthermore, we call σ[V ′ ] a realization of the pair (α, β).
We show first that in bounded degree graphs the problem of deciding whether a vertex is a realizing vertex for a pair of positive integers (α, β) is fixed-parameter tractable with respect to the parameter α.

Lemma 38 Checking whether a vertex v is a realizing vertex for a pair of positive integers (α, β) can be done in ∆ O(α 2 ) time, where ∆ is the maximum degree.

Proof. The algorithm solving the problem checks for all vertex subsets V ′ of size α in N 2α-1 [v] whether V ′ activates at least β vertices and whether σ[V ′ ] is connected. Since we consider unanimity thresholds it follows that σ

[V ′ ] ⊆ N 2α [v].
The correctness of this algorithm results from the exhaustive search. We study in the following the running time: The (2α -1) th neighborhood of any vertex contains at most ∆(∆ 2α )/(∆ -1) + 1 ≤ 2∆ 2α vertices. Hence, there are 2 α ∆ (2α)α possibilities to choose the α vertices forming V ′ . For each choice of V ′ the algorithm has to check how many vertices are activated by V ′ . Since this can be done in linear time and there are O(∆∆ 2α ) edges, this gives another O(∆ 2α+1 ) term. Altogether, we obtain a running time of

O(2 α ∆ 2α 2 +2α+1 ) = ∆ O(α 2 ) .
Consider in the following the Connected Influence problem that is Influence with the additional requirement that G[σ[S]] has to be connected. Note that with Lemma 38 we can show that Connected Influence parameterized by k is fixed parameter tractable on bounded degree graphs. Indeed, observe that two vertices in σ(S) cannot be adjacent since we consider unanimity thresholds. From this and the requirement that

G[σ[S]] is connected, it follows that G[σ[S]
] has a diameter of at most 2k. Hence, the algorithm for Connected Influence checks for each vertex v ∈ V whether v is a realizing vertex for the pair (k, ℓ). By Lemma 38 this gives an overall running time of ∆ O(k 2 ) • n.

We can extend the algorithm for the connected case to deal with the case where G[σ[S]] is not connected. The general idea is as follows. For each connected component C i of G[σ[S]] the algorithm guesses the number of vertices in S ∩ C i and in σ(S) ∩ C i . This gives an integer pair (k i , ℓ i ) for each connected component in G[σ [S]]. Similar to the connected case, the algorithm will determine realizations for these pairs and the union of these realizations give S and σ(S). Unlike the connected case, it is not enough to look for just one realization of a pair (k i , ℓ i ) since the realizations of different pairs may be not disjoint and, thus, vertices may be counted twice as being activated. To avoid the double-counting we show that if there are "enough" different realizations for a pair (k i , ℓ i ), then there always exist a realization being disjoint to all realizations of the other pairs. Now consider only the integer pairs that do not have "enough" different realizations. Since there are only "few" different realizations possible, the graph induced by all the vertices contained in all these realizations is "small". Thus, the algorithm can guess the realizations of the pairs having only "few" realizations and afterwards add greedily disjoint realizations of pairs having "enough" realizations. See Algorithm 1 for the pseudocode.

Theorem 39 Algorithm 1 solves

Influence with unanimity thresholds in 2 O(k 2 log(k∆)) • n time, where ∆ is the maximum degree of the input graph.

Proof. Let S be a solution set, that is, S ⊂ V , |S| ≤ k and σ(S) ≥ ℓ. In the following we show that Algorithm 1 decides whether such set S exists or not in 2 O(k 2 log(k∆)) • n time. We remark that the algorithm can be adapted to also give such set S if it exists. First we prove the correctness of the algorithm and then show the running time bound.

Correctness: We now show that a solution set S exists if and only if the algorithm returns "yes". Initialize c 1 = c 2 = . . . = c x ← 0 ⊲ one counter for each integer pair (k i , ℓ i )

5:
for each vertex v ∈ V do ⊲ determine realizing vertices 6:

for i ← 1 to x do 7:
if v is a realizing vertex for the pair (k i , ℓ i ) then ⊲ see Lemma 38

8:

c i ← c i + 1 9:
T (v, i) = "yes" 10:

else 11:

T (v, i) = "no" 12:
initialize X ← ∅ ⊲ X stores all pairs with "few" realizations 13:

for i ← 1 to x do 14:

if c i ≤ 2 • x • ∆ 4k then 15:
X ← X ∪ {i} 16:

for each vertex v ∈ V do ⊲ remove vertices not realizing any pair in X 17:

if ∀i ∈ X : T (v, i) = "no" then 18:

delete v from G.

19:

if all pairs (k i , ℓ i ), i ∈ X, can be realized in the remaining graph then return 'no' "⇒:" Assume that S is the solution set. Observe that G[σ[S]] consists of at most k connected components and, thus, the guesses in Lines 2 and 3 are correct. Clearly, in the solution set S there is a realization for each pair (k i , ℓ i ). Furthermore observe that in Line 13 it holds that X ⊆ {1, . . . , x} and that in the loop starting in Line 16 only vertices that cannot realize any pair corresponding to X are deleted. Hence, there exists a realization for the pairs corresponding to X in the remaining graph. Since the checking in Line 19 is done by trying all possibilities, the algorithm returns "yes".

"⇐:" Now assume that the algorithm returns "yes". Observe that this implies that in Line 19 there exists a realization for the all the pairs corresponding to X. Hence, it remains to show that for each pair (k j , ℓ j ) where j ∈ {1, . . . , x} \ X there exists a realization in G. (Clearly, if all pairs are realized then the union of the realizations form the vertex set σ[S] such that |S| = k.) To see that there exist realizations for these pairs observe the following: The (4k) th neighborhood of any vertex contains at most 2∆ 4k vertices. Thus, if in the case of two pairs (k 1 , ℓ 1 ), (k 2 , ℓ 2 ) the value of the second counter is c 2 > 2∆ 4k , then we can deduce that for every realizing vertex v 1 for (k 1 , ℓ 1 ) there exists a realizing vertex v 2 for (k 2 , ℓ 2 ) such that the distance d between v 1 and v 2 is more than 4k. Since d > 4k, it follows that the realizations for (k 1 , ℓ 1 ) and (k 2 , ℓ 2 ) do not overlap. (If two realizations would overlap then some vertices in σ(S) may be counted twice.) Generalizing this argument to x integer pairs (k 1 , ℓ 1 ), . . . , (k x , ℓ x ) yields the following: If there exists an i ∈ {1, . . . , x} such that c i > x • 2 • ∆ 4k , then for any realization of the pairs (k j , ℓ j ) with i = j there exists a non-overlapping realization of (k i , ℓ i ). Thus, we can ignore the pair (k i , ℓ i ) where c i > x • 2 • ∆ 4k in the remaining algorithm and can assume that (k i , ℓ i ) is realized.

Observe that from the Lines 5 to 16 it follows that for all j ∈ {1, . . . , x} \ X we have c j > x • 2 • ∆ 4k . Thus, from the argumentation in the previous paragraph it follows that there exist non-overlapping realizations for all pairs corresponding to {1, . . . , x} \ X. Thus, there exists a solution set S as required.

Running time: Observe that ℓ ≤ ∆k as described in the proof of Lemma 35. Thus, the guessing in Lines 2 and 3 can clearly be done in O(k

• k k (∆k) k ) = O(k 2k+1 ∆ k ).
By Lemma 38 the checking in Line 7 can be done in ∆ O(k 2 i ) time. Thus, the loop in Line 5 requires n

• x i=1 ∆ O(k 2 i ) ≤ ∆ O(k 2 ) • x • n time.
) O(k 2 ) • n = 2 O(k 2 log(k∆)) • n.

Conclusion and open problems

In this chapter, we investigate the parameterized complexity of Target Set Selection with respect to several graph parameters. We also established results concerning the polynomial-time and fpt-time approximability of two related problems, namely Max Open Influence and Max Closed Influence. We conclude this chapter with the following research directions.

(1) The parameterized tractability of Target Set Selection with respect to "cluster vertex deletion number" for majority thresholds, "distance to clique" for general thresholds, and "clique cover number" for majority and constant thresholds are open.

(2) Determine whether, in terms of computational complexity, Target Set Selection with majority thresholds is basically as hard as for general thresholds but significantly easier for constant thresholds. This last fact does not hold from the polynomial-time approximation and parameterized approximation point of view.

(3) It would be interesting to incorporate further natural parameters as the "graph diameter" or the "number of activation steps".

(4) An interesting open question consists of determining whether Max Open Influence is constant approximable in fpt-time. Such a positive result would improve the approximation in fpt-time for Densest k-Subgraph.

(5) Another interesting open question is to study the approximation of Min Target Set Selection (the minimization version of Target Set Selection) in fpt-time with respect to the optimum value or other structural parameters.

(6) Due to applications in social networks, the identification of tractable special cases in scale-free graphs, that is, graphs with power law degree distributions, would be of particular interest.

I

n the previous chapter, we tried to find a subset of the most "influential" vertices. We now ask the natural converse question: find the maximum subset of "less influential" vertices, namely a harmless set. Informally speaking, the problem asks to find the largest subset of vertices with the property that if any set of vertices gets activated in it then there will be no propogation at all i.e. no new vertex can be activated by the application of the same propagation rule as for Target Set Selection (see Chapter 3). In this context, the threshold of a vertex represents its reliability regarding its neighborhood; that is, how many neighbors need to be activated before a vertex gets himself actived.

In the graph below, if any vertex gets activated inside the dashed rectangle (harmless set) then no new vertices are activated. The thresholds are indicated inside the vertices. In this chapter, we study the parameterized complexity and the approximability of the problem.

The content of this chapter is based on in the following paper.

Introduction

Previously, we emphasize the strong intractability nature of maximizing the spread of information through networks. Therefore, with regards to this hardness, it is an interesting question to ask the complexity of the converse problem: find the largest subset of vertices such that if any set of vertices gets activated in it then there will be no consequence for all the other vertices. In this context, there is no propagation process involved like previously stated. On the contrary we want to prevent such phenomenon here. More formally, we introduce the Harmless Set problem defined as follows. Given the same input and propagation rule as for Target Set Selection (see Chapter 3), the task is to determine whether there exists a subset of vertices, called harmless set, of size at least k such that every vertex v of the input graph has less than thr(v) neighbors in this set. The reason for this definition follows from the fact that if any subset of a harmless set is activated then there will be no propagation at all since every vertex has a number of activated neighbors below its threshold. Another motivation for this problem arises from the context of containing the spread of dangerous ideas or epidemics which is also a fruitful research area [START_REF] Kimura | Blocking links to minimize contamination spread in a social network[END_REF][START_REF] Finbow | The firefighter problem: a survey of results, directions and questions[END_REF][START_REF] Chalermsook | Resource minimization for fire containment[END_REF] and further justifies the meaning of harmless set. Finally, it is worth pointing out that our problem can also be related to the generalized domination problem (σ, ρ)-Dominating Set introduced by Telle [100]: Given a graph G = (V, E), two sets of non-negative integers σ and ρ, and an integer k ≥ 1, the objective is to find a set S ⊆ V of size at most k such that |S ∩ N (v)| ∈ σ for every vertex v ∈ S, and |S ∩ N (v)| ∈ ρ for every vertex v ∈ S. The set S is then called a (σ, ρ)-dominating set of size at most k. As a matter of fact, if all thresholds are equal then Harmless Set is equivalent to (σ, ρ)-Dominating Set Of Size k [START_REF] Golovach | Parameterized complexity of generalized domination problems[END_REF] (this version asks to find a (σ, ρ)-dominating set of size exactly k) where σ = ρ = {0, . . . , thr max } and thr max is the maximum threshold value. Since this last problem is in W [START_REF] Aazami | Approximation algorithms and hardness for domination with propagation[END_REF] [61], this implies that Harmless Set is in W [START_REF] Aazami | Approximation algorithms and hardness for domination with propagation[END_REF] if all thresholds are equal. In this work, we extend this result to the case where the thresholds are bounded by a constant. Up to our knowledge, this is the only known result that could possibly carry over.

We study the parameterized complexity of Harmless Set and the approximation of the associated maximization problem (denoted Max Harmless Set). The chapter is organized as follows. In Section 4.2 we give the definitions and terminology. In Section 4.3 we establish parameterized complexity results with respect to various threshold functions (see Table 4.1). We show that the parametric dual problem (denoted Dual Harmless Set and asks the existence of a harmless set of size at least nk where n is the size of the graph) is fixed-parameter tractable for a large family of threshold functions. In Section 4.4 we give polynomial-time algorithms to solve the problem for graphs of bounded treewidth. In Section 4.5, we turn our attention to the maximization version. We establish that it is hard to approximate the problem within n 1-ε for any ε > 0 even when all thresholds are at most two. If each threshold is equal to the degree of the vertex, we show that the problem is APX-complete. Moreover it has a polynomial-time approximation scheme on planar graphs. Conclusion and open problems are given in Section 4.6.

Harmless

Problem definitions & terminology

Before defining the problems, we have to specify the notion of harmless vertices in a graph.

Reduction rule 40 to obtain the new equivalent instance (G, thr ′ , k). Then construct the following Turing machine M from (G, thr ′ , k) (see Figure 4.1). We create n + 1 tapes denoted by T 0 , T v 1 . . . , T vn . The tapes alphabet is V ∪ {×} plus the blank symbol . Initially, every tape is filled with . The transition function is defined hereafter. First, M non-deterministically chooses k vertices and write them on tape T 0 , that is, if M picks up a vertex v ∈ V then it writes symbol v on T 0 and move T 0 's head one step to the right. The previous procedure is done in k + 1 steps. Next, for each i = 1, . . . , k + 1, the Turing machine writes a symbol × on each tape T j and moves T j 's head one step to the right if v j has a threshold greater or equal to i.

According to Reduction rule 40, this phase takes at most k + 1 steps. During the third phase, M checks whether the selected set is a harmless set as follows. First, the machine moves all heads one step to the left. If T 0 's head reads symbol v then for every u i ∈ N (v), we simply move T i 's head one step to the left. We repeat the previous procedure until T 0 's head reads a blank symbol. If all the other tapes read a × symbol then M goes in accepting state; otherwise it goes to a rejecting state. This checking phase can be done in at most k + 1 steps. Finally, the input word x is empty and k ′ = 3k + 3. It is not hard to see that (G, thr ′ , k) is a yes-instance if and only if M accepts in at most k ′ steps. Given (G, k) an instance of Red/Blue Dominating Set, we construct an instance (G ′ = (V ′ , E ′ ), thr, k) of Harmless Set as follows. We consider the graph Ḡ obtained from G as follows. Two vertices u ∈ R and v ∈ B are adjacent in Ḡ if and only if they are not adjacent in G. Moreover, the sets R and B remain independent sets. The graph G ′ is obtained from this last graph by attaching max{kdeg Ḡ(v), 1} forbidden edges to each vertex v ∈ B. Finally, set thr(v) = k for every vertex v ∈ B and thr(v) = 1 for every vertex v ∈ R. Adding several forbidden edges to the vertices of B make sure that the threshold of these vertices is less than or equal to their degree as required in the problem definition.

4 v 3 2 v 5 1 v 6 2 v 1 2 v 2 2 v 4 T 0 . . . v 1 v 2 v 4 . . . T 1 . . . × × . . . T 2 . . . × × . . . T 3 . . . × × × × . . .
We claim that (G, k) is a yes-instance if and only if (G ′ = (V ′ , E ′ ), thr, k) is a yes-instance.

"⇒": Assume that (G, k) has a solution R ′ ⊆ R of size k. One can see that R ′ is also a solution for (G ′ , thr, k) since every vertex in B is not adjacent to at least one vertex in R ′ .

"⇐": Conversely, suppose that there is a harmless set S ⊆ V ′ of size k in G ′ . Since S is harmless, S cannot contain any vertex from B because of the forbidden edges, and thus S is entirely contained in R. Moreover, every vertex v in B is adjacent in G ′ to at most thr(v) -1 = k -1 vertices in S. Hence, every vertex in B is adjacent in G to at least one vertex in S. Therefore, S is a solution of size k for (G, k).

In the next two theorems, we show that Harmless Set parameterized by k goes one level down in the W-hierarchy when all thresholds are bounded by a constant.

Theorem 43 Harmless Set with constant thresholds is in W [START_REF] Aazami | Approximation algorithms and hardness for domination with propagation[END_REF] with respect to the parameter k.

Proof. Let (G, thr, k) be an instance of Harmless Set where thr(v) ≤ c, ∀v ∈ V for some constant c > 0. We construct in O(n c ) time, where n is the number of vertices of G, a boolean circuit C (see Definition 18) of depth 3 and weft 1 as follows. We identify the inputs of the circuit with the vertices of G. Connect a gate N OT to every input. For all v ∈ V and all subsets S ′ ⊆ N (v) of size thr(v), add a gate OR connected to every gate N OT adjacent to an input in S ′ . Finally, add a large gate AN D connected to every gate OR. It is not hard to see that G admits a harmless set of size k if and only if there is an assignment of weight k that satisfies C.

We establish the W[1]-hardness of Harmless Set parameterized by k by an fptreduction from the W[1]-hard problem Clique (see Appendix A).

Theorem 44 Harmless Set is W[1]-complete with respect to the parameter k even 1. For bipartite graphs and constant majority threshold.

2. For split graphs and threshold two for every vertex.

Proof. Membership follows from Theorem 43. We now prove the W[1]-hardness.

(1): Given (G = (V, E), k) an instance of Clique, we construct an instance (G ′ = (V ′ , E ′ ), thr, k) of Harmless Set as follows. V ′ is obtained from V by adding for each edge uv ∈ E, an edge-vertex e uv and edges ue uv and e uv v to E ′ . Remove every edge in E. Finally, attach a forbidden edge p uv q uv to each edge-vertex e uv . Set thr(v) = ⌈ deg (v) 2 ⌉, ∀v ∈ V ′ . We claim that (G, k) is a yes-instance if and only if (G ′ , thr, k) is a yes-instance. "⇒": Let C ⊆ V be a clique of size at least k. Then C is clearly a harmless set in G ′ since no edge-vertex has more than one neighbor in C.

"⇐": Conversely, let C ′ ⊆ V ′ be a harmless set in G ′ . Because of the forbidden edges, C ′ cannot contain an edge-vertex e uv and p uv , q uv and thus C ′ ⊆ V . Moreover, since thr(e uv ) = 2, C ′ cannot contain u and v such that uv / ∈ E and thus C ′ is a clique of size at least k in G.

(2): Let (G = (V, E), k) be an instance of Clique, we construct an instance (G ′ = (V ′ , E ′ ), thr, k) of Harmless Set as follows. As previously, for each edge uv ∈ E, add an edge-vertex e uv and the edges ue uv and e uv v. Add edges to make the set of all edge-vertices a clique. Remove every edge in E. Finally, set thr(v) = 2 for all v ∈ V ′ . Without loss of generality we may assume that k ≥ 2 and every vertex in V has minimum degree two.

We claim that (G, k) is a yes-instance if and only if (G ′ , thr, k) is a yes-instance. "⇒": Let C ⊆ V be a clique of size at least k. One can easily verify that C is a harmless set in G ′ .

"⇐": Conversely, suppose that there is a harmless set C ′ ⊆ V ′ of size k. Notice that C ′ ⊆ V since otherwise we would not have been able to take more than one vertex in G ′ . Indeed, if there are two vertices u, v ∈ C ′ with v ∈ V ′ \ V then there is always a vertex w ∈ V ′ \ V -{u, v} adjacent to both v and u. Thus, C ′ is entirely contained in V . From now on, it is not hard to see that C ′ is a clique of size at least k in G.

It is interesting to note that the ratio between the number of vertices with unbounded threshold over the number of vertices with bounded threshold in the proof of Theorem 42 can be made arbitrarily small by adding many forbidden edges. This implies a sharp dichotomy between the W[2]-and W[1]-completeness of Harmless Set parameterized by k regarding the thresholds.

Unanimity threshold. We consider now the Harmless Set problem with unanimity thresholds. First, we start with the following easy observation. In the case of unanimous threshold, any harmless set is the complement of a total dominating set. Recall that a total dominating set S is a set of vertices such that every vertex in the input graph has at least one neighbor in S. Moreover, we have the following theorem.

Theorem 45 (Cockayne et al. [START_REF] Cockayne | Total domination in graphs[END_REF]) If G is a connected graph of order at least 3 then there is a total dominating set of size at most 2n/3. This implies that, in the unanimity case, there is always a harmless set of size at least n/3 when n ≥ 3. The consequence of this result is that we directly get a linear kernel of size 3k.

Theorem 46 Harmless Set with unanimity thresholds admits a kernel with 3k vertices.

Indeed, let (G, k) be an instance of Harmless Set with unanimity thresholds, if k ≤ n/3 then the answer is yes. If k > n/3 then the instance (G, k) is a kernel of size at most 3k. However, the parameter k is "large" in this last case. This suggests to look for other parameterizations. One possibility is to decide the existence of solutions of size at least ⌈ n 3 ⌉ + k. Another one is to decide the existence of solutions of size at least nk. We study in the following this last problem.

Parametric dual. We show that Dual Harmless Set parameterized by k is in FPT for a large family of threshold functions. Toward this goal, we provide a kernelization by making use of the following reduction rule.

Reduction rule 47 Let (G, thr, k) be an instance of Dual Harmless Set. If there is a vertex v such that deg(v) ≥ k + thr(v) -1 then remove v and decrease by one the threshold of every vertex in N (v) to get a new equivalent instance (G ′ , thr ′ , k).

Regarding the correctness of the above rule, let S be a harmless set of size at least nk. Notice that if there is a vertex v with deg(v) ≥ k + thr(v) -1 then v must be in S since otherwise it will have at most k -1 neighbors outside S and then at least thr(v) neighbors in S.

We can now state the main result.

Theorem 48 Dual Harmless Set admits a kernel with O(k 2 ) vertices if for every vertex v in the input graph thr

(v) = ⌈α v deg(v) βv + γ v ⌉ for any constants α v , β v ∈ [0, 1], α v β v = 1, and γ v ∈ Q.
Proof. Let (G, thr, k) be an instance of Dual Harmless Set. Exhaustively apply Reduction rule 47 to get (G ′ , thr ′ , k). Assume that there exists a solution S ⊆ V of size at least nk. Because of Reduction rule 47, we have

deg(v) < k + thr(v) -1 = k + ⌈α v deg(v) βv + γ v ⌉ -1 ≤ k + α v deg(v) βv + γ v (4.1) We claim that deg(v) ≤ θ v (k) for all v ∈ V ′ where θ v (k) =      k + α v + γ v if β v = 0 k+γv 1-αv if β v = 1 k+γv 1-βv + (1/β v ) 1 1-βv otherwise
The first two cases are straightforward. Suppose now that β v ∈ (0, 1). First, it is not hard to show that the following holds:

x ε ≤ εx if and only if x ≥ (1/ε) 1 1-ε for any x ≥ 1 and ε ∈ (0, 1). Hence, if deg(v) ≥ (1/β v ) 1 1-βv then, together with (Eq. 4.1), we obtain deg(v) ≤ k + α v β v deg(v) + γ v and thus deg(v) ≤ k+γv 1-αvβv ≤ θ v (k). Otherwise deg(v) < (1/β v ) 1 1-βv ≤ θ v (k).
Since every vertex from S has at least one neighbor in V ′ -S then |S| has at most |V ′ -S| deg max ≤ kθ max (k) vertices where θ max (k) = max v∈V ′ θ v (k) and deg max is the maximum degree of vertices in V ′ -S.

The kernelization procedure is then defined as follows. From an instance (G, thr, k) of Dual Harmless Set, exhaustively apply Reduction rule 47 to get an instance (G ′ , thr ′ , k). If |V ′ | > kθ max (k)+k then return a trivial no-instance. Otherwise, return the instance (G ′ , thr ′ , k).

Notice that if α v = β v = 1 and γ v = 0, ∀v ∈ V then the Dual Harmless Set problem is exactly the Total Dominating Set problem which is known to be W [START_REF] Ackerman | Combinatorial model and bounds for target set selection[END_REF]-hard [START_REF] Golovach | Parameterized complexity of generalized domination problems[END_REF].

Algorithms for trees and tree-like graphs

In this section we establish a O(thr 3 tw max n)-time algorithm for Max Harmless Set and a O((k + 1) 3 tw n)-time algorithm for Harmless Set where thr max is the maximum threshold and tw the treewidth of the input graph. We decided to describe first a linear-time algorithm for trees. Besides to be more efficient in this case, it will introduce the underlying ideas used later for the general algorithm.

Theorem 49 Max Harmless Set is solvable in linear time on trees.

Proof. Let (T = (V, E), thr) be an instance of Max Harmless Set where T is a tree rooted at r. We describe a dynamic programming algorithm as follows. We denote by T v the subtree of T rooted at v. Moreover, we denote by C(v) the set of children of a vertex v in T . For each v ∈ V and each b ∈ {0, 1}, we compute A v [b], the optimal solution for the subtree T v with the additional constraint that the threshold of v is set to thr(v)b. Set A v [0] = {v} and A v [START_REF] Aazami | Approximation algorithms and hardness for domination with propagation[END_REF] = ∅ for every leaf v of T . Then for all v ∈ V and all b ∈ {0, 1}, we compute A v [b] as follows

A v [b] = arg max S∈{Iv[b],Ev[b]} |S| (4.2)
where

I v [b] (resp. E v [b]
) is the optimum solution of T v when thr(v) is decreased by b and v is included (resp. excluded). They are computed as follows

I v [b] =        ∅ if ∃c ∈ C(v) : A c [0] = ∅ {v} ∪ c∈C(v)\Ac[1] A c [1] ∪ thr(v)-b-1 i=1 B πv(i) [1] otherwise (4.3) E v [b] = c∈C(v)\Ac[0] A c [0] ∪ thr(v)-b-1 i=1 B πv(i) [0] (4.4)
In the above formulas, for any v ∈ V and b ∈ {0, 1}, we set

B v [b] = A v [b] if v ∈ A v [b],
∅ otherwise. Furthermore, we denote by

π v : {1, . . . , |C(v)|} → C(v) the permutation of the children of v such that |B πv(1) [b]| ≥ . . . ≥ |B πv(|C(v)|) [b]| for b ∈ {0, 1}.
To get the optimal solution for the tree T , return A r [0].

To see that the above algorithm is correct, first observe that when we make a decision for a vertex v, we know the decision made for all its children but we do not know about its parent, denoted by p v . We then have to compute two optimal solutions for the subtree T v by considering two cases: one where p v is in the solution and the other one when it is not. The first case can be done by computing an optimal solution with the threshold of v set to thr(v) -1 i.e., b = 1. For the second case, we compute another optimal solution without modifying v's threshold i.e., b = 0. Notice that in each case, the optimal solution for T v takes either v (Eq. 3) or not (Eq. 4). We then keep the best of these two solutions (Eq. 2). We now explain Eq. 3 (the other equation works analogously). First, observe that if there is a child c of v such that A c [0] = ∅ then this means that T c has only vertices with thresholds one and thus there exists no harmless set in T if v is part of the solution. Thus we set I v [b] = ∅. If it is not the case then the optimal solution for T v is the union of the following three sets: the singleton {v} since v is forced to be in the solution; the optimal solution of each subtree rooted at a child of v with the condition that this child is not taken; and the (thr(v)b -1)-th largest solutions of the subtrees rooted at a the children of v that belong to the solution. Now, we present the general algorithm for solving Max Harmless Set.

Theorem 50 Max Harmless Set is solvable in O(thr 3 tw max •n) time where thr max is the maximum threshold and tw is the treewidth of the input graph.

Proof. Let (G = (V, E), thr) be an instance of Max Harmless Set. Assume that we are given a nice tree decomposition T = (T = (X, F ), H) of G of width at most tw that can be found in fpt-time with respect to the treewidth [START_REF] Bodlaender | A linear-time algorithm for finding tree-decompositions of small treewidth[END_REF]. Let T x be the subtree of T rooted at some node x ∈ X. We denote by G x = (V x , E x ) the subgraph induced by the vertices from y∈Tx H y . We describe a dynamic programming algorithm to solve (G, t, k) using T .

Description. The general idea of the algorithm is as follows. For each node x ∈ X, we store a set of optimal solutions for the subgraph G x in a table denoted by A x . These tables are updated using a bottom-up procedure that starts from the leaves and ends to the root of T . In order to update the table of a node, we need to define an "update rule" according to the type of the node, i.e., insert, forget and join. Moreover, we also have to know what kind of solutions we should store in each table, or, equivalently, what information do we need in a node to update the table of its parent. In our algorithm, we use a two-entries table A x [ t, c] where t : H x → {1, . . . , thr max } and c : H x → {0, 1}. An entry A x [ t, c] corresponds to a maximum harmless set in G x that contains all the vertices in {v ∈ H x : c(v) = 1} and by imposing the threshold thr(v) = t(v), ∀v ∈ H x . We set A x [ t, c] = ∅, if no such harmless set is possible. To see why this table is sufficient for our purpose, consider the updating step occuring in a join node. Let x ∈ X be a join node with children y and z such that H x = H y = H z . The nodes y and z have their respective tables A y and A z already computed by dynamic programming and we want to compute the table A x . First, we introduce the role of the function t.

Let B = N G (H x ) \ V x .
Notice that, at the current stage of the algorithm, we do not know what vertices in B will be in the maximum harmless set. Thus, to compute the table A x one has to take into consideration that any subset S ⊆ B might be in the optimal solution. Hence, we have to compute a maximum harmless set in G x for each subset S ⊆ B considering S as part of a harmless set. This can be seen as equivalent to finding a maximum harmless set in G x for every possible thresholds t ∈ {1, . . . , thr max } Hx of vertices in H x . Indeed, picking a vertex v from G into a harmless set is equivalent of removing v from G and decreasing the thresholds of the vertices in N (v) by one. Moreover every set S only affects vertices in H x since no vertex in B can be adjacent to a vertex in V x \ H x . Hence we can compute A x solely from tables A y and A z . To do this, we have to take care of the following fact. Consider the optimal solutions S y and S z in G y and G z respectively and by imposing some thresholds t to H y and H z . Notice that we cannot directly make the union of S y and S z to compute the optimal solution S x of G x by imposing t on H x . Indeed, consider a vertex u ∈ H x . It may happen that u might have less than t(u) neighbors in S y and S z but more than t(u) in S y ∪ S z . To avoid this problem, we have to consider each union S y ∪ S z for every pair ( t 1 , t 2 ) ∈ {1, . . . , thr max } Hx such that t 1 + t 2 = t (where t 1 + t 2 is the classical vector component addition) and take the largest one. The final problem we have to deal with is that, whenever we make the previous union, we do not take into consideration that the sets H y and H z are equal. The consequence is that a vertex v in G y might have a number of neighbors in H y ∩ (S y ∪ S z ) that sums over its threshold. That's where the vector c comes into play. According to the definition of A x [ t, c], this vector ensures that the same vertices in both H y and H z are in the solution. This completes the description of the algorithm, we now give the details.

Algorithm. We denote by f ⊕ (a, b) the extended function f ′ such that ∀x,

f ′ (x) = f (x) if x = a and f ′ (a) = b.
Initialization step. We initialize all the tables A x where x is a leaf of T as follows. For each leaf x of T , t ∈ {1, . . . , thr max } Hx and c ∈ {0, 1} Hx . Let S = {v ∈ H x :

c(v) = 1} A x [ t, c] = S
if S is a harmless set for G x according to t and c ∅ otherwise

The running time of this step is thr |Hx| max 2 |Hx| n = O(thr 2 tw max n). Updating step. Starting from the leaves, we apply the following rules to each node x ∈ X we visit until we reach the root.

Case 1 (insert node). Suppose that x is an insert node with child y such that H x = H y ∪ {u}. Following the above discussion, we update the table A x as follows. For all t ∈ {1, . . . , thr max } Hy , c ∈ {0, 1} Hy and i = 1, . . . , thr max

A x [ t ⊕ (u, i), c ⊕ (u, 0)] =    A y [ t, c] if A y [ t, c] is a harmless set in G x with thr(u) = i ∅ otherwise A x [ t ⊕ (u, i), c ⊕ (u, 1)] =    A y [ t, c] ∪ {u} if A y [ t, c] ∪ {u} is a harmless set in G x with thr(u) = i ∅ otherwise
The running time is thr |Hy| max 2 |Hy| thr max n = O(thr 2 tw max n). Case 2 (forget node). Suppose that x is a forget node with child y such that H x = H y -{u}. Let t ∈ {1, . . . , thr max } Hy . Notice that vertex u has its neighbors entirely contained in G x . Hence, the maximum harmless set for G x where thr(v) = t(v) ∀v ∈ H x is exactly the maximum harmless set for G y where thr(v) = t(v) ∀v ∈ H y and such that u has threshold thr(u). Formally, we update the table A x as follows. For all t ∈ {1, . . . , thr max } Hx and c ∈ {0, 1} Hx

A x [ t, c] = max i∈{0,1} {A y [ t ⊕ (u, thr(u)), c ⊕ (u, i)]}
The running time is thr |Hx| max 2 |Hx| n = O(thr 2 tw max n). Case 3 (join node). Suppose that x is a join node with children y and z such that H x = H y = H z = {u 1 , . . . , u |Hx| }. According to the above discussion, we update the table A x as follows. For all t ∈ {1, . . . , thr max } Hx and c ∈ {0, 1} Hx , perform the following two steps 1.

( t * 1 , t * 2 ) = arg max t 1 , t 2 ∈{1,...,thrmax} Hx : t 1 + t 2 = t |A y [ t 1 , c] ∪ A z [ t 2 , c]| 2. A x [ t, c] = A y [ t * 1 , c] ∪ A z [ t * 2 , c]
The running time of this step is thr

Approximability

In this section, we first show that Max Harmless Set is strongly inapproximable even for small constant thresholds on bipartite graphs. However, we provide a polynomial-time approximation scheme when the graph is planar. Furthermore, in the case of unanimity thresholds we prove that Max Harmless Set is APX-hard and admits a linear-time 3-approximation algorithm.

We start by showing the hardness result on bipartite graphs by an E-reduction from the Max Clique problem (see Appendix A).

Theorem 52 For any ε > 0, Max Harmless Set with thresholds at most two cannot be approximated within n 1-ε in polynomial time even on bipartite graphs, unless NP = ZPP.

Proof. We construct an E-reduction (see Definition 31) from Max Clique. Let G be an instance of Max Clique. Consider the constructed instance I ′ = (G ′ , thr) from G as it is defined in Theorem 44. Let C be a harmless set in G ′ . From the proof of Theorem 44, we know that C is a clique in G. Thus, it is not hard to see that opt(I ′ ) = opt(G) and ε(G, C) = ε(I ′ , C). Since Max Clique is not approximable within n 1-ε for any ε > 0 unless NP = ZPP [START_REF] Håstad | Clique is hard to approximate within n 1-ǫ[END_REF], the result follows.

In the following we propose a polynomial-time approximation scheme on the class of planar graphs, using the previous polynomial-time algorithm for graphs of bounded treewidth.

Theorem 53 Max Harmless Set is in PTAS on planar graphs.

Proof. Let I = (G, thr) be an instance of Max Harmless Set. Given a planar embedding of an input graph, we consider the set of the vertices which are on the exterior face, they will be called level 1 vertices. By induction we define level k as the vertices which are on the exterior face when we have removed the vertices of levels smaller than k [START_REF] Baker | Approximation algorithms for NP-complete problems on planar graphs[END_REF]. A planar embedding is k-level if it has no nodes of level greater than k. If a planar graph is k-level, it has a k-outerplanar embedding.

If we want to achieve an approximation within 1+ε, let us consider k = 2(1+ 1 ε ). Let X t be the set of vertices of level t and let H i , 0 ≤ i ≤ k -1, be the graph obtained from G by considering the subgraphs formed by the set of vertices t+1≤j≤t+k X j , for t ≡ i (mod (k -2)). The subgraph containing exactly t+1≤j≤t+k X j is k-outerplanar, and so is H i , too.

Since H i is k-outerplanar, it has treewidth at most 3k -1 [START_REF] Hans | A partial k-arboretum of graphs with bounded treewidth[END_REF]. We construct graph H ′ i from H i by attaching a forbidden edge to each vertex on the boundary (that means vertices in X t+1 , X t+2 , X t+k-1 X t+k with t ≡ i (mod (k -2))). Thus, in each subgraph of H ′ i the vertices in X t+1 , X t+2 , X t+k-1 X t+k cannot take part from any harmless set. On applying Theorem 50, we can efficiently determine an optimal harmless set in each subgraph of H ′ i . Denote by S i the union of these harmless sets. Clearly S i is a harmless set on H i .

Among S 0 , . . . , S k-1 we choose the best solution that we denote S and we are going to prove that S is an (1 + ε)-approximation of the optimal value on G. We can easily show that there is at least one r, 0 ≤ r ≤ k -1 such that at most 2 k of vertices in an optimal solution S opt of G are on levels X t+1 , X t+2 , X t+k-1 X t+k with t ≡ r (mod (k -2))). This means that the solution S r obtained by deleting the vertices from levels X t+1 , X t+2 , X t+k-1 X t+k from S opt will have at least

|S opt |(1-2 k ) = k-2
k opt(I) vertices. According to our algorithm, |S| ≥ |S r | ≥ opt(I) 1+ε . The overall running time of the algorithm is k times what we need for graphs of treewidth at most k, that is O(k thr 6k-2 max n) = n O(1/ε) where thr max is the maximum threshold.

Unanimity thresholds. As previously observed in the section dedicated the parameterized complexity of Harmless Set, it seems like unanimity thresholds make the problem more "tractable". We confirm this fact from an approximation perspective by showing the APX-completeness of Max Harmless Set with unanimity thresholds.

Lemma 54 Max Harmless Set with unanimity thresholds is 3-approximable in linear time.

Proof. Let I be an instance of Max Harmless Set consisting of a graph G and unanimity thresholds. The algorithm consists of the following two steps:

1. Compute a spanning tree T of G.

2. Compute an optimal solution S of T using Theorem 49 with unanimity threshold. Observe that any feasible solution S for T is also a solution for G. Indeed, if a vertex v in T is such that N T (v) ⊆ S then we have N G (v) ⊆ S. Moreover, using Theorem 45, we get |S| ≥ n/3 ≥ opt(I)/3.

Theorem 55 Max Harmless Set with unanimity thresholds is APX-complete.

Proof. Membership follows from Lemma 54. In order to prove the APX-hardness we provide an L-reduction (see Definition 30) from the APX-hard problem Max E2Sat-3 (see Appendix A).

Given a formula φ of Max E2Sat-3 with n variables and m clauses (notice that m = 3n/2), we construct an instance I of Max Harmless Set consisting of a graph G = (V, E) and unanimity thresholds as follows (see Figure 4.2). For every variable x i , we construct the complete bipartite graph K 3,3 (x i ) = (V -(x i ), V + (x i )) in which every edge uv is replaced by an edge-vertex e uv and two edges ue uv and e uv v. We denote by E(x i ) this set of edge-vertices. The vertices in V + (x i ) (resp. V -(x i )) represents the positive (resp. negative) literals of x i . We denote by A the set of all vertices added so far. For every clause c j in φ add two adjacent clause-vertices cj and c′ j . For every variable x i , if x i appears positively (resp. negatively) in a clause c j then add an edge between c′ j and a vertex of V -(x i ) (resp. V + (x i )). Thus, vertex cj represents the complement of the clause c j in φ. Finally, add two adjacent vertices c and c ′ . For every vertex v ∈ V -(x i ) ∪ V + (x i ), if v is not adjacent to a clause-vertex then add the edge vc ′ . This completes the construction.

The optimal value in I is bounded by the number of vertices of G and thus, opt(I) ≤ 15n + 2m + 2 ≤ 16 opt(φ) + 2 ≤ 18 opt(φ) since opt(φ) ≥ 3/4m and opt(φ) ≥ 1.

Moreover, let x * be an optimal assignment for φ and let

S = i:x * i =1 V + (x i ) ∪ i:x * i =0 V -(x i ) ∪ n i=1 E(x i ) ∪ {c j : c j is satisfied by x * } ∪ {c}.
We can easily verify that S is a harmless set and |S∩(V -(x i )∪V + (x i )∪E(x i )| = 12 and thus |S ∩ A| = 8m and then opt(I) ≥ |S| = 8m + opt(φ) + 1.

Let S be a harmless set for I. We show in the following how to construct an assignment a S for φ from the solution S such that cost(φ, a S ) = |S|-8m -1. For each variable x i , S cannot contain vertices from both V -(x i ) and V + (x i ) since otherwise an edge-vertex has both neighbors inside S. Notice also that S cannot contain any vertex c′ j , since c′ j is adjacent to the degree one vertex cj . Similarly c ′ / ∈ S. If S contains for every i = 1, . . . , n the set E(x i ) and one of the sets V -(x i ) or V + (x i ) then |S ∩ A| = 8m and we can defined the following assignment a S :

x i = 1 ⇔ |S ∩ V + (x i )| = 0.
In this case, a clause-vertex is in S if and only if the corresponding clause is satisfied by a S . Thus, the number of clauses satisfied by a S is exactly cost(φ, a S ) = |S| -8m -1.

Assume now that |S ∩ A| < 8m. We show that there exists an other solution S ′ with |S ′ | ≥ |S| such that |S ′ ∩A| = 8m. If a vertex v ∈ E(x i )\S for some i ∈ {1, . . . , n}, we can add v in S since v cannot have both neighbors in S. Similarly, if c is not in S, then we add c in S. Furthermore, we may assume that for each i = 1, . . . , n, we have either

|V -(x i ) ∩ S| ≥ 1 or |V + (x i ) ∩ S| ≥ 1. Indeed, there is always a vertex v ∈ V -(x i ) ∪ V + (x i ) ∩ N (c ′ ) that can be added to S since c ′ ∈ S. Moreover, we have |V -(x i ) ∩ S| < 3 and |V + (x i ) ∩ S| < 3 for some i ∈ {1, . . . , n} since |S ∩ A| < 8m. Let i ∈ {1, . . . , n} be such that 1 ≤ |V -(x i ) ∩ S| ≤ 3 (the case 1 ≤ |V + (x i ) ∩ S| ≤ 3 is symmetric).
There must exist a vertex v ∈ V -(x i ) \ S which is either adjacent to c ′ or to a clause-vertex c′ j . In the first case we add v in S. In the second case, we denote N (c ′ j ) = {v, v ′ , cj }. If v ′ ∈ S and cj ∈ S then remove cj from S and add v instead, otherwise add v in S. Thus, we obtain a new solution S ′ such that |S ′ | ≥ |S| and |S ′ ∩ A| = 8m. Similarly to the above case, we can obtain an assignment a S ′ such that |S ′ |cost(φ, a S ′ ) = 8m + 1. In particular, if S ′ is an optimal solution, then opt(φ) ≥ cost(φ, a S ′ ) = opt(I) -8m -1 and thus, we have opt(I)opt(φ) = 8m + 1 and then opt(φ)cost(φ,

a S ′ ) = opt(I) -|S ′ |. . . . . . . c′ 1 c1 c′ m cm c ′ c x 1 x n A Figure 4.2:
The construction of G. The subgraph in a dashed box corresponds to a gadget K 3,3 (x i ) of a variable x i . A grey box represents either V + (x i ) or V -(x i ).

Conclusion and open problems

In this chapter, we introduced the harmless set problem. We established positive and negative results concerning its parameterized tractability and polynomial-time approximability. However, several questions remain open.

(1) Is Harmless Set fixed-parameter tractable for the parameter treewidth?

(2) We showed that there exists a ptas for Max Harmless Set on planar graphs. However, the parameterized complexity of Harmless Set parameterized by k in this class of graphs is open.

(3) There is room enough for improving the approximability of Max Harmless Set with unanimity thresholds.

(4) Another interesting open question is whether Harmless Set with unanimity thresholds is fixed-parameter tractable with respect to the parameter k when we ask to determine the existence of a harmless set of size at least ⌈ n 3 ⌉ + k. ollowing the idea of preventing an undesirable thing to propagate through a network, we investigate the complexity of the firefighter problem. This problem has been introduced by Hartnell in 1995 [START_REF] Hartnell | Firefighter! an application of domination[END_REF] and concerns a deterministic model of "fire" spreading through a graph via its edges. Of course, the term fire can be replaced by any other relevant term depending of the application context (e.g. rumor, virus, ideas, etc.). Formally, it consists of a graph with an initially burned vertex. At each time step, we are first allowed to protect one vertex and then the fire spreads to every unprotected vertex adjacent to the fire. When no new vertex can burn, the process stops. The goal is to minimize the number of burned vertices. This can be seen as a "dynamic" cut problem since protecting a vertex has the same effect as removing it from the graph. In the figure below, we depict an example of such process. In the figure, the burned vertices are represented by black vertices. A protected vertex is represented by a circle with a cross inside.

Step 1

Step 2

Step 3

In this chapter, we study the classical complexity, parameterized complexity and approximability of a generalized version of this problem where b > 1 vertices can be protected at each step as well as some variants.

The content of this chapter is based on the following papers.

Introduction

In the Chapter 3, we analyzed the computational complexity of maximizing the spread of information through a network. Motivated by applications such as the containment of a disease, we may modify the propagation process defined in that chapter by allowing the possibility of "vaccinate" some vertices. Once a vertex has been vaccinated, it can no longer be activated. Different objectives may then be of interest, for instance minimizing the total number of activated vertices by vaccinating some particular vertices, or making sure that some specific subset of vertices does not get activated at all, etc. In the literature, these questions have been referred to as the firefighter problem. In the context of this problem and, consequently, throughout this chapter, an activated vertex will be called a burned vertex and a vaccinated vertex is said to be protected. The firefighter problem was introduced by Hartnell [START_REF] Hartnell | Firefighter! an application of domination[END_REF] and further received considerable attention [START_REF] Anshelevich | Approximation algorithms for the firefighter problem: Cuts over time and submodularity[END_REF][START_REF] Cai | Firefighting on trees: (1 -1/e)-approximation, fixed parameter tractability and a subexponential algorithm[END_REF][START_REF] Develin | Fire containment in grids of dimension three and higher[END_REF][START_REF] Finbow | The firefighter problem for graphs of maximum degree three[END_REF][START_REF] Hartnell | Firefighter! an application of domination[END_REF][START_REF] Hartnell | Firefighting on trees: how bad is the greedy algorithm?[END_REF][START_REF] Iwaikawa | Improved approximation algorithms for firefighter problem on trees[END_REF][START_REF] King | The firefighter problem for cubic graphs[END_REF][START_REF] Macgillivray | On the firefighter problem[END_REF][START_REF] Kah | A generalization of the firefighter problem on ZxZ[END_REF]. Originally, the problem was defined as follows. Initially, a fire breaks out at some vertex s of a graph. At each time step, we have to choose one vertex which will be protected by a firefighter. Then the fire spreads to all unprotected neighbors of the vertices on fire. The process ends when the fire can no longer spread, and then all vertices that are not on fire are considered as saved. The objective consists of choosing, at each time step, a vertex which will be protected by a firefighter such that a maximum number of vertices in the graph is saved at the end of the process.

The firefighter problem was proved to be NP-hard for bipartite graphs [START_REF] Macgillivray | On the firefighter problem[END_REF], cubic graphs [START_REF] King | The firefighter problem for cubic graphs[END_REF] and unit disk graphs [START_REF] Fomin | Making life easier for firefighters[END_REF]. Finbow et al. [START_REF] Finbow | The firefighter problem for graphs of maximum degree three[END_REF] showed that the problem is NPhard even on trees. More precisely, they proved the following dichotomy theorem: the firefighter problem is NP-hard even for trees of maximum degree three and it is solvable in polynomial-time for graphs with maximum degree three, provided that the fire breaks out at a vertex of degree at most two. Furthermore, the firefighter problem is polynomial-time solvable for caterpillars and P-trees [START_REF] Macgillivray | On the firefighter problem[END_REF]. 1 From the approximation point of view, the problem is e e-1 -approximable on trees ( e e-1 ≈ 1.5819) [START_REF] Cai | Firefighting on trees: (1 -1/e)-approximation, fixed parameter tractability and a subexponential algorithm[END_REF] and it is not n 1-ε -approximable on general graphs for any ε > 0, if P = NP [START_REF] Anshelevich | Approximation algorithms for the firefighter problem: Cuts over time and submodularity[END_REF]. Moreover for trees where vertices have at most three children, the firefighter problem is 1.3997-approximable [START_REF] Iwaikawa | Improved approximation algorithms for firefighter problem on trees[END_REF]. Finally, Cai et al. [START_REF] Cai | Firefighting on trees: (1 -1/e)-approximation, fixed parameter tractability and a subexponential algorithm[END_REF] gave fixed-parameter tractable algorithms and polynomial-size kernels for trees for each of the following parameters: "number of saved leaves", "number of burned vertices", and "number of protected vertices". The problem on grid graphs of dimension two and higher has been particularly investigated [START_REF] Develin | Fire containment in grids of dimension three and higher[END_REF][START_REF] Kah | A generalization of the firefighter problem on ZxZ[END_REF][START_REF] Macgillivray | On the firefighter problem[END_REF][START_REF] Wang | Fire control on graphs[END_REF]. There are also several variants of the firefighter problem. For instance, Anshelevich et al. [START_REF] Anshelevich | Approximation algorithms for the firefighter problem: Cuts over time and submodularity[END_REF] considered the "spreading vaccination model" where both the fire and the firefighters propagate through the network. In this case, the problem admits a polynomial-time 2-approximation algorithm and a randomized e e-1 -approximation algorithm. The politician's firefighter, introduced by Scott et al. [START_REF] Scott | Politician's firefighting[END_REF], is a more "localized" version: At each time step, we are allowed to put as many firefighters as the number of burned vertices -we invest more ressources as the situation gets more threatening. Moreover, these firefighters can only be placed next to burned vertices -one would prefer to fight the threat closely than miles away. On the contrary to the classical problem, this variant is polynomial-time solvable on trees and fixed-parameter tractable for general graphs when parameterized by the "number of burned vertices" [START_REF] Scott | Politician's firefighting[END_REF]. However, it has been shown that the problem remains NP-hard even for planar graphs [START_REF] Scott | Politician's firefighting[END_REF]. Another problem related to the firefighter problem was introduced by King and MacGillivray [START_REF] King | The firefighter problem for cubic graphs[END_REF]. It consists of deciding whether there is a strategy of choosing a vertex to be protected at each time step such that all vertices of a given set S are saved. Similarly to the classical problem, it has been proved NP-complete even for trees of maximum degree three in which every leaf is at the same distance from the vertex where the fire starts and S is the set of leaves. However, it is polynomial-time solvable for trees of maximum degree three if the initially burned vertex has degree at most two. For a more complete overview and open questions about the firefighter problem, the reader is referred to the recent survey of Finbow and MacGillivray [START_REF] Finbow | The firefighter problem: a survey of results, directions and questions[END_REF]. In this thesis, we consider the version of the firefighter problem which allows us to protect b ≥ 1 vertices at each step (the value b is called budget). We then denote by Firefighter the decision problem that asks for the existence of a protection strategy with respect to a budget b > 0 such that at least k vertices are saved. We also consider its parametric dual Dual Firefighter that asks to save at least nk b vertices for some integer k b , and the Bounded Firefighter problem as well. This last problem is defined similarly to Firefighter except that we are allowed to protect a total of at most k p ≥ 1 vertices where k p is an integer given in the input. We are also interested in the generalized version of the Save problem introduced by King and MacGillivray [START_REF] King | The firefighter problem for cubic graphs[END_REF] that asks for the existence of a protection strategy with respect to a budget b > 0 such that no vertex of a given vertex subset is burned. This chapter is organized as follows. First, we provide the formal definitions of the previous problems in Section 5.2 as well as some preliminaries. In Section 5.3, we show that the Firefighter problem is NP-complete for trees of bounded degree b+3 and for trees of bounded pathwidth three for any fixed budget b ≥ 1. However, we prove that the problem is polynomial-time solvable for the class of graphs of both bounded degree and bounded pathwidth, and thus for graphs of bounded bandwidth. We also design a polynomialtime algorithm for solving the problem (and the corresponding weighted version) on a subclass of trees of pathwidth two, namely k-caterpillars. In Section 5.4, we establish the parameterized complexity lower and upper bounds of the problems Firefighter, Dual Firefighter, and Bounded Firefighter on general graphs with respect to their respective standard parameterization (see Figure 5.2). In Section 5.5, we give parameterized algorithms that answer several open questions from Cai et al. [START_REF] Cai | Firefighting on trees: (1 -1/e)-approximation, fixed parameter tractability and a subexponential algorithm[END_REF] and refine some of their results as well. In Section 5.6, we provide other positive parameterized complexity results with respect to the parameter vertex cover (see Figure 5.1). In Section 5.7, we observe that the minimum version of the Firefighter problem where the goal is to minimize the number of burned vertices is not n 1-ε -approximable on trees for any ε > 0 and any b ≥ 1 if P =NP. We also answer negatively an open question of Finbow and MacGillivray [START_REF] Finbow | The firefighter problem: a survey of results, directions and questions[END_REF]. Conclusion and open problems are given in Section 5.8. Diameter [START_REF] Fomin | Making life easier for firefighters[END_REF] Treewidth Girth (Th.77)

Figure 5.1: The parameterized complexity of the Firefighter problem with respect to some structural graph parameters (see Section 2.3.4). A dotted rectangle means FPT for this parameter, a gray rectangle indicates a W[1]-hardness result for any fixed budget, and a thick rectangle means that the problem is NP-hard for constant values of this parameter and any fixed budget. A parameter written in italic followed by a question mark indicates an open question. For the parameter "diameter", Fomin et al. [START_REF] Fomin | Making life easier for firefighters[END_REF] showed that Firefighter is in XP. 5.2: Summary of our parameterized complexity results including standard parameterizations. Each column is associated to a problem with its standard parameter and each line corresponds to a parameter as well (see the next section for the problems definition). The intersection of a line with a column corresponds to a parameterized complexity result with combination of the line's parameter and column's parameter. The first line corresponds to parameterized complexity results solely based on the standard parameters.

Firefighter Bounded Firefighter Dual Firefighter k k p k b W[1]-hard W[1]-hard W[1]-

Problem definitions and preliminaries

Before defining the problems, we need to explain the spreading process at stake. Let G = (V, E) be a graph of order n with a vertex s ∈ V and b > 0 be an integer called budget. At step t = 0, a fire breaks out at vertex s and s starts burning. At any subsequent step t > 0, the following two phases are performed in sequence:

1. Protection phase : Protect (or defend) at most b vertices not yet on fire.

2. Spreading phase : Every unprotected vertex which is adjacent to a burned vertex starts burning.

Burned and protected vertices remain burned and protected until the process stops, respectively. The process stops when in a step no new vertex can be burned. We call a vertex saved if it is either protected or if all paths from any burned vertex to it contains at least one protected vertex. Notice that, until the propagation process stops, there is at least one new burned vertex at each step. This leads to this obvious lemma.

Lemma 56

The number of steps before the propagation process stops is less or equal to the total number of burned vertices.

A protection strategy (or simply strategy) indicates which vertices to protect at each step until the propagation process stops. Since there can be at most n burned vertices, it follows from Lemma 56 that the propagation unfolds in at most n steps. We thus denote a strategy as a sequence Φ = (D 1 , D 2 , . . . , D n ) where D i ⊆ V and |D i | ≤ b for every i = 1, . . . , n. Each set D i contains the vertices of G defended at step i.

In an algorithmic perspective, it is more convenient to use the following equivalent protection phase definition:

Constrained protection phase : Protect (or defend) vertices not yet on fire under the following restrictions:

1. Each protected vertex must have a neighbor which is on fire.

2. After i steps of propagation at most ib vertices are protected.

The following lemma shows the equivalence between the original protection phase and the constrained protection phase.

Lemma 57 For a given strategy complying with the original protection phase, there exists a strategy that respects the constrained protection phase and saves exactly the same set of vertices. The converse is true.

Proof. Let D be the set of protected vertices of a strategy Φ complying with the original protection phase. We construct a new strategy Φ ′ that protects, during the i th step, exactly those vertices of D which have a neighbor on fire. Clearly after i steps at most ib vertices will be protected, since each vertex of D is protected by the strategy Φ ′ not earlier than by the strategy Φ. Thus Φ ′ respects the rules of the constrained protection phase. Moreover, it saves exactly the same set of vertices. Conversely, let D be the set of protected vertices of a strategy Φ ′ complying with the constrained protection phase. We construct a strategy Φ as follows. Let (v 1 , . . . , v |D| ) be a sequence of vertices of D sorted by the step in which a vertex is protected by Φ ′ (breaking ties arbitrarily). In the i-th step of strategy Φ, we protect the vertices v (i-1)b+1 , . . . , v ib . The vertex v j , j ∈ {(i -1)b + 1, . . . , ib}, is not on fire in the i th step, because in the strategy Φ ′ it is protected not earlier than in the i-th step. Thus Φ respects the rules of the original protection phase and saves exactly the same set of vertices.

Unless otherwise stated, we assume to use the original protection phase. We are now in position to give the formal definitions of the investigated problems.

Firefighter

Input: A graph G = (V, E), a vertex s ∈ V , and two integers b and k. Question: Is there a strategy with respect to budget b such that at least k vertices are saved when a fire breaks out at s?

In the above problem, we may assume without loss of generality that b < k since otherwise the problem is trivial: At time step one, if it is possible to protect b ≥ k vertices then the answer is "yes", otherwise the answer is "no". The parametric dual, denoted by Dual Firefighter, asks for the existence of a strategy such that at least nk b vertices are saved or, equivalently, at most k b vertices are burned. We denote by Max Firefighter the maximization version where the goal is to save the maximum number of vertices. The minimization version will be denoted by Min Firefighter and consists in finding a strategy that minimizes the number of burned vertices.

We also studied the following variant where the total number of protected vertices is bounded.

Bounded Firefighter

Input: A graph G = (V, E), a vertex s ∈ V , and three integers k, b and k p . Question: Is there a strategy with respect to the budget b that saves at least k vertices by protecting a total of at most k p vertices when a fire breaks out at s?

The corresponding maximization version, denoted by Max Bounded Firefighter, asks for finding a strategy that saves the maximum number of vertices by protecting a total of at most k p vertices. We may assume without loss of generality that b ≤ k p in any instance of Bounded Firefighter and Max Bounded Firefighter. Indeed, having a budget b ≥ k p is useless since we can protect a total of at most k p vertices. In other words, the answer of any instance with a budget b ≥ k p does not change if we set b = k p .

We investigate moreover the following generalized version of the problem introduced by King and MacGillivray [START_REF] King | The firefighter problem for cubic graphs[END_REF].

Save

Input: A graph G = (V, E), a "critical" set C ⊆ V , a vertex s ∈ V , and an integer b. Question: Is there a strategy with respect to budget b saving all the vertices that belong to C when a fire breaks out at s? It is worth noting that the above problem was originally entitled "Fire" [START_REF] King | The firefighter problem for cubic graphs[END_REF]. However, we find the name "Save" more relevant regarding the objective of the problem.

Finally, suppose that we are given a weight w(v) for each vertex v of a graph G. These weights may for instance reflect the importance of the vertices: if w(v 1 ) > w(v 2 ), vertex v 1 is considered as more important than vertex v 2 . Then we may define the following weighted version of the Max Firefighter problem.

Max Weighted Firefighter

Input: A graph G = (V, E), a vertex s ∈ V , a weight function w : V → N, and an integer b. Output: A strategy with respect to budget b that maximizes the total weight of the saved vertices in G when a fire breaks out at s.

When the weights are either 0 or 1, we denote the Max Weighted Firefighter problem as the Max 0-1 Firefighter problem.

When dealing with trees, we use the following observation which is a straightforward adaptation of the one by MacGillivray and Wang for the case b > 1 [82, Section 4.1].

Lemma 58 Among the strategies that maximizes the number (resp. the total weight) of saved vertices for a tree, there exists one that protects vertices adjacent to a burned vertex at each time step.

In other words, the Lemma 58 implies that we can only consider strategies that protect vertices adjacent to a burned vertex at each time step on a tree.

It is worth pointing out that the previous lemma does not hold for general graphs. Indeed, an optimal solution might protect some vertex away from the fire at some step (see for instance Figure 5.3).

v 2 v 4 v 6 v 1 v 3 v 5 s (a) v 2 v 4 v 6 v 1 v 3 v 5 s (b)

The importance of bounded degree and "path-likeness"

In this section, we emphasize the importance of the parameters pathwidth and maximum degree in the computational complexity of the Firefighter problem. More precisely, we prove that if both are bounded then the problem is polynomial-time solvable, but it is NP-complete if only one of these two parameters is bounded.

Graphs of bounded degree

It has been shown in [START_REF] Finbow | The firefighter problem for graphs of maximum degree three[END_REF] that Save is NP-complete for trees of maximum degree three when one firefighter is available at each step (i.e. b = 1). Furthermore, it is polynomialtime solvable for trees of maximum degree three if the fire breaks out at a vertex of maximum degree two. In the following, we extend these results for any fixed budget b ≥ 1. We then rely on these last results to show the NP-completeness of Firefighter on trees of bounded degree.

Maximum degree b + 3. We first show the NP-completeness of Save on trees for any fixed budget b ≥ 1. Before doing that, we need the following lemma.

Lemma 59 Let T be a tree with a burning root s together with a strategy that saves all the leaves of T . Suppose that there exist levels i and j > i containing b i ≤ b -1, for some budget b ≥ 1, and b j ≥ 1 firefighters, respectively. Then there exists a new strategy saving all the leaves of T and such that levels i and j contain b i + 1 and b j -1 firefighters, respectively.

Proof. Let v j be a protected vertex by the strategy at a level j > i, and let v i be the ancestor of v j at level i. It follows from Lemma 58 that we may assume that v i is not protected. We transform this strategy into a new one as follows (see Figure 5.4): protect v i at time step i and do not protect v j at time step j. Since v i is an ancestor of v j , it follows that using the new strategy, we save a subset of vertices that contains the vertices saved by using the former strategy. Since level i contains at most b -1 firefighters it follows that the new strategy is valid, saves all the leaves of T and levels i and j contain respectively b i + 1 and b j -1 firefighters. Proof. Clearly, Save belongs to NP. In order to prove its NP-hardness for any fixed budget b ≥ 1, we reduce in polynomial time every instance of Save with budget b to a new instance of the same problem with budget b + 1. Since Save is NP-hard for b = 1 [START_REF] King | The firefighter problem for cubic graphs[END_REF], it follows that Save is NP-complete for any fixed b ≥ 1.

s level i s level i level j level j Saved Saved v i v j
Let I = (T, C, s, b) be an instance of Save where T is a tree rooted at s of maximum degree b + 2, height h, and size n. We may assume without loss of generality that C is the set of leaves of T . Otherwise for each non-leaf vertex v ∈ C, since we have to save v, we can remove the subtree rooted at v such that v becomes a leaf. We construct a new instance

I ′ = (T ′ = (V ′ , E ′ ), C ′ , s ′ , b + 1)
of the same problem consisting of a tree T ′ of maximum degree b + 3 rooted at vertex s ′ , and the critical set C ′ which corresponds to the leaves of T ′ as follows (see Figure 5.5).

• Add a vertex s ′ .

• Add two paths {y 1 y 2 , . . . , y h-2 y h-1 }, {x 1 x 2 , . . . , x h-1 x h }, make y 1 , x 1 adjacent to s ′ and make y h-1 adjacent to s.

• Add vertices v 1 , . . . , v b+1 and make them adjacent to s ′ .

• For every vertex y i , i = 1 . . . , h -1, add vertices v i,1 , . . . , v i,b+1 and make them adjacent to y i .

• For i = 1, . . . , h add a path {w i,1 w i,2 , . . . , w i,h-1 w i,h } and make w i,1 adjacent to x i . This clearly gives us a tree T ′ of maximum degree b + 3 rooted at vertex s ′ and the set of leaves

C ′ ⊂ V ′ is given by C ′ = C ∪ h-1 i=1 {v i,1 , . . . , v i,b+1 } ∪ {w 1,h , . . . , w h,h } ∪ {v 1 , . . . , v b+1 }.
We claim that there exists a strategy for I that saves all the vertices in C if and only if there exists a strategy for I ′ that saves all the vertices in C ′ .

"⇒": Suppose there exists a strategy Φ = (D 1 , D 2 , . . . , D n ) for I that saves all the vertices in C. In order to save all vertices in C ′ , we will apply the strategy defined as follows: at time step t = 1, we have to protect the vertices v 1 , . . . , v b+1 ; at each time step 2 ≤ t ≤ h, we have to protect the vertices v t-1,1 , . . . , v t-1,b+1 ; thus after time step h, vertex s is burning as well as vertices w 1,h-1 , w 2,h-2 , . . . , w h-1,1 , x h ; at each time step h + 1 ≤ t ≤ 2h, we protect the vertices in T belonging to D t-h and we use the additional firefighter to protect the leaf w t-h,h . This clearly gives us a valid strategy saving all the vertices in C ′ .

"⇐": Suppose now that there exists a strategy Φ ′ for I ′ that saves all the vertices in C ′ . At time step t = 1, this strategy necessarily consists in protecting vertices v 1 , . . . , v b+1 . Furthermore, at each time step 2 ≤ t ≤ h, we have to protect the vertices v t-1,1 , . . . , v t-1,b+1 . It follows from Lemma 58 that we may assume that Φ ′ is a strategy which, at each time step, protects vertices adjacent to burning vertices. Thus Φ ′ protects, at each time step i, at most b + 1 vertices at level i in T ′ for i = h + 1, . . . , 2h. Let b T (i) be the number of firefighters in the subtree T of T ′ at level i used by Φ ′ and let

B T = {i : b T (i) = b + 1}. If B T = ∅, then for any i, b T (i) ≤ b
and thus the strategy Φ ′ , restricted to the tree T , is a valid strategy for I that saves all the leaves of T . So we may assume now that B T = ∅.

Let i ℓ be the ℓ th smallest value in B T . Consider the case ℓ = 1. Suppose that for any i < i 1 , b T (i) ≥ b. From the definition of i ℓ , it follows that we cannot have b T (i) = b + 1, thus b T (i) = b for every i < i 1 . By construction, this means that, at each time step i < i 1 , the additional firefighter protects the vertex w i-h,h , i ≥ h + 1. At time step i 1 , since b T (i 1 ) = b + 1, the vertex w i 1 -h,h is not protected and burns which is a contradiction. Thus there exists a level i < i 1 such that b T (i) < b. It follows from Lemma 59 that there exists a strategy saving the leaves of T ′ such that b T (i) ≤ b and b T (i 1 ) = b. Applying this argument iteratively for i 2 , . . . , i |B T | , we obtain a new strategy Φ ′′ that saves all the vertices in C ′ and such that for any level i, b T (i) ≤ b. Thus, the strategy Φ ′′ restricted to the tree T is a valid strategy that saves all the leaves of T . This completes the proof.

In what follows, we use the previous the result to show the NP-completeness of Firefighter on trees of bounded degree.

Theorem 61 For any fixed budget b ≥ 1, Firefighter is NP-complete for trees of maximum degree b + 3.

Proof. We construct a polynomial-time reduction from Save to Firefighter. In this reduction, we will make use of the following gadget. We denote by T (r, h, d) a complete tree of height h and root r such that every non-leaf vertex has exactly d children and every leaf is at the same distance from the root. For such a tree we obtain the following property. Consider a complete tree T (r, h, b + 1). If the fire breaks out at r, then at least one leaf will not be saved. Indeed, since each non-leaf vertex has exactly b + 1 children, it follows that at each time step there will be at least one new burning vertex. Thus at the end of the process, at least one leaf will be burned.

x 1 level 2 level h level h + 1 level 2h v 1,1 v 1,b+1 T y 1 y 2 level 1 v 1 v b+1 s ′ s v h-1,1 v h-1,b+1 x 2 y h-1 level h -1 x h w 2,h w h,h w 1,h w 1,1 w 2,1 w h,1 w h-1,1 w h-1,h
Let I be an instance of Save consisting of a tree T = (V, E) of maximum degree b + 2 with |V | = n, a burned vertex s ∈ V , and a subset C ⊆ V which corresponds to the set of leaves. We construct an instance I ′ of Firefighter consisting of a tree T ′ = (V ′ , E ′ ), and a positive integer k as follows (see Figure 5 We will prove that there exists a strategy for I that saves all the vertices in C if and only if there exists a strategy for I ′ that saves at least k vertices in C ′ .

"⇒": Suppose there exists a strategy for I that saves all the vertices in C. Since C is the set of all leaves in T , it follows that this strategy applied to T ′ saves all the vertices of the trees T i,ℓ . Notice that we have (b + 2)|C| such trees. Thus the number of saved vertices in T ′ is at least k = (b + 2)|C||T |.

"⇐": Conversely, suppose that no strategy for I can save all the vertices in C. Thus, at least one leaf of T is burned at the end. This necessarily implies that for any strategy Φ ′ for I ′ there is at least one vertex, say ℓ, of C which is burned. It follows from the construction of T ′ and the property of each tree T i,ℓ , that in this case there are at least |T | vertices which will be burned for strategy Φ ′ . Thus Φ ′ saves at most

n -1 + (b + 2)|C||T | -|T | ≤ (b + 2)|C||T | -1 < k vertices.
Maximum degree b + 2 and deg(s) ≤ b + 1. The following theorem shows that the sharp separation between the NP-hardness and polynomiality of Save on trees pointed out in [START_REF] King | The firefighter problem for cubic graphs[END_REF] is preserved for any fixed b ≥ 1. Lemma 62 Let b ≥ 1 be any fixed integer and T a tree of maximum degree b + 2. If the fire breaks out at a vertex of degree at most b + 1 then all the leaves of T can be saved by a strategy with respect to budget b if and only if T is not complete.

Proof. Notice that for trees of maximum degree b + 2, we can protect the vertices in such a way that there is at most one new burning vertex v at each time step. Moreover, the fire stops when the vertex v has degree at most b + 1.

Suppose that T is not complete. Then there exists a non-leaf vertex v of degree at most b + 1. From the previous remark we can direct the fire from s to v and stop it. Hence all the leaves of T are saved.

Suppose that T is complete. Then at each time step, there is at least one new burning vertex. Thus there will be a leave which will burn at the end of the process.

Since verifying whether a tree is complete can be done in polynomial-time, we deduce the following theorem.

Theorem 63 For any fixed budget b ≥ 1, Save is polynomial-time solvable for trees of maximum degree b + 2 if the fire breaks out at a vertex of degree at most b + 1.

It is worth noting that Theorem 63 also holds for Firefighter. It suffices to direct the fire to a vertex of degree at most b + 1 such that the number of burned vertices is minimum.

Theorem 64 For any fixed budget b ≥ 1, Firefighter is polynomial-time solvable for trees of maximum degree b+2 if the fire breaks out at a vertex of degree at most b+1.

Graphs of bounded pathwidth

In this section, we show that the Firefighter problem is NP-complete even on trees of pathwidth three. Then, we provide a polynomial-time algorithm for solving Max Weighted Firefighter on a subclass of trees of pathwidth two, namely k-caterpillars.

Pathwidth ≥ 3. We start with the NP-completeness result. For that purpose we reduce from the NP-hard [START_REF] Moore | Hard tiling problems with simple tiles[END_REF] problem Cubic Monotone 1-In-3-Sat (see Appendix A).

Theorem 65 Firefighter is NP-complete even on trees of pathwidth three and budget one.

Proof.

Clearly, Firefighter belongs to NP. For convenience, we provide a polynomial-time reduction from Cubic Monotone 1-In-3-Sat to Dual Firefighter. This will obviously imply the NP-hardness for Firefighter.

Let φ be a formula of Cubic Monotone 1-In-3-Sat with n variables {x 1 , . . . , x n } and m initial clauses {c 1 , . . . , c m }. Notice that a simple calculation shows that n = m. First, we extend φ into a new formula φ ′ by adding m new clauses as follows. For each clause c j we add the clause cj by taking negation of each variable of c j . A satisfying assignment for φ ′ is then a truth assignment such that each clause c j has exactly one true literal and each clause cj has exactly two true literals. It is not hard to see that φ has a satisfying assignment if and only if φ ′ has one. Now we construct an instance I ′ = (T, s, 1, k b ) of Dual Firefighter from φ ′ as follows (see Figure 5.7). In this construction, adding a guard-vertex g to a vertex v means making a copy of a star with center g and k b leaves such that g is made adjacent to v. Thus, if we want to burn at most k b vertices then a guard-vertex needs to be saved or protected. We start with the construction of the tree T .

• Start with a path {su 2 = u 1 u 2 , u 2 u 3 , . . . , u p-1 u p } where p = 2n -1 and add two degree-one vertices v x i and v xi adjacent to u 2i-1 for every i ∈ {1, . . . , n}. In the following two steps for each i ∈ {1, . . . , n}:

• Add a guard-vertex g i (resp. ḡi ) to v x i (resp. v xi ).

• At each vertex v x i (resp. v xi ) root a path of length 2 • (n -i) at v x i (resp. v xi )
in which the endpoint is adjacent to three degree-one vertices (called literalvertices) denoted by ℓ x i 1 , ℓ x i 2 , and ℓ x i 3 (resp. ℓ xi 1 , ℓ xi 2 , and ℓ xi 3 ). Each literalvertex corresponds to an occurence of the variable x i in an initial clause of φ ′ . Analogously, the literal-vertices ℓ xi 1 , ℓ xi 2 , and ℓ xi 3 represent the negative literal xi that appears in the new clauses.

Notice that each leaf of the constructed tree so far is at distance exactly p + 1 from s.

• For each variable x i , i ∈ {1, . . . , n}, choose a clause c j , j ∈ {1, . . . , m} containing

x i (resp. xi ). Then root a path Q x i j (resp. Qx i j ) of length j -1 at ℓ x i 1 (resp. ℓ xi 1 ), and add a guard-vertex g x i j to the endpoint of Q x i j (resp. add two degreeone vertices, named dummy-vertices, adjacent to the endpoint of Qx i j ). Repeat the same for two other clauses with x i (resp. xi ) and root a path at

ℓ x i 2 , ℓ x i 3 (resp. ℓ xi 2 , ℓ xi 3 ).
To finish the construction, set

k b = p + n i=1 (3 + 2(n -i) + 1) + 3 m-1 i=1 i + 4m.
In what follows, we use Lemma 58 and thus we only consider strategies that protect a vertex adjacent to a burned vertex at each time step. Recall that the budget is set to one in the instance I ′ . Now we show that there is a satisfying assignment for φ ′ if and only if there exists a strategy for I ′ such that at most k b vertices in T are burned.

"⇒" : Suppose that there is a satisfying assignment τ for φ ′ . We define the following strategy Φ τ from τ . At each step t from 1 to p + 1, if t is odd then protect v x⌈t/2⌉ if x ⌈t/2⌉ is true otherwise protect v x ⌈t/2⌉ . If t is even then protect the guard-vertex g ⌈t/2⌉ if v x⌈t/2⌉ has been protected, otherwise protect ḡ⌈t/2⌉ . At time step p + 1, the number of burned vertices is exactly p + n i=1 (3 + 2(ni) + 1). Moreover, the literal-vertices that are burned in T corresponds to the true literals in φ ′ . Thus, by construction and since τ statisfies φ ′ , the vertices adjacent to a burning vertex are exactly one guard-vertex, four dummy vertices and 3(n -1) other vertices. Since we must protect the guard vertex, we know that 3(n -1) + 4 vertices are going to burn at the next time step. More generally, at any subsequent time step t from p + 1 to p + m the stategy Φ τ must protect a guard-vertex g x i t-p for some x i , and four dummy-vertices together with 3(n -(tp)) vertices get burned. It follows that the number of burned vertices is p+m t=p+1 [3(n -(tp)) + 4] = m t=1 [3(nt) + 4] = 3 m-1 t=1 t + 4m leading to a total of k b burned vertices.

"⇐": Conversely, assume that there exists a strategy Φ for I ′ such that at most k b vertices in T are burned. Observe first that this strategy protects either v x i or v xi for each i ∈ {1, . . . , n}. As a contradiction, suppose that there exists an i ∈ {1, . . . , n} such that Φ does not protect v x i and v xi . Then in some time step both v x i and v xi get burned. Hence, it is not possible to protect both g i and ḡi and at least one will burn implying that more than k b vertices would burn, a contradiction. Now consider the situation at step p + 1. As previously, the number of burned vertices so far is exactly p + n i=1 (3 + 2(ni) + 1). Let n g and n d be the number of guardvertices and dummy-vertices adjacent to a burned vertex, respectively. Following the discussion, we know that n g = 3n d /2 with 1 ≤ n g ≤ 3 and 1 ≤ n d ≤ 6. In what follows, we will show that exactly one guard-vertex is adjacent to a burned vertex that is n g = 1. First, we must have n g ≤ 1 since otherwise more than k b vertices would burn. Suppose now that n g = 0. Hence we have exactly n d = 6 -2n g = 6 dummy vertices adjacent to burned vertices. Therefore the number of burned vertices would be at least 3 m-1 t=1 t + 4(m -1) + 6 = 3 m-1 t=1 t + 4m + 2 giving us a total of at least k b + 2 burned vertices, a contradiction. Hence, we have n g = 1 and n d = 4. This implies that exactly the three literal-vertices ℓ

x i 1 1 , ℓ xi 2 1 , ℓ xi 3 1 (resp. ℓ xi 1 1 , ℓ x i 2 1 ,ℓ x i 3
1 ) are burned (resp. saved) where x i 1 , x i 2 , x i 3 belong to the clause c 1 . Thus if we set x i 1 to true, x i 2 to false and x i 3 to false in φ ′ , we have exactly one true literal in c 1 and exactly two true literals in c1 . Applying this argument iteratively from step p + 2 to p + m, we arrive at the following satisfying assignment for φ ′ : if the vertex v x i is protected then set x i to false, otherwise set it to true.

It remains to prove that the pathwidth of T is at most three. To see this, observe that any subtree rooted at x i or xi has pathwidth two. Let P x i and P xi be the paths of the path-decompositions of these subtrees, respectively. We construct the pathdecomposition for T as follows. For every i ∈ {1, . . . , n -1}, define the node B i = {u 2i-1 , u 2i , u 2i+1 }. Extend all nodes of the paths P x i and P xi to P ′

x i and P ′ xi by adding the vertex u 2i-1 inside it. Finally, connect the paths P ′

x 1 , P ′ x1 and the node B 1 to form a path and continue in this way with P ′

x 2 , P ′ x2 , B 2 , P ′ x 3 , P ′ x3 , B 3 , . . . , B n-1 , P ′ xn , P ′ xn . This completes the proof.

The above proof can also be regarded as a simpler proof of the NP-completeness of the Firefighter problem on trees. We can generalize the previous result to any fixed budget b ≥ 1 as follows.

Corollary 66 For any fixed budget b ≥ 1, Firefighter is NP-complete even on trees of pathwidth three.

= (x 3 ∨ x 4 ∨ x 5 ) ∧ (x 1 ∨ x 2 ∨ x 3 ) ∧ (x 2 ∨ x 4 ∨ x 5 ) ∧ (x 1 ∨ x 3 ∨ x 6 ) ∧ (x 1 ∨ x 4 ∨ x 6 ) ∧ (x 2 ∨ x 6 ∨ x 5
). Guard vertices are represented by a dot within a circle.

Proof. We start from the reduction of Theorem 65 and alter the tree T as follows. Add a path {sw 2 = w 1 w 2 , w 2 w 3 , . . . , w 2n-1 w 2n } to T together with b-1 guard-vertices added to each w i . First, one can easily check that the pathwidth remains unchanged since the added component has pathwidth two and is only connected to the root s. Second, it can be seen that at each time step, only one firefighter can be placed "freely" as the other b -1 firefighters must protect b -1 guard-vertices. It follows that we end up to a similar proof as for Theorem 65. This completes the proof. k-caterpillars. We now present a subclass of trees of pathwidth two for which Max Weighted Firefighter is polynomial-time solvable. We first prove the result for Max 0-1 Firefighter and further extend it to the general weighted version.

MacGillivray and Wang [START_REF] Macgillivray | On the firefighter problem[END_REF] showed that the degree greedy algorithm (at each time step, protect the vertex adjacent to a burned vertex with the highest degree) gives an optimal solution for Max Firefighter on caterpillars when the budget is b = 1. However, this result does not hold anymore for k-caterpillars as can be seen in Figure 5.8.

In order to prove our main result of this section we first need to state the following result.

Theorem 67 For any k ≥ 1 and any budget b ≥ 1, Max 0-1 Firefighter is polynomial-time solvable for k-stars.

Proof. We construct a polynomial-time reduction from Max 0-1 Firefighter to the Min Cost Flow problem (see Appendix A) which is known to be polynomial-time solvable [START_REF] Ahuja | Network Flows: Theory, Algorithms, and Applications[END_REF]. Let I = (G = (V, E), s, w, b) be an instance of Max 0-1 Firefighter For each vertex v ij in these paths, we define S ij = {v ij , v (i+1)j , . . . v p j j } ∩ S. Notice that we may assume that every path P j contains at least one vertex of S (otherwise we may delete V (P j ) \ {s}). We construct the instance I ′ = (G ′ = (V ′ , A), α, β, γ) of Min Cost Flow as follows.

• Construct the digraph G ′ = (V ′ , A) (see Figure 5.9), where

V ′ = {L 1 , . . . , L p } ∪ {C 1 , . . . , C d } ∪{ℓ, r} and A = {(L i , C j ) : v ij ∈ P j } ∪ {(ℓ, L i ) : i = 1, . . . , p} ∪ {(C j , r) : j = 1, . . . , d}. • Associate with each arc (L i , C j ) of G ′ , a cost β(L i , C j ) = -|S ij |.
All other arcs have cost zero.

• Associate with each arc (ℓ, L i ) a capacity α(ℓ, L i ) = b, with each arc (L i , C j ) a capacity α(L i , C j ) = 1 and with each arc (C j , r) a capacity α(C j , r) = 1.

• Associate a supply of value d with vertex ℓ and a demand of value -d with vertex r (all other vertices have a supply and a demand equal to zero). This completes the construction and clearly I ′ can be obtained from I in polynomial time. Now we claim that solving the instance I of Max 0-1 Firefighter is equivalent to solving the instance I ′ of Min Cost Flow.

"⇒": Consider a feasible solution of Max 0-1 Firefighter in I of value ν. We may assume without loss of generality (see Lemma 58) that at most one vertex is protected in each path P j , j ∈ {1, . . . , d}, and (see Lemma 58) that at most b vertices are protected in each set V i = {v i1 , v i,2 , . . . , v id }, i ∈ {1, . . . , p} (notice that some of these vertices v ij , j = 1, . . . , d, may not exist in G).

Let D = {v ij : v ij is protected, i ∈ {1, . . . , p}, j ∈ {1, . . . , d}}. Thus ν = v ij ∈D |S ij |. Consider now some vertex v ij ∈ D.
Then in G ′ , we will use one flow unit on the path ℓ-L i -C j -r. Repeating this procedure for every vertex in D, we obtain a flow in G ′ of value |D| and of cost

v ij ∈D β(L i , C j ) = v ij ∈D -|S ij | = -ν.
Since at most b vertices are protected in each set V i , it follows that at most b units of flow use the arc (ℓ, L i ), for i ∈ {1, . . . , p}. Furthermore, since exactly one vertex is protected in each path P j , it follows that exactly one flow unit uses the arc (C j , r) for j ∈ {1, . . . , d}. Hence, we obtain a feasible solution of Min Cost Flow in I ′ of value -ν.

"⇐": Conversely, consider now a feasible solution of Min Cost Flow in I ′ of value -µ. Let A be the set of arcs (L i , C j ) used by a flow unit, i ∈ {1, . . . , p}, j ∈ {1, . . . , d}. Thus -µ = (L i ,C j )∈A -|S ij |. For each flow unit on a path ℓ-L i -C j -r, we choose vertex v ij in G to be protected, for i ∈ {1, . . . , p}, j ∈ {1, . . . , d}. Since the capacity of an arc (ℓ, L i ) is b, at most b vertices in V i will be chosen to be protected, i ∈ {1, . . . , p}. Let us denote by V * i the set of vertices in V i chosen to be protected. Furthermore, since the capacity of an arc (C j , r) is one, exactly one vertex in each path P j will be chosen to be protected, j ∈ {1, . . . , d}. Thus, if we protect at each time step i the vertices in V * i , we obtain a feasible solution of Max 0-1 Firefighter in

I of value i v ij ∈V * i |S ij | = µ.
Finally, we have to consider the case when s is not the center of G. The case when s has degree one is trivial. Thus we may assume now that deg G (s) = 2. If b ≥ 2, we are done. Thus we may assume now that b = 1. If both neighbors of s are in S, then the optimal solution value is clearly |S| -1. If both neighbors of s are not in S, then the optimal solution value is clearly |S|. Hence the only case remaining is when exactly one neighbor of s is in S. Let u 1 , u 2 be the neighbors of s such that u 1 ∈ S, u 2 ∈ S. If u 2 is not the center of G, the optimal solution value is clearly |S|. Thus we may assume now that u 2 is the center of G. Let Q denote the set of vertices of the unique maximal path starting at vertex u 2 and containg u 1 . In that case we have to compare the value of two solutions: (i) |S| -1 which is the value of the solution obtained by protecting first u 2 and then, during the second time step, we protect the neighbor of u 1 which is not s (if it exists); (ii) the value of the solution obtained by protecting first u 1 and then applying our algorithm described above to the graph G -(Q \ {u 2 }) (i.e., by reducing our problem to a Min Cost Flow problem).

C 2 C 3 C 1 -2 L 1 C 1 L 2 L 3 -1 -1 -1 -1 -2 -2 -1 2 1 1 2 1 L 1 L 2 L 3 s L 4 ℓ [b] [b] [b] L 4 [b] 0 2 1 1 0 r [1] C 2 [1] C 3 [1]
Figure 5.9: The digraph G ′ (right) obtained from G (left). For the graph G ′ , the capacities are indicated within brackets while the other numbers correspond to the costs. In the graph G, the squares correspond to vertices with weight one. Furthermore, the value associated to each vertex is equal to the number of saved vertices if this vertex is protected.

Notice that the polynomial reduction from Max 0-1 Firefighter to Min Cost Flow described in the proof of Theorem 67 is still valid if the number of vertices that can be protected at each time step is not constant (for instance if we are allowed to protect at most b 1 vertices during the first time step, b 2 vertices during the second time step, etc.). In that case we just need to adapt the capacity of the arcs (ℓ, L i ) accordingly.

Furthermore the polynomial reduction remains valid in the case where some of the vertices in a set V i are not allowed to be protected during time step i. In this case we Using this result and adapting the case when s / ∈ V (P ) according to the weights, it is straightforward that we obtain the following result.

Theorem 71 For any k ≥ 1 and any budget b ≥ 1, Max Weighted Firefighter is polynomial-time solvable for k-caterpillars.

Path-like graphs of bounded degree

As previously shown, for any fixed budget b ≥ 1, Firefighter is NP-complete on trees of bounded degree b+3 (Theorem 61) and on trees of bounded pathwidth three (Theorem 65). It is thus natural to ask the complexity of the problem when both the degree and the pathwidth of the input graph are bounded. In what follows, we answer this question positively.

Theorem 72 Consider a graph of pathwidth pw and maximum degree ∆. If the number of initially burned vertices is bounded by g 1 (pw, ∆) for some function g 1 then there exists a protection strategy where at most g 2 (pw, ∆) vertices get burned for some function g 2 .

Proof. We first prove the following claim: Consider a graph of cutwidth cw. If the number of initially burned vertices is bounded by g 1 (cw) for some function g 1 then there exists a protection strategy where at most g 2 (cw) vertices get burned for some function g 2 . We will prove this by induction on cw.

The claim is obviously true when cw = 0 since the graph cannot contain any edge. Suppose now that the property is true for any graph of cutwidth at most k, k > 0. We show that it also holds for a graph of cutwidth k + 1. Let H = (V, E) be such a graph, and F ⊆ V be the set of initially burned vertices with |F | ≤ g 1 (cw(H)) for some function g 1 . Consider a linear layout L = (v 1 , . . . , v n ) of H such that for every i = 1, . . . , n -1, there are at most k + 1 edges with one endpoint in {v 1 , . . . , v i } and the other in {v i+1 , . . . , v n }. For every s ∈ F , we define the following sets.

R i (s) = {s = v k , v k+1 , . . . , v k ′ } if ∃v k ′ ∈ N i [s] : v k ′ = arg max v∈N i [s] d L (s, v) R i-1 (s) otherwise (5.1) L i (s) = {s = v k , v k-1 , . . . , v k ′ } if ∃v k ′ ∈ N i [s] : v k ′ = arg min v∈N i [s] d L (s, v) L i-1 (s) otherwise (5.2)
We are now in position to define the set B i (s), called a bubble, by B i (s) = L i (s)∪R i (s).

Informally speaking, a bubble B i (s) corresponds to the effect zone of s after i steps of propagation i.e. every burned vertex inside the bubble is due to the vertex s. The idea of the proof is then to show that every bubble can be "isolated" from the rest of the graph in a bounded number of steps by surrounding it with firefighters (see Figure 5.11). We then show that the inductive hypothesis can be applied on each bubble which will prove the theorem. Let s 1 , s 2 ∈ F . We say that two bubbles B i (s 1 ) and B j (s 2 ) for some i, j ≥ 0 overlap if there exists an edge uv ∈ E with u ∈ B i (s 1 ) and v ∈ B j (s 2 ). In this case, we can merge the two bubbles into one i.e. we create a new bubble which is the union of B i (s 1 ) and B j (s 2 ).

Let us consider an initially burned vertex s ∈ F and its bubble B

to |F | • ∆ 2•cw(H) H ≤ g 1 (cw(H)) • (2 • cw(H)) 2•cw(H) = h(cw(H)) for some function h. Let us consider the subgraph H ′ = H[B ′ 2•cw(H) (s)].
Observe that we can safely remove every edge uv from H ′ for which u, v ∈ F ′ . Indeed, such edge cannot have any influence during the subsequent steps of propagation. By the definition of a bubble, this implies that the cutwidth of H ′ is decreased by one and thus is now at most k. Therefore, we can apply our inductive hypothesis to H ′ which tells us that there is a strategy for H ′ such that at most g ′ 2 (cw H ′ ) vertices are burned for some function g ′ 2 . By Lemma 56, this strategy uses at most g ′ 2 (cw H ′ ) steps to be applied. It follows that the number of burned vertices in H after applying this strategy is at most the number of burned vertices from step 1 to 2

• cw(H) + g ′ 2 (cw(H ′ )) which is |F |•∆(H) 2•cw(H)+g ′ 2 (cw H ′ ) ≤ g 1 (cw(H))•(2•cw(H)) 2•cw(H)+g ′ 2 (cw(H ′ )) = h(cw(H))
for some function h. From now on, one can see that the previous argument can be applied iteratively to each bubble. Since the number of bubbles is bounded by g 1 (cw(H)) (there is at most one bubble for each vertex initially on fire), we deduce that the total number of burned vertices is bounded by some function of cw(H). This concludes the proof of the claim.

We are now in position to show the theorem. Let G be a graph. Suppose that the number of initially burned vertices in G is at most g 1 (pw(G), ∆(G)) for some function g 1 . We know that pw(G) ≤ cw(G) and ∆(G) ≤ 2 • cw(G) [START_REF] Korach | Tree-width, path-width, and cutwidth[END_REF]. Thus the number of burned vertices is at most g ′ 1 (cw(G)) for some function g ′ 1 . From the above claim, we deduce that there exists a strategy such that at most g ′ 2 (cw(G)) vertices get burned. Since cw(G) ≤ pw(G) • ∆(G) [START_REF] Fan | Graphs with small bandwidth and cutwidth[END_REF], it follows that the number of burned vertices is bounded by g 2 (pw(G), ∆(G)) for some function g 2 . This completes the proof. We choosed to present the following parameterized complexity results here rather than in the later dedicated section. The reason is that these results complete quite well the complexity analysis conducted so far. In the next section (Section 5.4), we show that Dual Firefighter is fixed-parameter tractable with respect to the combined parameter k b and "maximum degree" of the input graph. Thus, using Theorem 72 together with Corollary 85, we deduce the following. • Connect a i,j to a i ′ ,j ′ (i ′ = i + 1) for all i, j, j ′ , connect a 1,j to s for all j, and connect a k-1,j to each vertex of V ′ = 1≤h≤b V h for all j.

s B 2 (s) B 3 (s) B 1 (s)
v 1 v 2 v 3 v 4 v 5 s c 1 a 1,1 a 1,2 a 1,3 a 2,1 a 2,2 a 2,3 v 1 1 v 1 2 v 1 3 v 1 4 v 1 5 e v 1 ,v 2 e v 2 ,v 3 e v 2 ,v 4 e v 3 ,v 4 e v 4 ,v 5
• Add b vertices c 1 , . . . , c b adjacent to all vertices of V ′ . To finish the construction, set k

′ = kb + k 2 + b. We claim that (G, k) is a yes-instance if and only if (G ′ , s, b, k ′ ) is a yes-instance. "⇒": Suppose that G has a clique C ⊆ V of size k.
Then the strategy that protects the vertices v 1 , . . . , v b for all v ∈ C saves the vertices e uv for all u, v ∈ C. Since C is a clique, these vertices e uv are indeed present in G ′ . Additionally, we can protect (and thus save) vertices c 1 , . . . , c b . It follows that this strategy saves at least k ′ vertices.

"⇐": Conversely, suppose that there exists a strategy Φ = {D 1 , . . . , D n } for (G ′ , s, k ′ , b) that saves at least k ′ vertices where D i = {p 1 i , . . . , p b i } for each i = 1, . . . , n. First observe that if p h t = a i,j for some i ,j, and h, then this vertex is not helpful, as there is always a vertex a i,j ′ that will be burned at time t and has the same neighborhood as a i,j . Hence we can assume that no vertex a i,j is protected by the strategy. This implies that all vertices of V ′ will be burned, except those that are protected by the strategy. But then protecting vertices of F does not save any further vertices. Since the fire will reach V ′ in k time steps, and thus F in k + 1 time steps, the vertices in (D

1 ∪ . . . ∪ D n ) ∩ V ′ are responsible for saving k 2 vertices, which is only possible if the vertices of (D 1 ∪ . . . ∪ D n ) ∩ V ′ induce a clique of size k in G.
Notice that the parameters in this reductions are quadratically related. Since Clique is W[1]-hard with respect to k and cannot be solved in time n o(k) unless ETH fails [START_REF] Chen | Strong computational lower bounds via parameterized complexity[END_REF], we obtain the desired lower bound.

The above reduction is also a NP-hardness reduction, and simpler than the original one on bipartite graphs [START_REF] Macgillivray | On the firefighter problem[END_REF].

Bounded Firefighter

We now consider the parameterized complexity of Bounded Firefighter parameterized by k p . To this end, we introduce the notion of valid protecting set. A valid protecting set is a subset of vertices D of a graph G such that there exists a strategy that protects exactly those vertices in D if the fire starts at some vertex s.

Lemma 78 Let G = (V, E) be a graph with an initially burned vertex s ∈ V . Verifying whether a subset D ⊆ (V \ {s}) is a valid protecting set can be done in linear time.

Proof. In this proof, we use the constrained protection phase. Let L i = {v ∈ V : dist G\ 0≤j≤i-1 L j ∩D (s, v) = i} for any i > 0 and L 0 = {s}. The graph G\ 0≤j≤i-1 L j ∩ D is obtained from G by removing protected vertices from time step 1 through i -1. Let r i = ib -| 0≤j<i L j ∩ D| be the number of available firefighters in step i. If there exists i ∈ {1, . . . , |V |} such that |D ∩ L i | > r i or r i < 0, then there is no strategy with respect to budget b that protects D.

We are now in position to state the following result.

Theorem 79 Bounded Firefighter is solvable in n O(kp) time.

Proof.

Let (G = (V, E), s, k, b, k p ) be an instance of Bounded Firefighter. Among all valid protecting set D ⊆ (V \ {s}) such that |D| ≤ k p , the algorithm simply chooses the first one that saves at least k vertices. From Lemma 78, the running time is n O(kp) .

By carrying out some modifications in the proof of Theorem 77, we can show that the above algorithm has nearly the best running time.

Theorem 80 For any fixed budget b ≥ 1, Bounded Firefighter is W[1]-hard with respect to the parameter k p and cannot be solved in n o( √ kp) time even on bipartite graphs unless ETH fails.

Proof.

Given an instance I = (G, k) of Clique we construct the instance I ′ = (G, s, k ′ , b, k p ) where G is the same bipartite graph as in the proof of Theorem 77. We set k p = kb + b and k ′ = kb + k 2 + b. Correctness now follows straightforwardly from the arguments in the proof of Theorem 77.

Dual Firefighter

In the following subsection, we consider the Dual Firefighter problem. We first show that Dual Firefighter can be solved in n O(k b ) time. Analogously to the previous result, we introduce the notion of valid burning set. A valid burning set is a subset B of vertices of a graph G such that there exists a strategy for which, at the end of the process, the burned vertices are exactly those in B if the fire starts at some vertex s.

Lemma 81 Let G = (V, E) be a graph with an initially burned vertex s ∈ V . Verifying whether a subset B ⊆ V is a valid burning set can be done in linear time.

Proof. In this proof, we use the constrained protection phase. First of all, if s ∈ B or G[B] is not connected then return "no". We now suppose that s ∈ B and G[B] is connected. Observe that the set of protected vertices must be exactly N (B). For any v ∈ N (B), let d s (v) be the length of a shortest path with endpoints s and v in G whose internal vertices are all in B. Then v has to be protected before or at time step d s (v). It follows that B is a valid burning set if and only if the number of vertices v for which d s (v) ≤ t is at most bt for every t = 1, . . . , k. Finally, we note that d s (v) can be easily computed using a breadth-first search. Hence we can determine whether B is a valid burning set in linear time.

Theorem 82 Dual Firefighter is solvable in n O(k b ) time.

Proof. Let (G = (V, E), s, b, k b ) be an instance of Dual Firefighter. Observe that if a vertex of G is burning and is at distance k b from s then we can deduce that at least k b + 1 are burned. Therefore, we can restrict valid burning sets to those which are subset of the k th b neighborhood of s. Having said that, exhaustively consider all subsets B ⊆ N k b [s] with |B| ≤ k b . If B is a valid burning set, then the answer is "yes". If no valid burning set is found, then the answer is "no". It follows from Lemma 81 that the running time is n O(k b ) . It is worth noting that this time can be rewritten as

∆(G) k 2 b • n since |N k b [s]| ≤ ∆(G) k b .
We now show that the above algorithm is likely to have the optimal running time. "⇐": Conversely, suppose that there is no clique of size k in G. At time step one, any valid strategy has to place the nk remaining firefighters on vertices that are not edge-vertices; otherwise at least k + 2 > k b vertices will burn. At time step two, since there is no clique of size k, there will be at least k∆ -k 2 + 1 edge-vertices adjacent to the k burned vertices. For the same reason as before, there remains b 2 = bb 1 firefighters which is not enough to protect these edge-vertices. Therefore, given any valid strategy there will be at least k + 2 > k b burned vertices.

We note that the parameters used in this reduction in are linearly related. Since Clique on regular graphs is W[1]-hard with respect to k and cannot be solved in time n o(k) unless ETH fails [START_REF] Mathieson | Parameterized graph editing with chosen vertex degrees[END_REF], we obtain the desired lower bound n o(k b ) .

s n -k = 1 b -(n -k) = 8

Parameterized algorithms

In this section, we start by showing the fixed-parameter tractability of Dual Firefighter with respect to the combined parameter k b and b. Then, we resolve several open questions of Cai et al. [START_REF] Cai | Firefighting on trees: (1 -1/e)-approximation, fixed parameter tractability and a subexponential algorithm[END_REF] on trees. We also refine and extend some of their results. Finally, we extend these results to the graphs of bounded treewidth. We further derive that Firefighter is in FPT with respect to the parameter k on graphs of bounded local treewidth, including planar graphs, graphs of bounded genus, apex-minor-free graphs, and graphs of bounded maximum vertex degree. Proof. In this proof, we consider the constrained protection phase. We present a simple branching algorithm. Assume that we are in the i-th time step and let B be the set of vertices which are currently on fire. Moreover, let P be the set of already protected vertices (in the first step we have B = {s} and P = ∅). Let a = ib -|P | and r = |N (B) \ P |. The algorithm does the following:

1. If |B| > k b , then we immediately answer "no".

If a ≥ r, then we can greedily protect the whole set N (B) \ P . Hence in this case we answer "yes".

3. In the last case, when a < r, we branch on all subsets of N (B) \ P of size at most a. Observe that the number of branches is equal to a j=0 r j ≤ 2 r -1, since we have a < r. The running time of the algorithm is as follows. We introduce a measure α = (k b -|B|) + (ib -|P |) which we use in our time bound. At the beginning of the first step of the burning process, we have α

= (k b -1) + (b -0) = k b + b -1. By T (α)
we denote the upper bound on the number of steps that our algorithm requires for a graph with measure value α. Observe that for α ≤ 0, we have T (α) = O(n). Let us assume that the algorithm did not stop in step 1 nor 2, and it branches into at most 2 r -1 choices of protected vertices. Observe that no matter how many vertices the algorithm decides to protect, the value of the measure decreases by exactly rb. Consequently, we have the inequality T (α) ≤ (2 r -1)T (αr + b) + O(n). Since the algorithm did not stop in steps 1 or 2, we infer that r ≥ b + 1. The time bound follows from the fact that the worst case for the inequality occurs when r = b + 1.

Using an easy observation, we can deduce the following corollary.

Firefighting on trees

In this section, we study the Firefighter, Bounded Firefighter and Dual Firefighter problems on trees. Previously known results on that class of graphs are from Cai et al. [START_REF] Cai | Firefighting on trees: (1 -1/e)-approximation, fixed parameter tractability and a subexponential algorithm[END_REF]. More specifically, they obtained the following parameterized algorithms on trees and budget b = 1. Proof. Let (T, s, b, k b ) be an instance of Dual Firefighter where T is a tree with root s. If s has at most b children, then we immediately answer "yes". We may assume that the root has exactly a children where a < b + k b since otherwise we simply answer "no". We use Lemma 58 and branch on every subset of b children of the root s. In each branch, we cut the subtree rooted at the protected vertex, identify all the vertices that are on fire after the first step, and decrease the parameter by ab. In this way, we obtain a new instance of the Dual Firefighter problem with parameter value equal to k b -(ab). The time bound follows from the inequality

• A randomized algorithm solving

T (k b ) ≤ a b T (k b -(a -b)) + O(n)
which is worst when a = b + 1.

The above theorem directly implies that Dual Firefighter admits a kernel of size (b+1) k b . However, we show in the following that this size cannot be reduced to (b+k b ) O (1) unless NP ⊆ coNP/poly. As a first step toward this goal, we provide the proof for b = 1 and then extend this result to any fixed b ≥ 1.

Theorem 87 There is no polynomial kernel for Dual Firefighter with respect to the parameter k b and budget one even for trees of maximum degree four unless NP ⊆ coNP/poly. Proof. We apply Theorem 4 (see subsection 2.3.2), where as the language L we use Dual Firefighter on trees of maximum degree three and budget one, which is NP-complete [START_REF] Finbow | The firefighter problem for graphs of maximum degree three[END_REF]. In the following, we show that Dual Firefighter on trees with maximum degree three and budget one cross-composes to Dual Firefighter parameterized by k b on trees with maximum degree four.

Observe that any polynomial equivalence relation is defined on all words over the alphabet Σ and for this reason we should also define how the relation behaves on words that do not represent instances of the problem. For the equivalence relation R we take a relation that puts all malformed instances into one equivalence class and all well-formed instances are grouped according to the number of vertices we are allowed to burn.

If we are given malformed instances, we simply output a trivial no-instance. Thus in the rest of the proof we assume we are given a sequence of instances (T i , s i , 1, k b ) t i=1 of Dual Firefighter, where each T i is of maximum degree three. Observe that in all instances we have the same value of the parameter k b . Without loss of generality, we assume that t = 2 h for some integer h ≥ 1. Otherwise we can duplicate an appropriate number of instances (T i , s i , 1, k b ).

We create an instance (T, s, 1, k ′ b ) of Dual Firefighter parameterized by k b where T is a tree defined as follows. We start with a full binary tree rooted at a vertex s with exactly t leaves . Now for each i = 1, . . . , t, we replace the i th leaf of the tree by the tree T i rooted at s i . Finally, we set k

′ b = k b + h = k b + log 2 t.
Observe that since each tree T i is of maximum degree three, the tree T is of maximum degree four. To prove correctness, it is enough to show that any strategy that minimizes the number of burned vertices protects exactly one vertex at each level 1, . . . , h, which follows from Lemma 58. Hence in any strategy that minimizes the number of burned vertices, there will be exactly one vertex s i which is on fire after h steps.

We now generalize the above result using the fact that Dual Firefighter is NPcomplete for trees of maximum degree b + 3 where b ≥ 2 is fixed (Theorem 61).

Theorem 88 For any fixed budget b ≥ 2, there is no polynomial kernel for the Dual Firefighter with respect to the parameter k b even for trees of maximum degree b + 4 unless NP ⊆ coNP/poly.

Proof. Let b ≥ 2 be any fixed constant. We use Theorem 4, where L is the Dual Firefighter problem on trees of maximum degree b + 3 and budget b. To give the cross-composition, we use the same proof from Theorem 87 with the following differences.

• For the equivalence relation R, we take a relation that puts all malformed instances into one equivalence class, and all well-formed instances are grouped according to the parameter value k b and the budget b.

• We assume that t = (b + 1) h for some integer h ≥ 1.

• The tree T is now constructed from a full (b + 1)-ary tree rooted at vertex s with exactly t leaves. Finally, we set k ′ b = k b + log b+1 t. This completes the proof.

Bounded Firefighter. Before we present the algorithm for this problem, we need to introduce the following notation. Let T = (V, E) be any rooted tree with n vertices. Use a pre-order traversal of T to number the vertices of T from 1 to n. We say that u ∈ V is to the left of v ∈ V if the number assigned to u is not greater than the number of v in the order. It is then easy to define what the leftmost or rightmost vertex is.

As a matter of fact we provide an algorithm for solving the optimization problem Max Bounded Firefighter. Obviously, this implies that Bounded Firefighter is also solvable with the same running time by applying this algorithm.

Theorem 89 Max Bounded Firefighter is solvable in O((b+1) (kp/b)+1 k p n) time on trees.

Proof. Let (T, s, b, k p ) be an instance of Max Bounded Firefighter on a tree T . Assume that T is rooted at s and let h = ⌈k p /b⌉. By Lemma 58, we can define a characteristic vector χ v of length h for each vertex v of the tree, which has value p i ∈ {0, . . . , b} at position i if and only if the optimal strategy protects p i vertices at level i in the part of the tree to the left of v. We use these vectors as the basis for a dynamic programming procedure. However, the vector cannot ensure that no ancestors of a protected vertex will be protected. To ensure this, we add another dimension to our dynamic programming procedure. The pre-order numbering ensures that no descendant is protected.

The dynamic programming algorithm is then as follows. Let L be the set of vertices in T that are at level at most h. For each v ∈ L, let P v denote the path in T between v and s. For each vector χ ∈ {0, . . . , b} h and each integer 0 ≤ i ≤ h, we compute A v (χ, i), the maximum number of vertices one can save when protecting at most χ(j) vertices at level j, where protected vertices must lie to the left of v but at level greater than i when lying on P v , and no protected vertex is an ancestor of another. Observe that s is the leftmost vertex of L. Now set A s (χ, i) = 0 for any χ and i. Then

A v (χ, i) = max A l(v) (χ, min{level(v) -1, i}), [χ(level(v)) ≥ 1 ∧ level(v) > i] • (r(v) + A l(v) (χ v , level(v) -1))
Here level(v) is the level where vertex v is lying, l(v) is the rightmost vertex in L which has strictly smaller value in the pre-order than v, and r(x) is the number of vertices saved when protecting only x. Moreover, χ v is the vector obtained from χ by reducing the number p level(v) by 1. In the formula we use Iverson's bracket notation, where [φ] is equal to one if φ is true and zero otherwise.

To see that the above formula is correct, observe that we can either protect the considered vertex v or not. If we do not protect v, then we must ensure that the value for the second dimension of our dynamic programming procedure does not exceed the length of P v , yet still captures the same forbidden part of P v . Correctness then follows from the fact that the parent of v is always on P l(v) . If we do protect v, we can protect v only if we are allowed to do so, i.e. if χ(level(v)) ≥ 1 and level(v) > i. Furthermore, we need to ensure that no ancestor of v is protected later. Therefore, we set the value for the second dimension of our dynamic programming procedure to level(v) -1.

To get the solution for the whole tree T , return A v * (χ * , 0), where v * is the rightmost vertex of L and χ * is a vector of length h with the h-th entry set to k p -(h -1)b and the other entries set to b. To obtain the claimed running time, first find L, and then l(v) for each vertex v ∈ L. This can be done in linear time by a depth-first search. We can also compute r(x) for each x ∈ V (T ) in linear time, as r(x) equals one plus the number of descendants of x. By traversing the vertices of L from left to right, the total running time is O

((b + 1) h k p n) = O((b + 1) (kp/b)+1 k p n).
Since we assumed that b ≤ k p (see the discussion in Section 5.2), we have the following immediate corollary.

Corollary 90 Bounded Firefighter is fixed-parameter tractable with respect to the parameter k p on trees.

The following result states that it is unlikely that Bounded Firefighter parameterized by k p admits a polynomial kernel on trees. Like we did for Dual Firefighter, we give the proof for b = 1 first and then extend it for any fixed b ≥ 2.

Theorem 91 There is no polynomial kernel for Bounded Firefighter with respect to the parameter k p and budget one even for trees of maximum degree four unless NP ⊆ coNP/poly.

Proof. There are only three differences compared to the proof for Theorem 87.

• The language L is now Bounded Firefighter on trees of maximum degree three and budget one.

• For the equivalence relation R, we take a relation that puts all malformed instances into one equivalence class, and all well-formed instances are grouped according to the number of vertices of the tree, the parameter value k p , and the value k.

• The instance (T, s, 1, k ′ , k ′ p ) of Bounded Firefighter parameterized by k p is as follows. The tree T is constructed as described in the proof of Theorem 87. The value k ′ p is set to k p + h, and the value of k ′ is equal to k + (t -1)n + (th -1), where n is the number of vertices in each of the trees T i . The additional summands are derived from the fact that any optimal strategy will ensure that after h steps exactly one vertex s i will be on fire and hence we save t-1 subtrees rooted at s i , each containing n vertices, and th -1 vertices of the full binary tree. This completes the proof.

Using the fact that Theorem 61 implies the NP-completeness of Bounded Firefighter for trees of maximum degree b + 3 for any fixed budget b ≥ 2, we may generalize the previous result as follows.

Theorem 92 For any fixed budget b ≥ 2, there is no polynomial kernel for Bounded Firefighter with respect to the parameter k p even for trees of maximum degree b + 4 unless NP ⊆ coNP/poly.

Proof. We start with the proof from Theorem 91 and modify it as follows.

• For the equivalence relation R, we take a relation that puts all malformed instances into one equivalence class, and all well-formed instances are grouped according to the number of vertices of the tree, the parameter value k p , the value k and the budget b.

• We assume that t = (b + 1) h for some integer h ≥ 1.

• The tree T is now constructed from a full (b + 1)-ary tree rooted at vertex s with exactly t leaves. This completes the proof. As we assumed that b ≤ k (see the discussion in Section 5.2), we get the following immediate corollary.

Corollary 94 Firefighter is fixed-parameter tractable with respect to the parameter k on trees.

A subexponential algorithm. To obtain a good subexponential algorithm for the Max Firefighter problem, we use the following lemma. A similar idea for b = 1 appeared independently in [START_REF] Floderus | Towards more efficient infection and fire fighting[END_REF]. This expresses that S can be saved by protecting b i vertices in step i. The sets B i contain all vertices that are burned by time step i, which is ensured by the formulas in lines 5.3 and 5.4. The set B contains vertices that are not saved (line 5.7) and all vertices of the sets B i (line 5.6). The vertices p 1 1 , . . . , p b 1 1 , . . . , p 1 ℓ , . . . , p b ℓ ℓ are the vertices that are protected. Line 5.5 ensures that the vertices we want to protect are not burned by the time we pick them. Then we want to find the largest set S such that Protect kp (S) := is true. Following a result of Arnborg, Lagergren, and Seese [START_REF] Arnborg | Easy problems for tree-decomposable graphs[END_REF], this can be done in f (tw, k p ) • n O(1) time using the above formula.

From the above Theorem, we immediately deduce the following.

Corollary 98 Bounded Firefighter is fixed-parameter tractable with respect to the combined parameter k p and treewidth of the input graph.

In the same way as Corollary 93, we then obtain the following.

Corollary 99 Firefighter is fixed-parameter tractable with respect to the combined parameter k and treewidth of the input graph.

This algorithm also works on graphs of bounded local treewidth.

Corollary 100 Firefighter is fixed-parameter tractable with respect to the parameter k on graphs of bounded local treewidth.

Proof. Let (G, s, b, k) be an instance of Firefighter. Observe that if the graph G has a vertex at distance more than k from the initially burned vertex s, then a strategy that protects any vertex at distance i from s in time step i will save at least k vertices, and we can answer "yes" immediately. From the previous discussion, we may assume that G = G[N k (s)]. Because G has bounded local treewidth, there exists a function f such that ltw k (G) ≤ f (k). Therefore, we have tw(G) ≤ f (k) since G = G[N k (s)] and thus tw(G) = ltw k (G).

The class of graphs having bounded local treewidth coincides with the class of apexminor-free graphs [START_REF] Eppstein | Diameter and treewidth in minor-closed graph families[END_REF], which includes the class of planar graphs.

Corollary 101 Firefighter is fixed-parameter tractable with respect to the parameter k on planar graphs.

Parameter "vertex cover number"

In this section, we show that the Firefighter problem is fixed-parameter tractable with respect to the parameter "vertex cover number".

Theorem 102 Firefighter admits a kernel with O(2 τ τ 2 ) vertices where τ is the vertex cover number of the input graph. From now on, we assume b < τ ′ . Observe that every two steps at least one vertex inside C ′ gets burned. It follows that after 2τ ′ steps, all the vertices in G are necessarily burned. Since the number of steps is at most 2τ ′ , there is a total of at most 2τ ′ b < 2τ ′2 protected vertices.

We now provide the kernelization algorithm. We call a set T ⊆ V a twins set if for every v, u ∈ T , v = u, we have N (u) = N (v) and uv / ∈ E. Consider the following reduction rule.

Reduction rule. If there exists a twins set T ⊆ V \ C ′ such that |T | ≥ 2τ ′2 + 1, then delete |T | -2τ ′2 -1 vertices of T .

As to the correctness of the rule, let Φ be a strategy such that at most k b vertices are burned in G and T ⊆ V \ C ′ be a twins set. Observe that if Φ protects a subset T 1 ⊆ T of vertices then protecting any subset T 2 ⊆ T instead of T 1 such that |T 2 | = |T 1 | leads to another strategy that saves exactly the same number of vertices. It follows that if Φ protects a vertex in a twins set that has been deleted by the reduction rule then we can protect any other non-deleted vertex in the same twins set instead, without changing the amount of saved vertices. Moreover, we have |T 1 | ≤ 2τ ′2 since at most 2τ ′2 vertices are protected. Therefore, it is enough to keep 2τ ′2 + 1 vertices in each twins set of V \ C ′ . Conversely, if there is a strategy for which at most k b vertices in G ′ are burned, then applying this strategy in G will result in at most k b burned vertices in G.

Let G ′ = (V ′ , E ′ ) be the graph obtained by iteratively applying the above reduction rule to every twins set in V \ C ′ . Notice that the procedure runs in polynomial time. The number of distinct twins sets in V \ C ′ is at most 2 τ (one for each subset of C ′ ). Moreover, each of these twins set has at most 2τ ′2 + 1 vertices. Therefore, the size of the reduced instance is O(2 τ τ 2 ). This completes the proof.

While the above result is useful as a complexity classification result, it might be more efficient to use the following kernelization instead, depending of the size of τ and k.

Theorem 103 Firefighter admits a kernel with O(2 τ k) vertices where τ is the vertex cover number of the input graph.

Proof. We use the same approach as in the proof of Theorem 102 with the following differences. Let (G, s, b, k) be an instance of Firefighter. We iteratively apply the following reduction rule to every twins set in G to obtain a new graph G ′ = (V ′ , E ′ ).

Reduction rule: If there exists a twins set T such that |T | ≥ k + 1, then delete |T |k -1 vertices of T .

Let C ⊆ V ′ be a minimum vertex cover and let D = V ′ \ C be an independent set. The number of distinct twins sets in D is at most 2 τ (one for each subset of C). Moreover, each twins set in G ′ has at most k + 1 vertices. Therefore, the size of the reduced instance is at most O(2 τ k).

The proof of correctness uses similar arguments as for Theorem 102 since we may assume that no strategy can protect more that k vertices otherwise the problem is trivial.

Approximability

Finbow and MacGillivray [START_REF] Finbow | The firefighter problem: a survey of results, directions and questions[END_REF] asked whether there exists a constant c > 1 such that the degree greedy algorithm that consists, at each time step, to protect a highest degree vertex adjacent to a burning vertex, gives a polynomial-time c-approximation for Max Firefighter for trees. The following proposition answers this question negatively.

Theorem 104 For any budget b ≥ 1, there exists no function α : N → (1, n) such that the degree greedy algorithm is a polynomial-time α(n)-approximation algorithm for Max Firefighter for trees.

Proof. Consider a tree T (r, h -1, b + 1) where h is a positive integer (see definition in the proof of Theorem 61). Add a vertex s adjacent to r and a vertex v 1 adjacent to s; for i = 2, . . . , h -1, add a path of length i -1 with endpoints u i and v i such that u i is adjacent to s; finally, for i = 1, . . . , h -1, add b + 2 vertices adjacent to v i (see Figure 5.14).

Notice that the degree greedy algorithm protects vertices in the following order: v 1 , . . . , v h-1 . Thus it saves g h = (h -1)(b + 2) vertices. However, it is not difficult to see that the optimal solution protects vertices in the following order: r, v 2 , . . . , v h-1 . Thus, in an optimal solution we save opt h = (h -2)(b + 2) + h-1 i=0 (b + 1) i vertices. Since opt h g h → +∞ when h → +∞, the result follows.

In contrast to Max Firefighter which is constant approximable on trees, the following theorem shows a strong inapproximability result for Min Firefighter even when restricted to trees.

Theorem 105 For any ε > 0 and any budget b ≥ 1, Min Firefighter is not n 1-εapproximable even for trees, unless P = N P . n 1-ε n 1 vertices are burned in V ′ . Using Theorem 8, we deduce that there is no polynomial-time n 1-ε -approximation algorithm for Min Firefighter unless P = NP.

Conclusion and open problems

In this chapter, we studied some generalizations and variants of the firefighter problem when more than one firefighter is available at each time step. We would like to conclude this study with the following research directions.

(1) The complexity of Firefighter and Save in the case when the number of firefighters at each time step depends on the number of vertices. The complexity of Firefighter for trees of maximum degree b + 2 is not established.

(2) The Max Firefighter problem is 2-approximable for trees. Establishing non approximability results or better approximability results is an interesting open question.

(3) We showed that the Firefighter problem is NP-complete for graphs of pathwidth at most three. While it is polynomial-time solvable for k-caterpillars, we do not know whether this algorithm can be extended to general graphs of pathwidth two.

(4) We have shown that Firefighter parameterized by k is in FPT on graphs of bounded local treewidth, and thus on planar graphs, by showing that the problem is in FPT parameterized by k and the treewidth of the input graph. While Bounded Firefighter parameterized by k p is also in FPT parameterized by k p and the treewidth, we do not know whether the problem is in FPT on planar graphs, and leave it as an open problem.

(5) We do not know if Dual Firefighter is fixed-parameter tractable when parameterized by k b and the treewidth. One could try to adapt the MSOL formula for Max Bounded Firefighter, but in its current form the size of the formula depends on k b and b. This is not the case for Theorem 97, as we can assume that b ≤ k p . However, it is not clear whether a similar assumption can be made for parameter k b .

(6) The Max Firefighter problem is solvable in subexponential time on trees. Is it solvable in time 2 o(n) on planar graphs? Even the case of outerplanar graphs is open.

(7) It is unknown if Firefighter admits a polynomial kernel for parameter "vertex cover number". While we have shown that Bounded Firefighter and Dual Fire-fighter do not admit a polynomial kernel with respect to the combined parameter k p and tw, and k b and tw, respectively, we do not know if it is also the case for Firefighter parameterized by k and tw. (9) Finally, we do not know if any of the three problems Firefighter, Dual Firefighter, and Bounded Firefighter are solvable in parameterized subexponential time on trees with respect to the standard parameters.

(10) An interesting research direction would be to complete the parameterized complexity landscape of the Firefighter problem with respect to structural graph parameters (see Figure 5.1).

Chapter 6 Conclusion

Motivated by the application of information diffusion through social networks, we analyzed the complexity of optimization problems involving a propagation process in a graph. Besides of being problems of practical relevance (social network analysis, epidemiology, . . . ), it seems to us that they are also of considerable theoretical interest.

Firstly, Target Set Selection generalizes well-known graph problems such as Dominating Set with thresholds [START_REF] Harant | On dominating sets and independent sets of graphs[END_REF], Vector Dominating Set [START_REF] Venkatesh Raman | Parameterized algorithms for generalized domination[END_REF], k-Tuple Dominating Set [START_REF] Klasing | Hardness results and approximation algorithms of k-tuple domination in graphs[END_REF], (all these variants allow for only one "activation step"), Vertex Cover [START_REF] Chen | On the approximability of influence in social networks[END_REF] (where the threshold value equals the vertex degree), Irreversible k-Conversion Set [START_REF] Dreyer | Irreversible k-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion[END_REF], r-Neighbor Bootstrap Percolation [START_REF] Balogh | Bootstrap percolation in high dimensions[END_REF] (where the threshold of each vertex is k or r, respectively), and so-called dynamic monopolies [START_REF] Peleg | Local majorities, coalitions and monopolies in graphs: a review[END_REF] (where the threshold of a vertex v with degree deg(v) equals ⌈deg(v)/2⌉). Furthermore, during our research we found out the existence of a close connection between the complementary optimization problem Max Open Influence and the Densest k-Subgraph problem. More precisely, we show that Max Open Influence with unanimity thresholds is at least as hard to approximate as Densest k-Subgraph. Due to this connection, we were able to provide an fpt-time approximation algorithm within any strictly increasing ratio with respect to the parameter k (independently established in [START_REF] Bourgeois | Exact and approximation algorithms for densest k-subgraph[END_REF]). Notably any improvement of the approximation ratio for Max Open Influence would automatically improve the ratio for Densest k-Subgraph. Recall that improving the approximation ratio of Densest k-Subgraph to a constant is a long open standing question. This makes Max Open Influence a problem of particular interest.

Secondly, it turns out that the previous two problems are also known as cardinality constrained problems [START_REF] Cai | Parameterized complexity of cardinality constrained optimization problems[END_REF]. Recall that a problem of this kind asks for finding a solution of k elements that optimizes an objective function. Several classical optimization problems admit a constrained version. For instance, Densest k-Subgraph is the constrained version of the Max Clique problem. From a classical complexity point of view, it is clear that an optimization problem shares the same complexity nature as its constrained version. However, from a parameterized and approximation perspective, this statement does not hold anymore. For example, the cardinality constrained version of Min Vertex Cover, denoted by Max k-Vertex Cover, asks to find a subset of at most k vertices that covers the maximum number of edges. The decision version has been proved W[1]hard with respect to k [START_REF] Cai | Parameterized complexity of cardinality constrained optimization problems[END_REF] while Vertex Cover is well known to be fixed-parameter tractable. Notably, Marx [START_REF] Marx | Parameterized complexity and approximation algorithms[END_REF] established an (1 + ε)-approximation algorithm for Max k-Vertex Cover with running time f (k, ε) • n O (1) for all ε > 0. As a consequence, it seems natural and interesting to ask the existence of an approximation algorithm for a cardinality constrained problem with running time f (k) • n O (1) where k is the number of elements in the solution. However, it comes to us that the study of parameterized approximation of these kind of problems, in a general perspective, is a largely unexplored area. We left this as an interesting research direction.

Finally, it is worth noting that combining the parameterized complexity with approximation, in the context of cardinality constrained problems, can be done in a straightforward way as the number of elements in the solution is a natural parameter candidate. However, the story is not as simple for classical optimization problems since it is not clear Chapter 6. Conclusion what could be the "standard" parameter. One might be tempted to choose the optimum value but this does not seem relevant since it is presumably hard to find it. To overcome this difficulty, the idea is to extend the input with an integer k > 0 (the value that an optimum should reach) which will play the role of the standard parameter. As a matter of fact, there exists several definitions of parameterized approximability based on that. In what follows, we will not review all of them and we refer the reader to the survey of Marx [START_REF] Marx | Parameterized complexity and approximation algorithms[END_REF] for that purpose. Though we will discuss some possible extensions of our work toward that direction. Following the definition of [START_REF] Chen | On parameterized approximability[END_REF], the Min Target Set Selection problem is parameterized α-approximable if the following holds: There exists a function α and an algorithm that, given any instance I = (G, thr) of Min Target Set Selection and an integer k > 0, computes in f (k) • |I| O(1) time a target set S such that |S| ≤ k • α(k) if a target set of size at most k exists; an arbitrary output otherwise. We left open the question whether there exists such algorithm. Up to our knowledge there are not many positive or negative results in the literature for other optimization problems. Marx and Razgon [START_REF] Marx | Constant ratio fixed-parameter approximation of the edge multicut problem[END_REF] provided a parameterized 2-approximation algorithm for Edge Multicut. However, this last problem has been proved fixed-parameter tractable [START_REF] Bousquet | Multicut is FPT[END_REF]. Nevertheless, Marx and Razgon introduced a first W [START_REF] Aazami | Approximation algorithms and hardness for domination with propagation[END_REF]-hard problem which is parameterized constant-approximable. As pointed out in [START_REF] Marx | Completely inapproximable monotone and antimonotone parameterized problems[END_REF][START_REF] Marx | Parameterized complexity and approximation algorithms[END_REF], the existing negative results in the literature [START_REF] Downey | Parameterized approximation of dominating set problems[END_REF][START_REF] Chen | On parameterized approximability[END_REF] essentially rely on the "non monotone" nature of the corresponding problems. A minimization (resp. maximization) problem is said monotone if every superset (resp. subset) of a solution set is still a feasible solution. It follows that if a problem is non monotone then it may happen that all feasible solutions have the same size, thus making the search of approximate solutions as hard as the search of optimal ones. However, Min Target Set Selection is clearly monotone and, as mentioned above, generalizes several classical optimization problems. This makes its parameterized approximability study an other interesting research direction.

Index

Mots clefs:

Optimisation combinatoire, théorie des graphes, algorithmes d'approximation, complexité paramétrée, approximation paramétrée, réseaux sociaux, problème du pompier, selection d'un ensemble cible.

Abstract. In this thesis, we investigate the computational complexity of optimization problems involving a "diffusion process" in a graph. More specifically, we are first interested to the target set selection problem. This problem consists of finding the smallest set of initially "activated" vertices of a graph such that all the other vertices become activated after a finite number of propagation steps. If we modify this process by allowing the possibility of "protecting" a vertex at each step, we end up with the firefighter problem that asks for minimizing the total number of activated vertices by protecting some particular vertices. In fact, we introduce and study a generalized version of this problem where more than one vertex can be protected at each step. We propose several complexity results for these problems from an approximation point of view and a parameterized complexity perspective according to standard parameterizations as well as parameters related to the graph structure.

Keywords: Combinatorial optimization, graph theory, approximation algorithms, parameterized complexity, parameterized approximation, social networks, firefighter problem, target set selection.

◮

  C. Bazgan, M. Chopin and M. R. Fellows, Parameterized complexity of the firefighter problem, Proceedings of the 22nd International Symposium on Algorithms and Computation (ISAAC 2011), LNCS 7074, pp. 643-652, 2011.

Figure 1 . 2 :

 12 Figure 1.2: Initially, Edward and Carol are offered an xphone. Then, by a "word-ofmouth" process, Alice and Dave are convinced (or "activated") and buy an xphone. Since Carol, Edward and now Dave have the mobile phone, Bob decides to get one.

Figure 2 . 1 :

 21 Figure 2.1: Shifting the combinatorial explosion into the parameter.

2 .

 2 For any finite set S ⊆ Σ * the equivalence relation R partitions the elements of S into at most (max x∈S |x|) O(1) classes.

3 .

 3 The size of each solution y ∈ sol(x) is polynomially bounded in |x| i.e. |y| ≤ |x| O(1) .

Definition 24 :

 24 APX, log-APX and poly-APX classes The class APX (resp. log-APX, poly-APX) corresponds to the class F -APX where F = O(1) (resp. O(log(n))

◮

  M. Chopin, A. Nichterlein, R. Niedermeier, and M. Weller, Constant Thresholds Can Make Target Set Selection Tractable, Proceedings of the 1st Mediterranean Conference on Algorithms (MedALG 2012), LNCS 7659, pp. 120-133, 2012. ◮ C. Bazgan, M. Chopin, A. Nichterlein and F. Sikora, Parameterized Approximability of Maximizing the Spread of Influence in Networks, Proceedings of the 19th Annual International Computing and Combinatorics Conference (COCOON 2013), LNCS 7936, pp. 543-554, 2013.

Figure 3 . 1 :

 31 Figure 3.1: Overview of our results for Target Set Selection when parameterized by structural parameters (see Section 2.3.4). The three rectangles below each parameter indicate a result with (from left to right:) constant, majority, and general threshold function. The white text on black background at the parameter "clique cover number" means NP-hard for constant values of this parameter, violet (dark gray) background in the other parameters means W[1]-hard, green (light gray) background means FPT, and white background indicates an open question.

  thr } the number of activation steps before the propagation stops. Notice that r(S) ≤ |V | since at least one vertex is activated at each step until the propagation process stops. Definition 32: Closed activated vertices Let G = (V, E) be a graph, a threshold function thr : V → N, and a set S ⊆ V . The closed activated vertices, denoted by σ G,thr [S], is the set of all activated vertices at the end of the propagation process provided that S is the set of initially activated vertices i.e. σ G,thr [S] = A r(S) G,thr (S). Definition 33: Open activated vertices Let G = (V, E) be a graph, a threshold function thr : V → N, and a set S ⊆ V . The open activated vertices, denoted by σ G,thr (S), is the set of newly activated vertices at the end of the propagation process provided that S is the set of initially activated vertices i.e. σ G,thr (S) = A r(S) G,thr (S) \ S. Definition 34: Target set

Reduction rule 12 (

 12 [START_REF] Nichterlein | On tractable cases of target set selection[END_REF] Reduction Rule 5]) Let (G, thr, k) be an instance of Target Set Selection reduced with respect to Reduction rule 11 and let

  1 a clique. The thresholds are set as follows. For every subset vertex w F i ∈ W F , set the Vertex

Figure 3 . 3 :

 33 Figure 3.3: Graph obtained after carrying out the modifications in the proof of Theorem 16 (with n = 4). The vertices inside an ellipse form a clique. The numbers in the vertices denote the thresholds. If no number is inside a vertex, the threshold is equal to the degree.

  and, thus, G contains at most thr O(2 ℓ thrmax) max ℓ vertices. The Reduction rules 11 and 20 can both be applied exhaustively in O(n + m) time and Reduction rule 23 can be applied exhaustively in O(n 2 ). Overall, the kernelization runs in O(n 2 ) time.

5 Figure 3 . 5 :

 535 Figure 3.5: The graph G ′ (right) obtained from G (left) after carrying out the modifications of Theorem 26. A thick edge represents an ℓ-edge for some ℓ > 0. A "star" vertex v represents a vertex adjacent to deg G ′ (v) 2pending-vertices.

5 Figure 3 . 6 :

 536 Figure 3.6: The graph G ′ (right) obtained from G (left) after carrying out the modifications of Theorem 27. A black arrow from u to v represents a directed edge gadget from u to v.A gray arrow from u to v indicates a directed tree where u is one of the leafs and v is the root.

  |N G[S∪σ(S)] (v)| = ∆ and N G[S∪σ(S)] (v) ⊆ S. Our fpt-algorithm solving Influence runs in two phases: Phase 1: Guess a graph H being isomorphic to G[S ∪ σ(S)].

Algorithm 1

 1 The pseudocode of the algorithm solving the decision problem Influence. The guessing part in the algorithm behind Lemma 38 is used in Line 7 as subroutine. The final check in Line 19 is done by brute force checking all possibilities.

1 : 2 : 3 :

 123 procedure solveInfluence(G, thr, k, ℓ) Guess x ∈ {1, . . . , k} ⊲ x: number of connected components of G[σ[S]] Guess (k 1 , ℓ 1 ), . . . , (k x , ℓ x ) such that x i=1 k i = k and x i=1 ℓ i = ℓ 4:

◮

  C. Bazgan and M. Chopin, The robust set problem: parameterized complexity and approximation, Proceedings of the 37th International Symposium on Mathematical Foundations of Computer Science (MFCS 2012), LNCS 7464, pp. 136-147, 2012.

T 4

 4

Figure 4 . 1 :

 41 Figure 4.1: A non-deterministic multi-tape Turing machine (right) accepting an instance of Harmless Set where k = 3 (left). The guessed solution is the set {v 1 , v 2 , v 4 }. We indicate the thresholds inside the vertices. Now in order to prove the W[2]-hardness of Harmless Set paramterized by k, we construct a simple fpt-reduction from the W[2]-hard problem Red/Blue Dominating Set (see Appendix A). Theorem 42 Harmless Set is W[2]-complete with respect to the parameter k even on bipartite graphs.

2|Hx| max 2

 2 |Hx| n = O(thr 3 tw max n). Final step. The optimal solution is then arg max c∈{0,1} Hr |A r [ t r , c]| where r is the root of T and t r (v) = thr(v) for all v ∈ H r . Using Reduction rule 40 together with Theorem 50, we immediately get the following. Corollary 51 Harmless Set is fixed-parameter tractable with respect to the combined parameter k and the treewidth tw of the input graph. The algorithm runs in O((k + 1) 3 tw • n) time.

( 5 ) 4 . 1

 541 The parameterized approximability of the problem has not been studied and thus, left as an open question. bounded degree and "path-likeness" 75 5.3.1 Graphs of bounded degree 75 5.3.2 Graphs of bounded pathwidth 79 5.3.3 Path-like graphs of bounded degree 87 5.4 Parameterized complexity in the general case 89 5.Firefighter 89 5.4.2 Bounded Firefighter 90 5.4.3 Dual Firefighter 91 5.5 Parameterized algorithms 93 5.5.1 Dual Firefighter parameterized by k b and b 93 5.5.2 Firefighting on trees 94 5.5.3 Firefighting on tree-like graphs 100 5.6 Parameter "vertex cover number" 101 5.7 Approximability 103 5.8 Conclusion and open problems 104

F

◮

  C. Bazgan, M. Chopin and M. R. Fellows, Parameterized complexity of the firefighter problem, Proceedings of the 22nd International Symposium on Algorithms and Computation (ISAAC 2011), LNCS 7074, pp. 643-652, 2011. ◮ C. Bazgan, M. Chopin and B. Ries, The firefighter problem with more than one firefighter on trees, Discrete Applied Mathematics 161(7-8), 899-908, 2013. ◮ C. Bazgan, M. Chopin, M. R. Fellows, F. V. Fomin, E. J. van Leeuwen and M. Cygan, Parameterized Complexity of Firefighting, submitted. ◮ J. Chlebíková and M. Chopin, The Firefighter Problem: A Structural Analysis, ongoing paper.

Figure 5 . 3 :

 53 Figure 5.3: In the figure (a) the applied strategy consists of protecting a vertex adjacent to the fire at each step. The strategy of figure (b) protects vertex v 3 at time step one and v 4 at time step two. The later strategy save one more vertex than the former one. The burned vertices are represented by black vertices. A protected vertex is represented by a circle with a cross inside.

Figure 5 . 4 :Theorem 60

 5460 Figure 5.4: Moving up a firefighter leads to a strategy that saves at least the same set of leaves.

Figure 5 . 5 :

 55 Figure 5.5: The construction of T ′ . The vertices from C are represented by a square.

  .6). For every leaf ℓ of T , add b + 2 copies T 1,ℓ , . . . , T b+2,ℓ of the tree T (r, ⌈log b+1 n + 1⌉, b + 1) such that the root r i,ℓ of T i,ℓ is adjacent to ℓ, for i ∈ {1, . . . , b + 2}. Let |T | denote the cardinality of each of those trees. Notice that each tree T i,ℓ has |T | ≥ n vertices. Set k = (b + 2)|C||T |.

Figure 5 . 6 :

 56 Figure 5.6: Construction of T ′ from the tree T for the case b = 1.

Figure 5 . 7 :

 57 Figure 5.7: An example of part of a tree constructed from the formula φ= (x 3 ∨ x 4 ∨ x 5 ) ∧ (x 1 ∨ x 2 ∨ x 3 ) ∧ (x 2 ∨ x 4 ∨ x 5 ) ∧ (x 1 ∨ x 3 ∨ x 6 ) ∧ (x 1 ∨ x 4 ∨ x 6 ) ∧ (x 2 ∨ x 6 ∨ x 5). Guard vertices are represented by a dot within a circle.

Figure 5 . 8 :

 58 Figure 5.8: A k-caterpillar for which the degree greedy algorithm (left) saves less vertices than the optimal solution (right).

Figure 5 .

 5 Figure 5.11: A linear layout of a graph of cutwidth two. Dashed ellipses represent the bubbles associated to an initially burned vertex s.

Figure 5 . 12 :

 512 Figure 5.12: An instance of Clique and the corresponding graph G ′ constructed in the proof of Theorem 77 for k = 3 and b = 1.

Theorem 83

 83 Dual Firefighter is W[1]-hard with respect to the parameter k b and cannot be solved in n o(k b ) time even on bipartite graphs unless ETH fails. Proof. We construct an fpt-reduction from the Clique problem (see Appendix A) on regular graphs to Dual Firefighter. Let (G, k) be an instance of Clique where G is a ∆-regular graph. We construct the instance (G ′ , s, b, k b ) of Dual Firefighter from (G, k) as follows. Add a new vertex s adjacent to all vertices of G. Set k b = k + 1 and b = b 1 + b 2 where b1 = k(nk) and b 2 = k∆ -k 2 .In what follows, attaching a guard vertex g to a vertex v means making a copy of a star with center g and b + k b leaves such that g is made adjacent to v. Thus, if we want to burn at most k b vertices then a guard vertex needs to be saved or protected. Now, attach b -(nk) guard vertices to s and nk guard vertices to every vertex in V as well. Remove every edge uv ∈ E and add an edge-vertex e uv adjacent to u and v (see Figure5.13). Notice that, at time step 1, there are only nk firefighters that can be placed freely because of the guard vertices.We claim that (G, k) is a yes-instance if and only if (G ′ , s, k b , b) is a yes-instance. "⇒": Suppose that we have a clique C ⊆ V of size k and consider the following strategy. At time step one, the strategy uses the n-k remaining firefighters to protect all the original vertices V in G ′ except those in the clique C. At time step two, all the k vertices of C are burned. Since there are nk guard vertices attached to each vertex in C, we need to protect b 1 = k(nk) vertices. Moreover, there are k∆ -k 2 edge-vertices adjacent to the vertices in the clique C. Since there remain b 2 = bb 1 firefighters, we can protect all of them. Hence, no more than k + 1 = k b vertices are burned at the end of the process.

Figure 5 . 13 :

 513 Figure 5.13: Construction of G ′ (right) from a regular graph of degree three (left). In this example, we have k = 3, n = 4, ∆ = 3, b 1 = 3, b 2 = 6 and b = 9. Guard vertices are represented by a dot within a circle.

5. 5 . 1

 51 Dual Firefighter parameterized by k b and b In contrast to what we have previously shown, we prove that Dual Firefighter becomes fixed-parameter tractable if we combine the parameter k b with the budget b. Theorem 84 Dual Firefighter is fixed-parameter tractable with respect to the combined parameter k b and b. It can be solved in O((2 b+1 -1) k b +b-1 n) time.

Corollary 85

 85 Dual Firefighter is fixed-parameter tractable with respect to the combined parameter k b and "maximum degree" ∆ of the input graph. It can be solved in O((2 ∆+1 -1) k b +∆-1 n) time. Proof. Let (G, s, b, k b ) be an instance of Dual Firefighter where G has maximum degree ∆. If b ≥ ∆ then protect all the vertices in N (s) at time step one. Otherwise, apply the algorithm from Theorem 84 that runs in O((2 ∆+1 -1) k b +∆-1 n) since b < ∆.

  Firefighter in time O(4 k + n), which can be derandomized to a O(n + 2 O(k) )-time algorithm. They also gave a polynomial kernel with respect to the parameter k. • A randomized algorithm for Dual Firefighter of running time O(4 k b n), which can be derandomized to a O(2 O(k b ) n log n)-time algorithm. They left as an open problem whether Dual Firefighter has a polynomial kernel with respect to the parameter k b . • For Bounded Firefighter, they gave a randomized algorithm of running time O(k O(kp) p n), which can be derandomized to a O(k O(kp) p n log n)-time algorithm. They left open whether the problem has a polynomial kernel with respect to the parameter k p , and asked whether there is an algorithm solving the problem in time 2 o(kp log kp) n O(1) . In what follows, we give a deterministic O((b + 1) k b n)-time algorithm for Dual Firefighter on trees which improves the O(4 k n) running time of the randomized algorithm from [27] for b = 1. We also answer the open question of Cai et al. by showing that Dual Firefighter parameterized by k b has no polynomial kernel on trees of maximum vertex degree four for b = 1, and no polynomial kernel on trees of maximum vertex degree b + 4 for any b ≥ 2. We provide a deterministic algorithm solving Bounded Firefighter on trees in O((b + 1) kp/b+1 k p n) time. This answers an open question of Cai et al. since it gives us a O(2 kp k p n)-time algorithm when b = 1. Furthermore, we show that the problem has no polynomial kernel with respect to the parameter k p on trees. This last result was independently obtained by Yang [104]. Based on the parameterized algorithm, we also give an exact subexponential-time algorithm, for the classical firefighter problem (i.e. the Firefighter problem with budget b = 1) on a tree in O((b + 1) √ 2n/b n 3/2 ) time, thus improving on the 2 O( √ n log n) running time from [27, 56] for b = 1. Finally, we show a deterministic algorithm solving Firefighter on trees in O((b + 1) k/b+3 kn) time, improving the running time O(4 k + n) of the randomized algorithm from [27] for b = 1. Dual Firefighter. Using a simple branching algorithm we get the following result. Theorem 86 Dual Firefighter is solvable in O((b + 1) k b n) time on trees.

Firefighter.

  Using the dynamic programming algorithm of Theorem 89, we can easily prove the following.Corollary 93 Firefighter is solvable in O((b + 1) (k/b)+2 kn) time on trees.Proof. Let (T, s, b, k) be an instance of Firefighter on a tree T . Using the same argument from Theorem 76, we run the dynamic programming algorithm for Max Bounded Firefighter (see Theorem 89) for all k p = 1, . . . , k. Furthermore, we note thatk i=1 ((b + 1) (i/b)+1 in) ≤ kn(b + 1) k i=1 (b + 1) i/b ≤ (b + 1) k/b+2 kn,implying that the worst-case running time is O((b + 1) k/b+2 kn).

Lemma 95

 95 If a vertex at level ℓ burns in an optimum strategy for an instance of the Max Firefighter problem on trees, then at least b 2 (ℓ 2 + ℓ) vertices are saved. Proof. Let (T, s, b) be an instance of the Max Firefighter problem on trees, and let v be a vertex of level ℓ that burns in an optimum strategy. Then the strategy protects b vertices at level ℓ, and by Lemma 58 it thus protects b vertices p 1 i , . . . , p b i at each level i for 1 ≤ i ≤ ℓ. For any i, each of the subtrees rooted at p 1 i , . . . , p b i should contain at least ℓi + 1 vertices, or it would have been better to protect the vertex at level i that is on the path from v to s. But then the strategy saves at least b ℓ i=1 (ℓi + 1) = b 2 (ℓ 2 + ℓ) vertices. Theorem 96 Max Firefighter is solvable in O((b + 1) √ 2n/b n 3/2 ) time on trees. Proof. Let (T, s, b) be an instance of the Max Firefighter problem on trees. Suppose that a vertex v at level 2n/b burns in an optimum strategy. Then, by Lemma 95, the strategy saves at least n + bn/2 > n vertices, which is not possible. It follows that all vertices at level 2n/b are saved in any optimum strategy. Since in any optimum strategy every protected vertex has a burned ancestor by Lemma 58, all protected vertices are at level at most 2n/b. Hence there is an optimum strategy that protects at most b 2n/b = √ 2bn vertices, and we can find the optimum strategy by running the algorithm of Theorem 89 with k p = √ 2bn.

Proof.

  Since Firefighter fpt-reduces with respect to the parameter vertex cover number to Dual Firefighter by setting k b = nk, we will prove the result for Dual Firefighter for convenience.Let (G = (V, E), s, b, k b ) be an instance of Dual Firefighter with |V | = n. We may assume that |N (s)| < b + k b , because otherwise the answer is clearly "no". First, we compute in polynomial-time a 2-approximate vertex cover C ′ ⊆ V where|C ′ | = τ ′ ≤ 2τ .If b ≥ τ ′ then the answer is "yes". Indeed, suppose that b ≥ τ ′ . If s / ∈ C ′ , then it is enough to protect all the vertices in C ′ at the first step to stop the fire.If s ∈ C ′ , then let N 1 (s) = N (s) ∩ C ′ and N 2 (s) = N (s) \ C ′ . If |N 2 (s)| < k b ,then the strategy that protects all the vertices in C ′ at the first step will result in at most k b burned vertices in G. If |N 2 (s)| ≥ k b , then consider the following strategy. At the first time step, protect all the vertices in N 1 (s), and b -|N 1 (s)| vertices of N 2 (s). Notice that |N 1 (s)| ≤ b since |N (s)| ≤ b + k b . At the second time step, protect all the vertices in C ′ \ N 1 [s]. Hence, the number of burned vertices using this strategy is at most k b .

Proof. 1 . 1 rFigure 5 . 14 :

 11514 Figure 5.14: Instance where the degree greedy algorithm gives no approximation guarantee for the case b = 1. Since here S = V , we did not represent vertices in S by squares.

( 8 )

 8 fighter do not admit a polynomial kernel with respect to the combined parameter k p and tw, and k b and tw, respectively, we do not know if it is also the case for Firefighter parameterized by k and tw.(8) We have proved that Firefighter and Dual Firefighter have no n o( √ k)

Table 1 :

 1 Tableau regroupant nos résultats d'approximation pour max open influence et max closed influence avec des seuils constants (Cons.), majoritaires (Maj.) et unanimes (Una.). Les résulats d'approximation paramétrés sont établis par rapport au paramètre k pour les deux problèmes. La valeur n correspond à la taille du graphe. ◮ M. Chopin, A. Nichterlein, R. Niedermeier, and M. Weller, Constant Thresholds Can Make Target Set Selection Tractable, Proceedings of the 1st Mediterranean Conference on Algorithms (MedALG 2012), LNCS 7659, pp. 120-133, 2012. ◮ C. Bazgan, M. Chopin, A. Nichterlein and F. Sikora, Parameterized Approximability of Maximizing the Spread of Influence in Networks, Proceedings of the 19th Annual International Computing and Combinatorics Conference (COCOON 2013), LNCS 7936, pp. 543-554, 2013.

	1-ε ,∀ε > 0
	(Th.26)

Table 2 :

 2 Nos résultats de complexité paramétrée pour harmless set et son dual dual harmless set où l'objectif est de trouver un ensemble inoffensif de taille au moins nk où n est la taille du graphe. Le paramètre est k pour les deux problèmes. Le résultat marqué par Proceedings of the 37th International Symposium on Mathematical Foundations of Computer Science (MFCS 2012), LNCS 7464, pp. 136-147, 2012.

		harmless set	dual harmless set
	General	W[2]-complete (Th.42)	W[2]-hard
	Constants	W[1]-complete (Th.44)	FPT (Th.48)
	Majoritaires	W[1]-hard (Th.44)	FPT (Th.48)
	Unanimes	FPT (Th.46)	W[2]-hard *

* est dû à l'équivalence entre dual harmless set et le problème total dominating set qui a été montré W[2]-hard

[START_REF] Golovach | Parameterized complexity of generalized domination problems[END_REF]

Les résultats présentés dans cette section sont basés sur le papier suivant: ◮ C. Bazgan and M. Chopin, The robust set problem: parameterized complexity and approximation,

  Overview of the relations between some structural graph parameters, see the paper of Komusiewicz and Niedermeier[START_REF] Komusiewicz | New races in parameterized algorithmics[END_REF] for more details. An arc from a parameter k 2 to a parameter k 1 means that there exists a constant c > 0 such that k 1 ≤ c • k 2 . Given a graph G, the "Distance to C" parameter where C is a class of graphs, corresponds to the minimum number of vertices to remove from G in order to obtain a new graph that belongs to C.

		Distance to Clique	Vertex Cover	Cluster Editing	Max Leaf Number
	Minimum Clique Cover	Distance to Co-Cluster	Distance to Cluster	Distance to Disjoint Paths	Edge Set Feedback	Bandwidth
							Maximum
	Maximum Independent Set	Distance to Cograph	Distance to Interval	Feedback Vertex Set	Pathwidth	Degree
	Minimum Dominating Set	Chordal Distance to		Bipartite Distance to	Outerplanar Distance to		h-index
	Max Diameter of Components	Perfect Distance to				Treewidth
							Degeneracy
	Min Diameter
	of Components					Chromatic Number	Minimum Degree
	Girth
					Maximum Clique	Distance to Disconnected	Domatic Number
	Figure 2.2:

  Clearly, the loop in Line 13 needs O(x) ≤ O(k) time. Furthermore, the loop in Line 16 needs O(k • n) time. For the checking in Line 19 observe the following. After deleting the vertices in the loop in Line 16 the remaining graph can have at most i∈X c i ≤ x • 2 • x • ∆ 4k vertices. Furthermore, i∈X k i ≤ k and, thus, there are at most (2 • x 2 • ∆ 4k ) k candidate subsets for the solution set S. Checking whether i∈X k i chosen vertices activate i∈X ℓ i other vertices can be done in (2 • x 2 • ∆ 4k ) 2 time. Hence, the checking in Line 19 can be done in ∆ O(k 2 ) time. Putting all together we arrive at a running time of (k∆

Table 4 .

 4 1: Our parameterized complexity results for Harmless Set and its parametric dual Dual Harmless Set where the parameter is k for both problems. The result marked with * is due to the equivalence between Dual Harmless Set and the Total Dominating Set problem proved W[2]-hard for parameter k in[START_REF] Golovach | Parameterized complexity of generalized domination problems[END_REF].

	Set	Dual Harmless Set

  2•cw(H) (s). First, merge B 2•cw(H) (s) with every other bubble B 2•cw H (s ′ ) with s ′ ∈ F that possibly overlap into a new one B ′ 2•cw(H) (s). By definition, we know that the number of edges with an endpoint in B ′ 2•cw(H) (s) and the other one in V \B ′ 2•cw(H) (s) is less or equal to 2•cw(H). Thus, we define the strategy that consists in protecting one vertex v ∈ V \ B ′ 2•cw(H) (s) adjacent to a vertex in B ′ 2•cw(H) (s) at each step t = 1, . . . , 2 • cw(H). Let F ′ be the set of vertices burned at step 2 • cw(H). Since ∆(H) ≤ 2 • cw(H), we deduce that |F ′ | is less or equal

  Résumé. Dans cette thèse, nous étudions la complexité algorithmique de problèmes d'optimisation impliquant un processus de diffusion dans un graphe. Plus précisément, nous nous intéressons tout d'abord au problème de sélection d'un ensemble cible. Ce problème consiste à trouver le plus petit ensemble de sommets d'un graphe à "activer" au départ tel que tous les autres sommets soient activés après un nombre fini d'étapes de propagation. Si nous modifions ce processus en permettant de "protéger" un sommet à chaque étape, nous obtenons le problème du pompier dont le but est de minimiser le nombre total de sommets activés en protégeant certains sommets. Dans ce travail, nous introduisons et étudions une version généralisée de ce problème dans laquelle plus d'un sommet peut être protégé à chaque étape. Nous proposons plusieurs résultats de complexité pour ces problèmes à la fois du point de vue de l'approximation mais également de la complexité paramétrée selon des paramètres standards ainsi que des paramètres liés à la structure du graphe.

		Index
	neighborhood	treewidth, 8
	close, 6	local, 8
	open, 6	vertex
	optimization problem, 17	activated, 29
	parameterized, 20	burned, 73
	APX, 19 log-, 19 parameterization standard, 11 poly-, 19 FPT, 12 XP, 12 protection strategy, 73 W[2], 16 constrained, 73 W[t], 15 protection phase, 72 W[1], 16 polynomial equivalence relation, 14 PTAS, 19 performance ratio, 19 PO, 18 pathwidth, 9 P, 10 nice, 9 NPO, 18 path decomposition, 9 hard, 10 problem, 11 complete, 10 complexity, 10 NP, 10 parameterized structural, 11	no-instance, 10 yes-instance, 10 protected, 73 saved, 73 complete, 7 cograph, 7 clique, see complete k-caterpillar, 7 caterpillar, 7 bipartite, 7 graph, 5 tion fpt-reduction, see parameterized reduc-fixed-parameter tractable, 12 error, 19 word, see string
	reduction activated vertices E-, 22 closed, 29 L-, 22 open, 29 gap-introducing, 20 alphabet, 9 parameterized, 21 approximation parameterized, 14 fpt-time, 20 polynomial-time, 10 polynomial-time, 19 approximation algorithm, 19 rule, 13	cycle, 5 degree, 6 maximum, 6 diameter, 5 digraph, see directed directed, 6 false twins, see twins in-neighbors, 6
	approximation scheme polynomial-time, 19 bandwidth, 6 boolean circuit, 15 depth, 15 spreading phase, 72 string, 9 length, 9 subgraph, 5 induced, 5	interval, 7 out-neighbors, 6 path, 5 regular, 7 ∆-regular, 7 star, 7
	gate, 15 large, 15 small, 15 target set, 29 tree, 7 t-ary, 7 input, 15 output, 15 ancestor, 7 child, 7 truth assignment, 15 (Hamming) weight, 15 weft, 15 budget, 72 complete, 7 descendant, 7 full, 7 height, 7	k-star, 7 true twins, see twins twins, 5 harmless, 57 kernel, 12 polynomial, 12 kernelization, 12
	leaf, 7 cross-composition, 14 level, 7 cutwidth, 5 rooted, 7	language, 9 linear layout, 5
	decision problem, 9 tree decomposition, 7	monadic second order logic, 13
	instance, 10 nice, 8	MSOL, see monadic second order logic

As hinted in Section

2.2, we do not discuss how to actually encode objects into strings of Σ * .

A graph is a cluster graph if and only if it contains no induced P3, that is, an induced path of three vertices. Using this characterization, the factor 3-approximation simply deletes all vertices occurring in an induced P3.

A P-tree[START_REF] Macgillivray | On the firefighter problem[END_REF] is a tree which does not contain the following configuration.level i level i + 1 level i +

Observe that in the i-th step we are allowed to protect at most min(a, r) vertices.

Remerciements

Definition 37: Harmless vertices

Let G = (V, E) be a graph, and thr : V → N a threshold function. A subset V ′ ⊆ V is called harmless if for every vertex v ∈ V ,

We define in the following the problems we study in this chapter.

Harmless Set

Input: A graph G = (V, E), a threshold function thr : V → N where 1 ≤ thr(v) ≤ deg(v) for every v ∈ V , and an integer k. Question: Is there a harmless set V ′ ⊆ V of size at least k?

We also consider the parametric dual problem Dual Harmless Set which asks for the existence of a harmless set of size at least nk where n denotes the number of vertices in the input graph.

The optimization version of Harmless Set is defined as follows.

Max Harmless Set

Input: A graph G = (V, E) and a threshold function thr :

As in the previous chapter, we consider majority thresholds i.e. thr(v) = ⌈ deg(v) 2 ⌉ for each vertex v, unanimity thresholds i.e. thr(v) = deg(v) for each vertex v, and constant thresholds i.e. thr(v) ≤ c for each vertex v and some constant c > 1.

Parameterized complexity

In this section, we consider the parameterized complexity of Harmless Set. In some reductions we make use of the following gadget: a forbidden edge denotes an edge uv where both vertices have threshold one. Attaching a forbidden edge to a vertex w means to create a forbidden edge uv and make w adjacent to u. Notice that none of the three vertices u, v or w can be part of a harmless set. Moreover, we need the following simple but useful reduction rule.

Reduction rule 40 Let (G, thr, k) be an instance of Harmless Set. If there is a vertex v such that thr(v) > k + 1 then set the threshold thr(v) to k + 1 to get a new equivalent instance (G, thr ′ , k).

To see that the above rule is correct, observe that if S ⊆ V is a harmless set of size at least k for (G, thr, k), then any subset of size k of S is a harmless set for (G, thr ′ , k). The converse is clear.

We now show that Harmless Set parameterized by k belongs to W[2] using the Turing way, that is, we reduce Harmless Set to the Short Multi-tape Nondeterministic Turing Machine problem (see Section 2.3.3).

Theorem 41 Harmless Set is in W [START_REF] Ackerman | Combinatorial model and bounds for target set selection[END_REF] with respect to the parameter k.

Proof. We construct an fpt-reduction from Harmless Set to Short Multi-tape Nondeterministic Turing Machine as follows. Let (G, thr, k) be an instance of Harmless Set with G = (V, E) and V = {v 1 , . . . , v n }. First, exhaustively apply simply do not put an arc from L i to the corresponding vertices C j in G ′ .

Consider now a k-caterpillar G = (V, E). Let P be the path in the caterpillar from which G has been obtained, which is induced by vertices of degree at least two. We will call P the spine of the k-caterpillar.

We are now ready to prove the main result of this section.

Theorem 68 For any k ≥ 1 and any budget b ≥ 1, Max 0-1 Firefighter is polynomial-time solvable for k-caterpillars.

Proof. Let G = (V, E) be a k-caterpillar and let P = {v 1 v 2 , v 2 v 3 , . . . , v p-1 v p } be the spine of G. First assume that s is a vertex of P , say s = v i , i ∈ {1, . . . , p}.

It follows from Lemma 58 that we may assume that at most one vertex is protected in P 1 and at most one vertex is protected in P 2 . Consider a strategy in which we decide to protect exactly two vertices of P , say vertex v j , for j ∈ {1, . . . , i -1} and vertex v q , for q ∈ {i + 1, . . . , p}. We may assume that v j is protected during time step ij and vertex v q is protected during time step qi (see Lemma 58). Notice that the vertices v j+1 , . . . , v i-1 , v i+1 , . . . , v q-1 will not be protected in this strategy. Construct a (k + p)-star G ′ as follows (see Figure 5.10):

• Delete all vertices v 1 , . . . , v j as well as the legs at these vertices (all these vertices are saved in our strategy).

• Delete all vertices v q , . . . , v p as well as the legs at these vertices (all these vertices are saved in our strategy).

• Delete all edges of P .

• For every r ∈ {j+1, . . . , i-1, i+1, . . . , q-1}, let u r 1 , . . . , u r deg G (vr)-2 be the neighbors of v r not belonging to P ; delete v r and replace it by deg

• Join every vertex v r ℓ , for r ∈ {j + 1, . . . , i -1} and ℓ ∈ {1, . . . , deg G (v r ) -2}, to v i by a path P rℓ of length ir.

• Join every vertex v r ℓ , for r ∈ {i + 1, . . . , q -1} and ℓ ∈ {1, . . . , deg G (v r ) -2}, to v i by a path P rℓ of length ri.

From the above construction it follows that G ′ is a (k + p)-star with center v i . Now in order to solve our initial problem, we need to solve Max 0-1 Firefighter in G ′ with the following additional constraints: for every r ∈ {j + 1, . . . , i -1, i + 1, . . . , q -1} and every ℓ ∈ {1, . . . , deg G (v r ) -2} we are not allowed to protect the vertices of V (P rℓ ). Indeed, since we decided to protect v j and v q , the vertices v j+1 , . . . , v i-1 , v i+1 , . . . , v q-1 will not be saved. Notice that these vertices are represented by the vertices of paths P rℓ in G ′ . Moreover, if ij = qj then at time steps ij and qj only b -1 firefighters are available (since we protect v j and v q at these time steps); if ij = qj then only b -2 firefighters are available at time step ij. It follows from Theorem 67 and previous discussion that this problem can be solved in polynomial time.

Since the number of choices of a pair of vertices (v j , v q ) to be protected on P is (i -1) × (pi), we can determine in polynomial time the best strategy to adopt if we want to protect exactly two vertices on P . Notice that a similar procedure to the one described above can be used if we decide to protect exactly one vertex on P respectively if we decide not to protect any vertex of P . Clearly the number of choices of exactly one vertex v j , j ∈ {1, . . . , i -1, i + 1, . . . , p}, to be protected on P is p -1. Thus we conclude that if s ∈ V (P ) we can determine an optimal strategy in polynomial time.

It remains the case when s ∈ V (P ). Similar to the proof of Theorem 67, we will distinguish several cases. The case when s has degree one is trivial. Thus we may assume now that deg G (s) = 2. If b ≥ 2, we are done. Thus we may assume now that b = 1. Let S be the set of vertices with weight equal to one. If both neighbors of s are in S, then the optimal solution value is clearly |S| -1. If both neighbors of s are not in S, then the optimal solution value is clearly |S|. Hence the only case remaining is when exactly one neighbor of s is in S. Let u 1 , u 2 be the neighbors of s such that u 1 ∈ S, u 2 ∈ S. If u 2 ∈ V (P ), the optimal solution value is clearly |S|. Thus we may assume now that u 2 ∈ V (P ). In this case we have to compare the value of two solutions: (i) |S| -1 which is the value of the solution obtained by protecting first u 2 and then, during the second time step, we protect the neighbor of u 1 which is not s (if it exists); (ii) the value of the solution obtained by protecting first u 1 and then applying our algorithm described above to the graph G -(Q \ {u 2 }), where Q is the set of vertices of the unique maximal path starting at u 2 and containing u 1 .

Figure 5.10: The construction of G ′ with i = 5, j = 2, and q = 7. The squares correspond to vertices with weight one.

Extension to general weights. We would like to mention that some of our positive results (Theorem 64, Theorem 67, and Theorem 68) may be generalized to a weighted version.

Using the strategy in the proof of Theorem 64, if we direct the fire to a vertex of degree at most b + 1 such that the total weight of the burned vertices is minimum then we get the following result.

Theorem 69 For any budget b ≥ 1, Max Weighted Firefighter is polynomialtime solvable for trees of maximum degree b + 2 if the fire breaks out at a vertex of degree at most b + 1. Now by replacing the costs β(L i , C j ) in the proof of Theorem 67 by β(L i , C j ) = -| v∈S ij w(v)| and adapting the case when s is not the center of G according to the weights, it is not difficult to see that we obtain the following.

Theorem 70 For any k ≥ 1 and any budget b ≥ 1, Max Weighted Firefighter is polynomial-time solvable for k-stars.

Theorem 73 Firefighter is fixed-parameter tractable with respect to the combined parameter "pathwidth" and "maximum degree" of the input graph.

From the claim in Theorem 73, we easily deduce the following.

Corollary 74 Firefighter is fixed-parameter tractable with respect to the parameter "cutwidth" of the input graph.

Finally, since for any graph G it holds that cw(G) ≤ bw(G)(bw(G)+1) 2 [START_REF] Hans | Classes of graphs with bounded tree-width[END_REF], we have the following result.

Corollary 75 Firefighter is fixed-parameter tractable with respect to the parameter "bandwidth" of the input graph.

Parameterized complexity in the general case

In this section, we give upper and lower bounds on parameterized complexity of the following three problems: Firefighter parameterized by k, Bounded Firefighter parameterized by k p , and Dual Firefighter parameterized by k b .

Firefighter

In the following, we will give the parameterized complexity upper and lower bounds of Firefighter with respect to its standard parameterization.

Proof. Let (G, s, b, k) be an instance of Firefighter. We run the n O(kp) -time algorithm of Theorem 79 for all k p = 1, . . . , k. Observe that it is possible to save k vertices of the graph if and only if the algorithm saves at least k vertices for some value of k p . This implies a running time of n O(k) .

Next, we prove in the following that the above algorithm is nearly optimal. Recall that ETH (Exponential-Time Hypothesis) is an assumption stating that the 3-Sat problem cannot be solved in subexponential time in the worst case [START_REF] Impagliazzo | The Complexity of k-SAT[END_REF]. Proof. We construct an fpt-reduction from Clique to Firefighter as follows. Let (G = (V, E), k) be an instance of Clique. We construct the instance (G ′ , s, b, k ′ ) of Firefighter as follows. (see Figure 5.12).

• For each edge uv ∈ E, we add a vertex e uv ; this set of vertices is denoted by F .

• Add an edge from e uv to both u h and v h for each uv ∈ E and each h = 1, . . . , b.

• Add a vertex s, and add vertices a i,j for all 1 ≤ i ≤ k-1 and 1 ≤ j ≤ (k-1)b+1.

This implies an O(2 √ 2n n 3/2 )-time algorithm for the classical firefighter problem (b = 1) on trees.

Firefighting on tree-like graphs

We generalize the above results by showing that Bounded Firefighter (resp. Firefighter) is fixed-parameter tractable when parameterized by k p (resp. k) and the treewidth of the underlying graph. To this end, we use Monadic Second Order Logic (see Section 2.3.1). As a matter of fact, we show that Max Bounded Firefighter can be solved in f (tw, k p )•n O (1) time where tw is the treewidth of the input graph. To do that, we actually need Linear Extended MSOL (LEMSOL) [START_REF] Arnborg | Easy problems for tree-decomposable graphs[END_REF], which allows the maximization over a linear combination of the size of unbound set variables in the MSOL formula. (The definition of LEMSOL in [START_REF] Arnborg | Easy problems for tree-decomposable graphs[END_REF] is slightly more general, but this suffices for our purposes.) 1) time where tw is the treewidth of the input graph and f is a function that solely depends on tw and k p .

Proof. Let (G, s, b, k p ) be an instance of Max Bounded Firefighter such that the treewidth of G is tw. Recall that we assumed b ≤ k p (see the discussion in Section 5.2). Use Bodlaender's Algorithm [START_REF] Bodlaender | A linear-time algorithm for finding tree-decompositions of small treewidth[END_REF] to find a tree decomposition of G of width at most tw. Consider the following MSOL formulae.

This expresses that if the vertices of B i-1 are burning by time step i -1, then the vertices of B i burn by time step i, assuming that vertices p 1 1 , . . . , p b 1 1 , . . . , p 1 i , . . . , p b i i have been protected so far, with b j ≤ b for j = 1, . . . , i.

Saved(S, B, p

This expresses that S is a set of saved vertices when B is a set of burned vertices and vertices p 1 1 , . . . , p b 1 1 , . . . , p 1 m , . . . , p bm m are protected, with b j ≤ b for j = 1, . . . , m.

∧ Saved(S, B, p

A Compendium of problems

We give here the list of problems that are used in this document.

Clique

Input: A graph G = (V, E) and a positive integer k. Question: Is there a clique C ⊆ V of size at least k?

Cubic Monotone 1-In-3-Sat Input: A CNF formula in which every clause contains exactly and only three positive literals and every variable appears in exactly three clauses. Question: Is there a truth assignment to the variables such that each clause has exactly one true literal (such assignment is called a satisfying assignment)?

Densest k-Subgraph Input: A graph G = (V, E) and a positive integer k.

Output: A subset S ⊆ V of cardinality k that induces a maximum number of edges.

Dominating Set

Input: A graph G = (V, E) and an integer k. Question: Is there a subset S ⊆ V , |S| ≤ k, such that for every vertex v ∈ V \ S there is a vertex u ∈ S with uv ∈ E?

Hitting Set

Input: A collection F of subsets of a finite set U and an integer k ≥ 0. Question: Is there a subset U ′ ⊆ U with U ′ ≤ k such that U ′ contains at least one element from each subset in F?

Min Cost Flow

Output: A feasible flow of minimum cost i.e. a function f : A → N that minimizes uv∈A f (uv) • β(uv) such that uv∈A f (uv) -vu∈A f (vu) ≥ γ(u) for all u ∈ V and f (uv) ≤ α(uv) for all uv ∈ A.

Appendix A. Compendium of problems

Min Vertex Cover

Input: A graph G = (V, E). Output: Find a subset S ⊆ V of minimum size such that for all uv ∈ E we have u ∈ S or v ∈ S.

Max Independent Set

Input: A graph G = (V, E). Output: Find a subset S ⊆ V of maximum size such that for every u, v ∈ S we have uv ∈ E.

Max Clique

Input: A graph G = (V, E). Output: Find a clique C ⊆ V of maximum size.

Max E2Sat-3

Input: A CNF formula in which every clause contains exactly two literals and every variable appears in exactly three clauses. Output: Find an assignment to the variables satisfying a maximum number of clauses.

Multicolored Clique

Input: A graph G = (V, E), an integer k, and a coloring col : V → {1, . . . , k}. Question: Does G contain a multicolored clique of size k, that is, a vertex subset V ′ ⊆ V with |V ′ | = k such that for all u, v ∈ V ′ it holds that uv ∈ E and col(u) = col(v)?

Red/Blue Dominating Set Input: A bipartite graph G = (R ∪ B, E) and a positive integer k. Question: Is there a set R ′ ⊆ R of cardinality k such that every vertex in B has at least one neighbor in R ′ ?

Vertex Cover

Input: A graph G = (V, E) and an integer k. Question: Is there a subset S ⊆ V , |S| ≤ k, such that for all uv ∈ E we have u ∈ S or v ∈ S?