Finding e cient codes is crucial for achieving reliable communication over noisy channels. While error-correcting codes for i.i.d. channels are well understood, the task of approximating the best code for a general channel, i.e. that maximizes the success probability of the encoding and decoding process, is much less developed. This thesis addresses this coding problem in several scenarios.

For point-to-point channels, optimal approximation algorithms for coding and list-decoding are known. We study a generalization of the latter with a variable list size, compensated by an increasing probability of failure. More generally, we study the coverage problem which entails maximizing a∈ [n] w a ϕ(|{i ∈ S : a ∈ T i }|) over subsets S ⊆ [m] of cardinality k, for a general nondecreasing concave function ϕ. We provide an approximation algorithm for that problem of ratio

ϕ(E[Poi(x)]) , which cannot be improved for sublinear ϕ if P = NP.

For multiple-access channels, we show that the coding problem cannot be approximated within any constant ratio under a complexity hypothesis on random k-SAT formulas. Generalizing and abstracting quantum entanglement, non-signaling correlations can be used to enhance communication. We show that optimal non-signaling assisted codes for multiple-access channels can be found in polynomial time in the number of copies of the channel. Applied to the binary adder channel, a non-signaling advantage on its capacity region is established. We provide a general single-letter outer bound on the non-signaling capacity region. When non-signaling assistance is not shared between encoders, we show that the capacity region is not changed.

Je tiens tout d'abord à remercier mon directeur de thèse Omar Fawzi, qui m'a accompagné durant toute la durée de ma thèse et m'a grati é de sa présence et de sa disponibilité tout au long de celle-ci.

Je tiens à remercier Mario Berta et Andrei Romashchenko d'avoir accepté de participer à mon jury de thèse en tant que rapporteurs, ainsi qu'à Jean-Marie Gorce et Alantha Newman en tant qu'examinateur et examinatrice.

Je tiens également à remercier Siddharth Barman pour les collaborations ainsi que pour les discussions autour des problèmes algorithmiques venant de l'économie.

Merci Stéphan et Pascal pour le temps passé tout au long de cette thèse autour de moments conviviaux et scienti ques. Un grand merci Subhayan pour la bonne humeur à toute épreuve, la compagnie dans l'e ort et l'aide généreuse fournie durant toutes ces années. Merci Rémi pour ces débats endiablés qui ont rythmé les semaines et brisé tout risque de monotonie au l des mois. Merci Peter pour toute l'aide scienti que et l'accueil chaleureux au début de ma thèse. Et merci Julien pour les pauses caféinées et cartonnées nécessaires à la recharge des batteries.

Merci Maïwenn de m'avoir convaincu de me lancer dans cette thèse malgré les hésitations initiales. Merci Lilian d'avoir été présent tout au long de la thèse, autour de restaurants, de musique, de cafés et débats partagés par delà nos laboratoires respectifs.

Un merci très spécial à Léo pour la décennie d'amitié et de soutien réciproque depuis notre rencontre à l

For broadcast channels, when restricted to deterministic channels, we provide a (1 -e -1 ) 2approximation algorithm for the unassisted coding problem, and we show that their capacity region is not changed with non-signaling assistance. In the value query model, we show that we cannot achieve a better approximation ratio than Ω 1 √ m for the general broadcast channel coding problem, with m the output size of the channel.

iii

Résumé

La recherche de codes e caces est essentielle pour obtenir des communications ables sur des canaux bruités. Alors que les codes correcteurs d'erreurs pour les canaux i.i.d. sont bien compris, le problème d'approximer le meilleur code pour un canal générique, c'est-à-dire qui maximise la probabilité de succès de la communication sur les canaux bruités, a été beaucoup moins étudié. Cette thèse aborde ce problème de codage dans plusieurs scénarios.

Pour les canaux point-à-point, des algorithmes d'approximation optimaux pour le problème du codage ainsi que le problème du décodage de liste sont connus. Nous étudions une généralisation de ce dernier avec une taille de liste variable, compensée par une probabilité d'échec croissante. Plus généralement, nous étudions le problème de couverture qui consiste à maximiser a∈ [n] w a ϕ(|{i ∈ S : a ∈ T i }|) sur les sous-ensembles S ⊆ [m] de cardinal k, pour ϕ une fonction croissante concave quelconque. Nous proposons un algorithme d'approximation pour ce problème de ratio α ϕ := min x∈N * E[ϕ(Poi(x))]

ϕ(E[Poi(x)]) , qui ne peut être amélioré pour ϕ sous-linéaire si P = NP.

Pour les canaux à accès multiple, nous montrons que le problème du codage ne peut être approximé avec un ratio constant sous une hypothèse de complexité sur des formules k-SAT aléatoires. En généralisant et en abstrayant l'intrication quantique, les corrélations non-signalantes peuvent être utilisées pour améliorer la communication. Nous montrons que les codes optimaux avec assistance non-signalante pour les canaux à accès multiples peuvent être trouvés en temps polynomial en le nombre de copies du canal. Appliqué au canal additionneur binaire, l'utilisation de corrélations non-signalantes étend sa zone de capacité. Nous fournissons une borne supérieure sur la zone de capacité avec assistance non-signalante. Lorsque l'assistance non-signalante n'est pas partagée entre les encodeurs, nous montrons que la zone de capacité n'est pas modi ée.

Pour les canaux de di usion, lorsque l'on se limite aux canaux déterministes, nous fournissons une (1 -e -1 ) 2 -approximation pour le problème du codage sans assistance, et nous montrons que leur zone de capacité n'est pas modi ée par l'assistance non-signalante. Dans le modèle d'accès par valeur, nous montrons que nous ne pouvons pas obtenir un meilleur ratio d'approximation que Ω 1 √ m pour le problème global du codage des canaux de di usion, où m est la taille de sortie du canal. the set of integers from 1 to k, i.e. {1, . . . , k} S k the set of permutations over [k] (x, y) the set of real numbers z such that x < z < y k, , n, m, . . . nonnegative integers x, y, z, . . . real numbers p, q, . . . real numbers between 0 and 1 x / x the ceiling/ oor of x log(x) the binary logarithm of x e x / exp(x) the exponential of x the zero-error capacity region of W for a success probability S x n , y n , z n , . . . words in X n , Y n , Z n , . . . x i , y i , z i , . . . the ith letter of x n , y n , z n , . . . x -i , y -i , z -i , . . . the words x n , y n , z n , . . . without their ith letter T n ε (X) the set of ε-typical words of n letters following X H(X) the entropy of X H(X|Y ) the conditional entropy of X given Y I(X : Y ) the mutual information of X and Y I(X : Y |Z) the conditional mutual information of X and Y given Z X → Y → Z a Markov chain

Introduction

With the growing number of connected devices and the explosion in the amount of exchanged data, the need for e cient and reliable communication has never been as critical as in today's world. Noise, coming from physical imperfection, signal interference, or even lepidopterans [START_REF] Murray | The rst bug[END_REF], is one of the major hurdles to overcome. The Theory of Information established by Shannon in his seminal work [START_REF] Shannon | A mathematical theory of communication[END_REF] provides clear and de nite answers to the amount of data that can be transmitted through noisy pointto-point channels. The asymptotic rate at which information can be sent through multiple independent copies of a channel is fully characterized by a mathematical quantity called capacity. Although one cannot hope to surpass this fundamental limit, developing codes achieving the channel's capacity is not a simple task in practice, as the code construction by Shannon was only probabilistic and the decoding is ine cient as a function of the number of copies of the channel.

Nonetheless, restricting the study to binary symmetric channels, the research in errorcorrecting codes led to capacity-approaching solutions with e cient encoding and decoding procedures, such as low-density parity-check codes [START_REF] Robert | Low-density parity-check codes[END_REF], turbo codes [START_REF] Berrou | Near optimum error correcting coding and decoding: turbo-codes[END_REF] or more recently polar codes [START_REF] Arikan | Channel polarization: a method for constructing capacity-achieving codes for symmetric binary-input memoryless channels[END_REF]; see [START_REF] Richardson | Modern Coding Theory[END_REF] for a general review on error-correcting codes.

However, in some settings, it cannot be assumed that the channel copies are independent. The eld of one-shot information theory [START_REF] Renner | The single-serving channel capacity[END_REF][START_REF] Tomamichel | A framework for non-asymptotic quantum information theory[END_REF][START_REF] Tomamichel | Quantum coding with nite resources[END_REF] addresses this problem, studying information theoretical quantities de ned for a unique copy of an arbitrary channel. Another approach, developed by Barman and Fawzi in [START_REF] Barman | Algorithmic aspects of optimal channel coding[END_REF], focuses on the algorithmic aspects of coding for a single copy of a channel. The objective shifts to maximizing the probability of successfully transmitting a message through that channel over all possible encoders and decoders, known as the channel coding problem. As this algorithmic task is NP-hard, they proposed a polynomial-time approximation algorithm achieving a ratio of 1 -e -1 , which cannot be increased if P = NP thanks to a connection to the maximum coverage problem [START_REF] Feige | A threshold of ln n for approximating set cover[END_REF]. Note that this approach di ers from algorithmic information theory [START_REF] Chaitin | Algorithmic information theory[END_REF], where information is de ned using Turing machines rather than probability theory.

Following this work, the -list-decoding task [START_REF] Elias | List decoding for noisy channels[END_REF][START_REF] John | List decoding[END_REF], where the decoder is only asked to output a list of possible guesses instead of giving the right input message, was studied from an algorithmic point of view in [START_REF] Barman | Tight approximation bounds for maximum multi-coverage[END_REF]. Similarly, they developed a 1 -e - ! -approximation algorithm running in polynomial time, and proved that this 1 1. I ratio cannot be increased under the unique games conjecture [START_REF] Khot | On the power of unique 2-prover 1-round games[END_REF]. More generally, a larger class of combinatorial optimization problems, called maximum -multi-coverage, falls in the analysis of [START_REF] Barman | Tight approximation bounds for maximum multi-coverage[END_REF].

A natural extension of the maximum -multi-coverage problem entails maximizing the quantity a∈ [n] w a ϕ(|{i ∈ S : a ∈ T i }|) over subsets S ⊆ [m] of cardinality k, for a general nondecreasing concave function ϕ; the maximum -multi-coverage problem is retrieved by taking ϕ(x) := min(x, ). In terms of channel coding interpretation, this corresponds to what we call ϕ-list-decoding, where the length restriction on the list of guesses of the decoder in -list-decoding is replaced by a probability ϕ( ) of correctly decoding a list of guesses of variable size . This problem will be the rst subject of study of the thesis.

Shaking the physical foundations of the world at the beginning of the twentieth century, the rise of quantum mechanics has revolutionized the comprehension of nature at atomic and subatomic scale. One of its most intriguing characteristics is the notion of entanglement.

Two entangled particles have the particularity of being correlated, even when spread apart, in a way that cannot be explained by non-quantum physics laws.

Einstein, Podolsky and Rosen [START_REF] Einstein | Can quantum-mechanical description of physical reality be considered complete?[END_REF] rst discovered that phenomenon, which implied either that the quantum-mechanical description of physical reality was not complete or that two incompatible physical quantities did not have simultaneously a concrete value. Surprisingly and contrary to the original goal of what has become known as the Einstein-Podolsky-Rosen paradox [START_REF] Einstein | Can quantum-mechanical description of physical reality be considered complete?[END_REF], it is the latter that describes correctly the atomic and subatomic physical reality.

Bell showed in [START_REF] Bell | On the Einstein Podolsky Rosen paradox[END_REF] that no hidden-variable theory, where we could associate unknown local variables to each of the entangled particles, would be enough to explain their correlation when spread apart. This nonlocal nature of quantum physics is in contradiction with the usual principle of locality of classical physics, which states that any object can be in uenced only by its immediate surroundings. The results in [START_REF] Bell | On the Einstein Podolsky Rosen paradox[END_REF] were generalized into the so-called CHSH inequality [START_REF] Clauser | Proposed experiment to test local hidden-variable theories[END_REF], which if violated, imply that a hidden-variable theory cannot explain the correlations between the particles. These inequalities violations were practically realized by Aspect et al. [START_REF] Aspect | Experimental test of Bell's inequalities using time-varying analyzers[END_REF], which nally proved the nonlocal nature of quantum physics.

Another particularity of quantum mechanics is that any measurement of physical properties of a quantum system actually disturbs it. Furthermore, in the case of entangled particles, measuring one of the particles will also a ect the other. This spooky action at a distance, as stated by Einstein [START_REF] Born | The Born-Einstein letters: correspondence between Albert Einstein and Max and Hedwig Born from 1916 to 1955[END_REF], does not violate special relativity, as one cannot actually transmit information through that process.

However, entanglement can be used to enhance communication. A qubit, the quantum basic unit of information, cannot by itself convey more than one bit of information [START_REF] Semenovich | Bounds for the quantity of information transmitted by a quantum communication channel[END_REF]. However, Bennett and Wiesner showed in [START_REF] Bennett | Communication via one-and two-particle operators on Einstein-Podolsky-Rosen states[END_REF] that if quantum entanglement is shared between the parties, in the form of what is called an EPR pair, then one can actually transmit two bits of information using only one qubit. This is known today as superdense coding. For point-to-point (classical) channels, quantum entanglement shared between the sender and the receiver can increase the optimal success probability of channel coding [CLMW10, PLM + 11], although it does not increase the channel capacity [BBC + 93, BSST99].

One can also abstract quantum entanglement into non-signaling correlations [START_REF] Cirel'son | Quantum generalizations of Bell's inequality[END_REF][START_REF] Popescu | Quantum nonlocality as an axiom[END_REF].

In order to understand those, it is useful to look at the game interpretation of the previously mentioned CHSH inequality. We consider a two-player game with Alice and Bob. A referee gives a uniformly random bit x (resp. y) to Alice (resp. Bob), and their common goal is to output respectively bits a and b such that a ⊕ b = x ∧ y without communicating, where ⊕ denotes the exclusive disjunction. It is easy to see that no strategy can lead to a better success probability than 3 4 , even assuming that hidden variables are shared between the players. However, as shown in [START_REF] Clauser | Proposed experiment to test local hidden-variable theories[END_REF], if Alice and Bob share an entangled EPR pair, they can apply well chosen measurements and achieve a success probability of cos 2 π 8 0.85, which in fact cannot be outperformed [START_REF] Cirel'son | Quantum generalizations of Bell's inequality[END_REF]. Note that the correlation P (ab|xy) describing the result of this quantum strategy con rms that no information is transmitted using quantum entanglement. Indeed, one can show that the marginal from a player is independent of the other player's input, that is to say that b P (ab|xy) = b P (ab|xy ) and a P (ab|xy) = a P (ab|x y) for all a, b, x, y, x , y . Non-signaling correlations are de ned as the set of all joint distributions P (ab|xy) that satisfy the previous equalities. They naturally include quantum strategies, but are in fact stronger, as one can even achieve a success probability of 1 for the CHSH game using a general non-signaling strategy, simply by de ning P (ab|xy) := 1 2 if a ⊕ b = x ∧ y, and P (ab|xy) := 0 otherwise [START_REF] Popescu | Quantum nonlocality as an axiom[END_REF]. General non-signaling correlations are not representing physical reality, as they imply stronger nonlocal behaviors than those occurring within quantum mechanics. However, they present a strong theoretical interest. Notably, the description of the set of non-signaling correlations is much simpler than with quantum ones, as they are characterized by simple linear constraints; see [BCP + 14] for a general review on nonlocality.

The approximation algorithm for the channel coding problem in [START_REF] Barman | Algorithmic aspects of optimal channel coding[END_REF] relies in fact on non-signaling assisted codes. As nding the best non-signaling assisted code for a channel is a linear program, it can be solved exactly in polynomial time. They have developed a strategy to transform that non-signaling assisted code into a classical one, losing at most a factor 1 -e -1 in the success probability. A more precise statement of this strategy actually implies that the capacity regions with or without non-signaling assistance are the same, retrieving back a result by [START_REF] Matthews | A linear program for the nite block length converse of Polyanskiy-Poor-Verdú via nonsignaling codes[END_REF]. It should also be noted that non-signaling assistance does not change the capacity region of the reverse task of channel simulation [START_REF] Michael | Channel simulation: Finite blocklengths and broadcast channels[END_REF]. This unexpected link between approximation algorithms for the channel coding problem and non-signaling correlations is the main motivation of the rest of the thesis. Speci cally, we will develop these analysis in the main network communication scenarios.

Network information theory, which aims at understanding communication over multiplesender multiple-receiver channels, was rst studied by Shannon in the particular case of two-way channels [START_REF] Shannon | Two-way communication channels[END_REF] (known today as interference channels). Later, Cover [START_REF] Thomas | Broadcast channels[END_REF] introduced broadcast channels, with multiple receivers but a single sender. Ahlswede [START_REF] Ahlswede | Multi-way communication channels[END_REF] and independently Liao [START_REF] Herng | Multiple access channels[END_REF] studied the reverse scenario of multiple-access channels, with multiple senders but a single receiver. These more complex channels allow a better modelization of real-life interconnected communication.

Contrary to the point-to-point setting, nonlocality can increase the capacity of network channels. Quek and Shor showed in [START_REF] Quek | Quantum and superquantum enhancements to twosender, two-receiver channels[END_REF] the existence of two-sender two-receiver interference channels with gaps between their classical, quantum-entanglement assisted and non-signaling assisted capacity regions. Following this result, Leditzky et al. [START_REF] Leditzky | Playing games with multiple access channels[END_REF][START_REF] Seshadri | On the separation of correlation-assisted sum capacities of multiple access channels[END_REF] showed that quantum entanglement shared between the two senders of a multipleaccess channel can strictly enlarge its capacity region.

Introduction

Avec le nombre croissant d'appareils connectés et l'explosion du volume de données échangées, le besoin de communication e cace et able n'a jamais été aussi critique que dans le monde d'aujourd'hui. Le bruit, qui provient d'imperfections physiques, d'interférences entre les signaux ou même de lépidoptères [START_REF] Murray | The rst bug[END_REF], est l'un des principaux obstacles à surmonter.

La théorie de l'information établie par Shannon dans son ouvrage fondateur [START_REF] Shannon | A mathematical theory of communication[END_REF] apporte des réponses claires et précises à la quantité de données pouvant être transmises par des canaux point à point bruités. Le taux asymptotique auquel les informations peuvent être envoyées par de multiples copies indépendantes d'un canal est entièrement caractérisé par une quantité mathématique appelée capacité. Bien que l'on ne puisse espérer dépasser cette limite fondamentale, le développement de codes permettant d'atteindre la capacité du canal n'est pas une tâche aisée en pratique, car la construction du code par Shannon n'était que probabiliste et le décodage est ine cace en fonction du nombre de copies du canal.

Néanmoins, en limitant l'étude aux canaux symétriques binaires, la recherche sur les codes correcteurs d'erreurs a conduit à des solutions approchant la capacité avec des procédures de codage et de décodage e caces, telles que les codes de contrôle de parité à faible densité [START_REF] Robert | Low-density parity-check codes[END_REF], les codes turbo [START_REF] Berrou | Near optimum error correcting coding and decoding: turbo-codes[END_REF] ou, plus récemment, les codes polaires [START_REF] Arikan | Channel polarization: a method for constructing capacity-achieving codes for symmetric binary-input memoryless channels[END_REF] ; voir [START_REF] Richardson | Modern Coding Theory[END_REF] pour une revue générale des codes correcteurs d'erreurs.

Toutefois, dans certains contextes, on ne peut pas supposer que les copies du canal sont indépendantes. Le domaine de la théorie de l'information à un coup [RWW06, Tom12, TBR16] aborde ce problème, en étudiant les quantités de la théorie de l'information dé nies pour une copie unique d'un canal arbitraire. Une autre approche, développée par Barman et Fawzi dans [START_REF] Barman | Algorithmic aspects of optimal channel coding[END_REF], se concentre sur les aspects algorithmiques du codage pour une copie unique d'un canal. L'objectif consiste alors à maximiser la probabilité de transmettre avec succès un message par ce canal pour tous les codeurs et décodeurs possibles, ce que l'on appelle problème du codage de canal. Cette tâche algorithmique étant NP-di cile, ils ont proposé un algorithme d'approximation en temps polynomial atteignant un ratio de 1 -e -1 , qui ne peut être augmenté si P = NP grâce à un lien avec le problème de la couverture maximale [START_REF] Feige | A threshold of ln n for approximating set cover[END_REF]. Il convient de noter que cette approche di ère de la théorie algorithmique de l'information [START_REF] Chaitin | Algorithmic information theory[END_REF], où l'information est dé nie à l'aide de machines de Turing plutôt qu'avec la théorie des probabilités. Suite à ces travaux, le problème du décodage de listes de paramètre [START_REF] Elias | List decoding for noisy channels[END_REF][START_REF] John | List decoding[END_REF], où l'on 1. I demande au décodeur de produire une liste de suppositions possibles au lieu de donner le bon message d'entrée, a été étudié d'un point de vue algorithmique dans [START_REF] Barman | Tight approximation bounds for maximum multi-coverage[END_REF]. De même, ils ont développé un algorithme d'approximation de ratio 1 -e - ! fonctionnant en temps polynomial, et ont prouvé que ce ratio ne peut pas être augmenté sous l'hypothèse de la conjecture des jeux uniques [START_REF] Khot | On the power of unique 2-prover 1-round games[END_REF]. Plus généralement, une classe plus large de problèmes d'optimisation combinatoire, appelée couverture multiple maximale de paramètre , relève de l'analyse de [START_REF] Barman | Tight approximation bounds for maximum multi-coverage[END_REF].

Une extension naturelle du problème de couverture multiple maximale de paramètre consiste à maximiser la quantité a∈ [n] w a ϕ(|{i ∈ S : a ∈ T i }|) sur les sous-ensembles S ⊆ [m] de cardinal k, pour ϕ une fonction croissante concave quelconque ; le problème de couverture multiple maximale de paramètre est retrouvé en prenant ϕ(x) := min(x, ).

En termes d'interprétation du codage de canal, cela correspond à ce que nous appelons le décodage de liste de paramètre ϕ, où la restriction de longueur sur la liste de suppositions du décodeur dans le décodage de liste de paramètre est remplacée par une probabilité ϕ( ) de décodage correct d'une liste de suppositions de taille variable . Ce problème sera le premier sujet d'étude de la thèse.

Au début du vingtième siècle, la mécanique quantique a ébranlé les fondements physiques du monde et révolutionné la compréhension de la nature à l'échelle atomique et subatomique. L'une de ses caractéristiques les plus fascinantes est la notion d'intrication. Deux particules intriquées ont la particularité d'être corrélées, même lorsqu'elles sont éloignées l'une de l'autre, d'une manière qui ne peut être expliquée par les lois de la physique non quantique.

Einstein, Podolsky et Rosen [START_REF] Einstein | Can quantum-mechanical description of physical reality be considered complete?[END_REF] ont été les premiers à découvrir ce phénomène, qui impliquait soit que la description de la réalité physique par la mécanique quantique n'était pas complète, soit que deux grandeurs physiques incompatibles n'avaient pas simultanément une valeur concrète. De manière surprenante et contrairement à l'objectif initial de ce qui est devenu le paradoxe d'Einstein-Podolsky-Rosen [START_REF] Einstein | Can quantum-mechanical description of physical reality be considered complete?[END_REF], c'est ce dernier qui décrit correctement la réalité physique atomique et subatomique.

Bell a montré dans [START_REF] Bell | On the Einstein Podolsky Rosen paradox[END_REF] qu'aucune théorie des variables cachées, où l'on pourrait associer des variables locales inconnues à chacune des particules intriquées, ne su rait à expliquer leur corrélation lorsqu'elles sont éloignées l'une de l'autre. Cette nature non locale de la physique quantique est en contradiction avec le principe de localité de la physique classique, qui stipule que tout objet ne peut être in uencé que par son environnement immédiat. Les résultats de [START_REF] Bell | On the Einstein Podolsky Rosen paradox[END_REF] ont été généralisés dans ce que l'on appelle les inégalités CHSH [START_REF] Clauser | Proposed experiment to test local hidden-variable theories[END_REF], qui, si elles sont violées, impliquent qu'une théorie des variables cachées ne peut pas expliquer les corrélations entre les particules. Les violations de ces inégalités ont été pratiquement réalisées par Aspect et al. [START_REF] Aspect | Experimental test of Bell's inequalities using time-varying analyzers[END_REF], ce qui a nalement prouvé la nature non locale de la physique quantique.

Une autre particularité de la mécanique quantique est que toute mesure des propriétés physiques d'un système quantique le perturbe. En outre, dans le cas de particules intriquées, la mesure de l'une des particules a ectera également l'autre. Cette action étrange à distance, comme l'a décrite Einstein [START_REF] Born | The Born-Einstein letters: correspondence between Albert Einstein and Max and Hedwig Born from 1916 to 1955[END_REF], ne viole pas la relativité restreinte, car il est impossible de transmettre des informations par ce biais.

Cependant, l'intrication peut être utilisée pour améliorer la communication. Un qubit, l'unité quantique de base de l'information, ne peut à lui seul transmettre plus d'un bit d'information [START_REF] Semenovich | Bounds for the quantity of information transmitted by a quantum communication channel[END_REF]. Cependant, [START_REF] Cirel'son | Quantum generalizations of Bell's inequality[END_REF]. Notons que la corrélation P (ab|xy) décrivant le résultat de cette stratégie quantique con rme qu'aucune information n'est transmise à l'aide de l'intrication quantique. En e et, on peut montrer que la marginale d'un joueur est indépendante de l'entrée de l'autre joueur, c'est-à-dire que b P (ab|xy) = b P (ab|xy ) et a P (ab|xy) = a P (ab|x y) pour tous a, b, x, y, x , y . Les corrélations non-signalantes sont dé nies comme l'ensemble des distributions conjointes P (ab|xy) qui satisfont les égalités précédentes. Elles incluent naturellement les stratégies quantiques, mais sont en fait plus fortes, car on peut même obtenir une probabilité de succès de 1 pour le jeu CHSH en utilisant une stratégie générale non-signalante, simplement en dé nissant P (ab|xy) := 1 2 si a ⊕ b = x ∧ y, et P (ab|xy) := 0 sinon [START_REF] Popescu | Quantum nonlocality as an axiom[END_REF].

Les corrélations non-signalantes générales ne représentent pas la réalité physique, car elles impliquent des comportements non locaux plus forts que ceux qui se produisent dans le cadre de la mécanique quantique. Cependant, elles présentent un grand intérêt théorique. Notamment, la description de l'ensemble des corrélations non-signalantes est beaucoup plus simple que celle des corrélations quantiques, car elles sont caractérisées par de simples contraintes linéaires ; voir [BCP + 14] pour une revue générale sur la non-localité. L'algorithme d'approximation pour le problème du codage de canal dans [START_REF] Barman | Algorithmic aspects of optimal channel coding[END_REF] s'appuie en fait sur des codes avec assistance non-signalante. Comme la recherche du meilleur code avec assistance non-signalante pour un canal est un programme linéaire, il peut être résolu exactement en temps polynomial. Ils ont mis au point une stratégie pour transformer ce code avec assistance non-signalante en un code classique, en perdant au maximum un facteur 1 -e -1 dans la probabilité de succès. Un énoncé plus précis de cette stratégie implique en fait que les zones de capacité avec ou sans assistance non-signalante sont les mêmes, retrouvant un résultat de [START_REF] Matthews | A linear program for the nite block length converse of Polyanskiy-Poor-Verdú via nonsignaling codes[END_REF]. Il convient également de noter que l'assistance non-signalante ne modi e pas la zone de capacité de la tâche inverse de simulation des canaux [START_REF] Michael | Channel simulation: Finite blocklengths and broadcast channels[END_REF]. Ce lien inattendu entre les algorithmes d'approximation pour le problème du codage de canal et les corrélations non-signalantes est la motivation principale du reste de la thèse. En particulier, nous développerons ces analyses dans les principaux scénarios de communication en réseau.

La théorie de l'information des réseaux, qui vise à comprendre la communication sur des 1. I canaux à émetteurs multiples et à récepteurs multiples, a été étudiée pour la première fois par Shannon dans le cas particulier des canaux deux-à-deux [START_REF] Shannon | Two-way communication channels[END_REF] (connus aujourd'hui sous le nom de canaux d'interférence). Plus tard, Cover [START_REF] Thomas | Broadcast channels[END_REF] a introduit les canaux de di usion, avec plusieurs récepteurs mais un seul émetteur. Ahlswede [START_REF] Ahlswede | Multi-way communication channels[END_REF] et indépendamment Liao [START_REF] Herng | Multiple access channels[END_REF] ont étudié le scénario inverse des canaux à accès multiple, avec plusieurs émetteurs mais un seul récepteur. Ces canaux plus complexes permettent une meilleure modélisation des communications interconnectées réelles.

Contrairement à la situation point-à-point, la non-localité peut augmenter la capacité des canaux en réseau. Quek et Shor ont montré dans [QS17] l'existence de canaux d'interférence à deux émetteurs et deux récepteurs avec des écarts entre leurs zones de capacité classique, avec assistance d'intrication quantique et avec assistance non-signalante. À la suite de ce résultat, Leditzky et al. [START_REF] Leditzky | Playing games with multiple access channels[END_REF][START_REF] Seshadri | On the separation of correlation-assisted sum capacities of multiple access channels[END_REF] ont montré que l'intrication quantique partagée entre les deux émetteurs d'un canal à accès multiple peut strictement agrandir sa zone de capacité.

Nous aborderons le problème du codage pour les canaux à accès multiple et les canaux de di usion. Nous étudierons l'impact des corrélations non-signalantes sur leurs capacités, ainsi que leurs liens avec les algorithmes d'approximation des problèmes de codage associés. ϕ(E[Poi(x)]) . En complément de cette garantie d'approximation, nous établissons un résultat de NP-di culté lorsque ϕ croît de manière sous-linéaire. Appliqué au codage de canal, et plus spéci quement au décodage de liste de paramètre ϕ, où la restriction de longueur sur la liste de suppositions du décodeur dans le décodage de liste de paramètre est remplacée par une probabilité ϕ( ) de décoder correctement une liste de suppositions de taille variable , nous obtenons une stricte garantie d'approximation α ϕ pour la classe de canaux W de la forme W (y|x) = 1 t pour y ∈ T x avec |T x | = t et W (y|x) = 0 ailleurs. Notre résultat dépasse ce cadre particulier et nous l'illustrons par des applications aux problèmes d'allocation de ressources distribuées, aux problèmes de maximisation de la prospérité et au vote basé sur l'approbation pour des règles générales.

Résumé des chapitres

Dans le Chapitre 4 (basé sur [START_REF] Fawzi | Beating the sum-rate capacity of the binary adder channel with non-signaling correlations[END_REF][START_REF] Fawzi | Multiple-access channel coding with non-signaling correlations[END_REF]), nous abordons le problème du codage pour les canaux à accès multiple. Nous montrons tout d'abord qu'il ne peut pas être approximé en temps polynomial avec un ratio constant, sous une hypothèse de complexité sur des formules k-SAT aléatoires. Ensuite, nous étudions l'in uence des corrélations non signalantes entre les parties. Nous développons un programme linéaire calculant la probabilité de succès optimale pour le codage sur n copies d'un canal d'accès multiple W dont la taille croît polynomialement en n. La résolution de ce programme linéaire nous permet d'obtenir des bornes inférieures pour les canaux à accès multiple. En appliquant cette méthode au canal additionneur binaire, nous montrons qu'en utilisant une assistance non-signalante, la somme des taux log 2 (72) 4 1.5425 peut être atteinte même avec une erreur nulle, ce qui dépasse la capacité maximale de la somme des taux de 1.5 dans le cas sans assistance. Pour les canaux bruités, où la région de capacité avec assistance non-signalante et sans erreur est triviale, nous pouvons utiliser des codes concaténés pour obtenir des points réalisables dans la zone de capacité. Appliqués à une version bruitée du canal additionneur binaire, nous montrons que l'assistance non-signalante améliore encore la capacité. En complément de ces résultats de faisabilité, nous donnons une borne supérieure à la zone de capacité avec assistance non-signalante qui a la même expression que la région sans assistance, sauf que les entrées du canal ne sont pas obligées d'être indépendantes. En n, nous montrons que la zone de capacité avec assistance non-signalante partagée uniquement entre chaque émetteur et le récepteur indépendamment est la même que sans assistance.

Dans le Chapitre 5, nous abordons le problème du codage pour les canaux de di usion. Pour la classe des canaux de di usion déterministes, nous décrivons un algorithme d'approximation de ratio (1 -e -1 ) 2 fonctionnant en temps polynomial, et nous montrons que la zone de capacité pour cette classe est la même avec ou sans assistance non-signalante. En n, nous montrons que dans le modèle d'accès par valeur, nous ne pouvons pas obtenir un meilleur ratio d'approximation que Ω 1 √ m en temps polynomial pour le problème général du codage des canaux de di usion, avec m la taille de l'une des sorties du canal.

C 2 Background

In this chapter, we recall brie y basic de nitions and properties, as well as more unusual results, used throughout the thesis. Rather than an introduction to each of those topics, one should see this chapter as a toolbox and use it when needed in the other chapters.

Probability Theory

Usual Distributions and Properties

De nition 2.1 (Usual Distributions). We will consider the following probability distributions:

1. The Bernouilli distribution Ber(p) is de ned on {0, 1} with the mass function:

(1 -p, p) .

2. The binomial distribution Bin(n, p) is de ned on {0, . . . , n} with the mass function:

n k p k (1 -p) n-k k∈{0,...,n} .
3. The Poisson distribution Poi(λ) is de ned on N with the mass function:

λ k e -λ k! k∈N .
Proposition 2.1 (Jensen's Inequality [START_REF] Ludwig | Sur les fonctions convexes et les inégalités entre les valeurs moyennes[END_REF]). For X random variable and ϕ concave, we have:

E [ϕ(X)] ≤ ϕ (E[X]) .
Proposition 2.2. For ϕ concave, and p ∈ [0, 1] m , we have:

E ϕ m i=1 Ber(p i ) ≥ E ϕ Poi m i=1 p i .

B

Proof. The notion of convex order discussed in [SS07] allows us to prove this result. We say that

X ≤ cx Y ⇐⇒ E[f (X)] ≤ E[f (Y )]
for any convex f . Thanks to Lemma 2.3 of [START_REF] Barman | Tight approximation bounds for maximum multi-coverage[END_REF], we have that for p ∈ [0, 1]:

Ber(p) ≤ cx Poi(p) .
Since this order is preserved through convolution (Theorem 3.A.12 of [SS07]), and the fact that m i=1 Poi(p i ) ∼ Poi ( m i=1 p i ), we have:

m i=1 Ber(p i ) ≤ cx Poi m i=1 p i .
Applying this result to -ϕ, which is convex, concludes the proof.

Negatively Associated Random Variables

We present a weaker notion of independence for random variables which is called negative association as introduced in [JDP83], for which the Cherno -Hoe ding bounds still hold.

De nition 2.2. Random variables X 1 , . . . , X n are said to be negatively associated if for every pair of disjoints subsets I, J of [n] and (coordinate-wise) increasing functions f, g, we have:

E[f ({X i : i ∈ I})) • g({X i : i ∈ J})] ≤ E[f ({X i : i ∈ I})] • E[g({X i : i ∈ J})] .
Proposition 2.3 (Property P 1 of [START_REF] Joag | Negative association of random variables with applications[END_REF]). A pair of random variable X, Y is negatively associated if and only if:

∀x ∈ X , ∀y ∈ Y, P XY (x, y) ≤ P X (x)P Y (y) .

Proposition 2.4 (Property P 4 of [START_REF] Joag | Negative association of random variables with applications[END_REF]). A subset of two or more negatively associated random variables is negatively associated.

Proposition 2.5 (Property P 5 of [START_REF] Joag | Negative association of random variables with applications[END_REF]). A set of independent random variables is negatively associated.

Proposition 2.6 (Property P 6 of [START_REF] Joag | Negative association of random variables with applications[END_REF]). Increasing functions de ned on disjoint subsets of a set of negatively associated random variables are negatively associated.

Proposition 2.7 (Property P 7 of [START_REF] Joag | Negative association of random variables with applications[END_REF]). The union of independent sets of negatively associated random variables is negatively associated.

De nition 2.3 (Permutation Distribution

). Let x = (x 1 , . . . , x k ) ∈ R k .
A permutation distribution is the joint distribution of the vector X = (X 1 , . . . , X k ) which takes as values all k! permutations of x with equal probabilities, each being 1 k! .

Proposition 2.8 (Theorem 2.11 of [START_REF] Joag | Negative association of random variables with applications[END_REF]). A permutation distribution is negatively associated.

Proposition 2.9 (Cherno -Hoe ding bound -version 1). Let X 1 , . . . , X n be negatively associated random variables such that 0 ≤ X i ≤ B and E[X i ] = µ. Then for η ∈ (0, 1):

P 1 n n i=1 X i -µ > η ≤ 2e -2( η B ) 2 n .
Proof. This usual version of the Cherno -Hoe ding bound for independent variables (Theorem 2 of [Hoe63]) can be extended to negatively associated random variables directly as pointed out by [DR98].

Proposition 2.10 (Cherno -Hoe ding bound -version 2). Let X 1 , . . . , X n be negatively associated Bernouilli random variables of parameter p. Then for 0 < ε ≤ 1 2 :

P 1 n n i=1 X i > (1 + ε)p ≤ e -
X i > (1 + ε)p ≤ e -D((1+ε)p||p)n ,
with D (x||y) := x ln x y + (1 -x) ln 1-x 1-y the Kullback-Leibler divergence between Bernoulli distributed random variables with parameters x and y. As D ((1 + ε)p||p) ≥ ε 2 p 4 for 0 < ε < 1 2 , we recover the expected bound.

Non-Signaling Probability Distributions

De nition 2.4. We say that a conditional probability distribution P (a n |x n ) de ned on

× n i=1 A i × × n i=1 X i is non-signaling if for all a n , x n , xn , we have ∀i ∈ [n], âi P (a 1 . . . âi . . . a n |x 1 . . . x i . . . x n ) = âi P (a 1 . . . âi . . . a n |x 1 . . . xi . . . x n ) . De nition 2.5. Let P (a n |x n ) a conditional probability distribution de ned on × n i=1 A i × × n i=1 X i and P (a n |x n ) de ned on × n i=1 A i × × n i=1 X i . We de ne P ⊗ P the tensor product conditional probability distribution de ned on × n i=1 (A i × A i ) × × n i=1 (X i × X i ) by (P ⊗ P ) (a 1 a 1 . . . a n a n |x 1 x 1 . . . x n x n ) := P (a n |x n ) • P (a n |x n ).
Proposition 2.11. If both P and P are non-signaling, then P ⊗ P is non-signaling.

Proof. Let a n ∈ × n j=1 A j , a n ∈ × n j=1 A j , x n ∈ × n j=1 X j , x n ∈ × n j=1 X j and xi ∈ X i , 13
2. B

x i ∈ X i . Using the fact that P, P are non-signaling, we have:

âi â i P (a 1 a 1 . . . âi â i . . . a n a n |x 1 x 1 . . . x i x i . . . x n x n ) = âi â i P (a 1 . . . âi . . . a n |x 1 . . . x i . . . x n ) • P (a 1 . . . â i . . . a n |x 1 . . . x i . . . x n ) =   âi P (a 1 . . . âi . . . a n |x 1 . . . x i . . . x n )   •   â i P (a 1 . . . â i . . . a n |x 1 . . . x i . . . x n )   =   âi P (a 1 . . . âi . . . a n |x 1 . . . xi . . . x n )   •   â i P (a 1 . . . â i . . . a n |x 1 . . . x i . . . x n )   = âi â i P ⊗ P (a 1 a 1 . . . âi â i . . . a n a n |x 1 x 1 . . . xi x i . . . x n x n ) ,
(2.1) so P ⊗ P is non-signaling.

Information Theory

Information Quantities

Here, X, Y, Z will denote random variables over nite sets X , Y, Z and we will denote their distributions by P X (x) = P(X = x), P X,Y (x, y) = P(X = x, Y = y), P X|Y (x|y) = P(X = x|Y = y), . . .

De nition 2.6 ((Shannon) Entropy H(X))

. The (Shannon) entropy H(X) of X is de ned by:

H(X) := E X [-log(P X (X))] = - x∈X P X (x) log(P X (x)) .
The entropy of two random variables X, Y is de ned as H(X, Y ) := H((X, Y )).

De nition 2.7 (Conditional Entropy H(X|Y )). The conditional entropy H(X|Y ) of X

given Y is de ned by:

H(X|Y ) := y∈Y P Y (y)H(X|Y = y) = y∈Y P Y (y) - x∈X P X|Y (x|y) log P X|Y (x|y) = - x∈X ,y∈Y P X,Y (x, y) log P X|Y (x|y) .
(2.2)

De nition 2.8 (Mutual Information I(X : Y )). The mutual information I(X : Y ) of X and Y is de ned by:

I(X : Y ) := x∈X ,y∈Y P X,Y (x, y) log P X,Y (x, y) P X (x)P Y (y) .
Proposition 2.12. Equivalently, we have:

I(X : Y ) = H(X) -H(X|Y ) = H(Y ) -H(Y |X) = H(X) + H(Y ) -H(X, Y ) .
De nition 2.9 (Conditional Mutual Information I(X : Y |Z)). The mutual information I(X : Y |Z) of X and Y given Z is de ned by:

I(X : Y |Z) := z∈Z P Z (z)I(X : Y |Z = z) = x∈X ,y∈Y,z∈Z P X,Y,Z (x, y, z) log P X,Y |Z (x, y|z) P X|Z (x|z)P Y |Z (y|z)
.

(2.3)

De nition 2.10 (Markov chain X → Y → Z). We say that X → Y → Z forms a Markov chain if Z is conditionally independent of X given Y , that is to say: ∀x ∈ X , ∀y ∈ Y, ∀z ∈ Z, P Z|X,Y (z|x, y) = P Z|Y (z|y) .

Typical Sets

We consider typical sets as de ned in Chapter 2.5 of [START_REF] El | Network Information Theory[END_REF]:

De nition 2.11 (Typical set and conditional typical set). We have the following de nitions:

1. T n ε (X) := {x n : |π(x|x n ) -P X (x)| ≤ εP X (x)
for all x ∈ X }, with the empirical distribution π de ned by π(x|x n ) := |{i:x i =x}| n . This de nition generalizes for any t-uple of variables.

∀y

n ∈ T n ε (Y ), T n ε (X|y n ) := {x n : (x n , y n ) ∈ T n ε (X, Y )}.
A crucial property of such typical sets is the typical average lemma:

Lemma 2.13 (Typical Average Lemma [START_REF] El | Network Information Theory[END_REF]). Let x n ∈ T n ε (X). Then for any nonnegative function g on X :

(

1 -ε)E[g(X)] ≤ 1 n n i=1 g(x i ) ≤ (1 + ε)E[g(X)] .
In particular, with this tool, we can derive the following properties:

Proposition 2.14 (Properties of typical sets [START_REF] El | Network Information Theory[END_REF]). We have, among others, the following statements about typical sets:

1. ∀x n ∈ T n ε (X), 2 -n(1+ε)H(X) ≤ P X n (x n ) ≤ 2 -n(1-ε)H(X) . 2. lim n→+∞ P (X n ∈ T n ε (X)) = 1. 3. |T n ε (X)| ≤ 2 n(1+ε)H(X) . 4. For n su ciently large, |T n ε (X)| ≥ (1 -ε)2 n(1-ε)H(X) . 5. If (x n , y n ) ∈ T n ε (X, Y ) then x n ∈ T n ε (X) and y n ∈ T n ε (Y ). 6. ∀y n ∈ T n ε (Y ), T n ε (X|y n ) ⊆ T n ε (X). 7. ∀(x n , y n ) ∈ T n ε (X, Y ), 2 -n(1+ε)H(X|Y ) ≤ P X n Y n (x n |y n ) ≤ 2 -n(1-ε)H(X|Y ) . 8. ∀y n ∈ T n ε (Y ), |T n ε (X|y n )| ≤ 2 n(1+ε)H(X|Y ) .
9. For ε < ε and n su ciently large, we have that

∀y n ∈ T n ε (Y ), |T n ε (X|y n )| ≥ (1 -ε)2 n(1-ε)H(X|Y ) .
Proof. We reproduce the proof of the last statement here to emphasize on the fact that there is an n 0 such that for all n ≥ n 0 and for all y n ∈ T n ε (Y ), the property holds. For any ε > ε > 0, let us show that there exists n such that we have:

∀y n ∈ T n ε (Y ), P ((X n , y n ) ∈ T n ε (X, Y )) ≥ 1 -ε ,
where X n are drawn from the distribution P X n |Y n =y n . This will imply the statement. Indeed, we have that: X|Y ) . In order to prove our result, we take the proof in Appendix 2A of [START_REF] El | Network Information Theory[END_REF]. We take y n ∈ T n ε (Y) and X n ∼ P X n |Y n (x n |y n ) = n i=1 P X|Y (x i |y i ). Applied to our choice of variables, we have that P ((X n , y n ) / ∈ T n ε (X, Y )) is equal to:

P ((X n , y n ) ∈ T n ε (X, Y )) = x n ∈T n ε (X|y n ) P X n |Y n (x n |y n ) ≤ |T n ε (X|y n )|2 -n(1-ε)H(X|Y ) , (2.4) since P X n |Y n (x n |y n ) ≤ 2 -n(1-ε)H(X|Y ) as (x n , y n ) ∈ T n ε (X, Y ). Thus, we have that |T n ε (X|y n )| ≥ (1 -ε)2 n(1-ε)H(
P (∃(x, y) : |π(x, y|X n , y n ) -P XY (x, y)| > εP XY (x, y)) ≤ x,y P (|π(x, y|X n , y n ) -P XY (x, y)| > εP XY (x, y)) = x,y P π(x, y|X n , y n ) P XY (x, y) -1 > ε = x,y P π(x, y|X n , y n ) P X|Y (x|y)π(y|y n ) π(y|y n ) P Y (y) -1 > ε ≤ x,y P π(x, y|X n , y n ) π(y|y n ) > 1 + ε 1 + ε P X|Y (x|y) + x,y P π(x, y|X n , y n ) π(y|y n ) < 1 -ε 1 -ε P X|Y (x|y) , (2.5) since y n ∈ T n ε (Y) and thus 1 -ε ≤ π(y|y n ) P Y (y) ≤ 1 + ε . However, since ε < ε, we have 1+ε
1+ε > 1 and 1-ε 1-ε < 1. We will show that for all x, y with P Y (y) > 0, we have

π(x,y|X n ,y n ) π(y|y n ) → n→+∞ P X|Y (x|y)
in probability, with a convergence rate independent of y n ∈ T n ε (Y ), which will be enough to conclude the proof. Let us x some x, y with P Y (y) > 0. Since y n ∈ T n ε (Y ), we have in particular that

(1 -ε )P Y (y) ≤ π(y|y n ) ≤ (1 + ε )P Y (y). Thus N := |{i : y i = y}| = nπ(y|y n ) ≥ (1 -ε )P Y (y)n. Then we have: π(x, y|X n , y n ) π(y|y n ) = 1 N i∈S Z i with Z i := 1 X i =x and S := {i : y i = y} .
Thus, all Z i with i ∈ S are independent and follow the same law:

Z i := 1 with probability P X|Y (x|y) , 0 otherwise .
Furthermore, we have E[Z i ] = P X|Y (x|y), and all Z i have the same variance σ 2 x|y < +∞ (depending only on X, Y, x, y). Thus we can apply Chebyshev inequality:

P 1 N i∈S Z i -P X|Y (x|y) ≥ η ≤ σ 2 x|y N η 2 .
However, since N ≥ (1 -ε )P Y (y)n, we get:

P π(x, y|X n , y n ) π(y|y n ) -P X|Y (x|y) ≥ η ≤ σ 2 x|y η 2 (1 -ε )P Y (y)n → n→+∞ 0 .
Thus, we have

Channels

Formally, a channel W is a conditional probability distribution depending on n inputs belonging to X n and m outputs belonging to Y m , so

W := (W (y m |x n )) x n ∈X n ,y m ∈Y m with: ∀x n , y m , W (y m |x n ) ≥ 0 and ∀x n , y m ∈Y m W (y m |x n ) = 1 .
We will denote such a channel by W :

X n → Y m . The tensor product of two channels W : X n → Y m and W : X n → Y m is denoted by W ⊗ W : (X n × X n ) → (Y n × Y n )
and de ned by:

(W ⊗ W )(y m y m |x n x n ) := W (y m |x n ) • W m |x n ) .
We denote by W ⊗n the nth tensor product of W , i.e. W ⊗n = W ⊗ . . . ⊗ W with n occurrences of W .

When n = m = 1, we will speak of regular or point-to-point channels. When n > 1, m = 1, we will speak of multiple-access channels. When n = 1, m > 1, we will speak of broadcast channels. If both n, m are greater than 1, we will speak of interference channels. More speci cally, in this thesis, we will focus on the cases of n = 2, m = 1 and n = 1, m = 2, which are at the core of the speci city of network channels.

B

Capacity Regions

De nition 2.12 (Capacity Region C[S](W ) for a success probability S(W,

k 1 , k 2 )). A rate pair (R 1 , R 2 ) is S-achievable (for the channel W ) if: lim n→+∞ S(W ⊗n , 2 R 1 n , 2 R 2 n ) = 1 .
We de ne the S-capacity region C[S](W ) as the closure of the set of all achievable rate pairs (for the channel W ).

De nition 2.13 (Zero-error Capacity Region

C 0 [S](W ) for S(W, k 1 , k 2 )). A rate pair (R 1 , R 2
) is S-achievable with zero-error (for the channel W ) if:

∃n 0 ∈ N * , ∀n ≥ n 0 , S(W ⊗n , 2 R 1 n , 2 R 2 n ) = 1 .
We de ne the zero-error S-capacity region C[S](W ) as the closure of the set of all achievable rate pairs with zero-error (for the channel W ).

Proposition 2.15 (Time-sharing). If the success probability S satis es:

∀W, W , ∀k 1 , k 2 , k 1 , k 2 , S(W ⊗ W , k 1 k 1 , k 2 k 2 ) ≥ S(W, k 1 , k 2 ) • S(W , k 1 , k 2 ) , then for all channels W , C[S](W ) and C 0 [S](W ) are convex. Proof. Let (R 1 , R 2 ) and (R 1 , R 2 )
, two pairs of S-achievable rational rates for W , i.e.:

S(W ⊗n , 2 R 1 n , 2 R 2 n ) → n→+∞ 1 and S(W ⊗n , 2 R 1 n , 2 R 2 n ) → n→+∞ 1 .
Let λ ∈ (0, 1) rational and de ne R λ,i :

= λ • R i + (1 -λ) • R i , let us show that (R λ,1 , R λ,2 )
is achievable with non-signaling assistance. Let us call respectively

k i := 2 R i , k i := 2 R i , k λ,i := 2 R λ,i = k λ i • k (1-λ) i . We have R λ,i n = λ • R i n + (1 -λ) • R i n = (λn) • R i + (1 -λ)n • R i
. This is the idea of time-sharing: for λn copies of the channel, we use the strategy with rate (R 1 , R 2 ) and for the (1 -λ)n other copies of the channel, we use the strategy with rate (R 1 , R 2 ). There exists some n such that λn, (1 -λ)n, λnR i , (1 -λ)nR i are integers, since everything is rational. This implies that k λn i , k

(1-λ)n i , k n λ,i are integers. Thus, by hypothesis, we have that S(W ⊗n , k n λ,1 , k n λ,2 ) is larger than or equal to:

S(W ⊗(λn) , k λn 1 , k λn 2 ) • S(W ⊗((1-λ)n) , k (1-λ)n 1 , k (1-λ)n 2 ) → n→+∞ 1 • 1 = 1 . Thus, since S(W ⊗n , k n λ,1 , k n λ,2 ) ≤ 1, we get the result S(W ⊗n , k n λ,1 , k n λ,2 ) → n→+∞ 1, so (R λ,1 , R λ,2
) is S-achievable for the channel W . Finally, since C[S](W ) is de ned as the closure of S-achievable rates for the channel W , we get that C[S](W ) is convex.

For zero-error capacity regions, since by hypothesis there exists ranks n 0 , n 0 such that

S(W ⊗n , 2 R 1 n , 2 R 2 n ) = 1 for n ≥ n 0 and S(W ⊗n , 2 R 1 n , 2 R 2 n ) = 1 for n ≥ n 0 ,
then in particular we get that for λn ≥ n 0 and (1 -λ)n ≥ n 0 that S(W ⊗n , k n λ,1 , k n λ,2 ) is larger than or equal to:

S(W ⊗(λn) , k λn 1 , k λn 2 ) • S(W ⊗((1-λ)n) , k (1-λ)n 1 , k (1-λ)n 2 ) = 1 .

Channel Coding for Point-to-Point Channels

It is in particular true for n ≥ n 0 := max

n 0 1-λ , n 0 λ . As S NS (W ⊗n , k n λ,1 , k n λ,2 ) ≤ 1, we have for all n ≥ n 0 that S NS (W ⊗n , k n λ,1 , k n λ,2 ) = 1, i.e. (R λ,1 , R λ,2
) is S-achievable with zero-error for the channel W . Finally, since C 0 [S](W ) is de ned as the closure of S-achievable rates with zero-error for the channel W , we get that C 0 [S](W ) is convex.

Channel Coding for Point-to-Point Channels

We recall here the main results from [START_REF] Barman | Algorithmic aspects of optimal channel coding[END_REF] on channel coding for point-to-point channels, i.e. with one input and one output. Let us rst recall the de nition of the maximum success probability S(W, k) of transmitting k messages using the channel W :

S(W, k) := maximize e,d 1 k i,x,y W (y|x)e(x|i)d(i|y) subject to x∈X e(x|i) = 1, ∀i ∈ [k] j∈[k] d(j|y) = 1, ∀y ∈ Y e(x|i), d(j|y) ≥ 0 (2.6)
Then, the following characterization of S(W, k) can be derived:

Proposition 2.16 (Proposition 2.1 of [BF18]). S(W, k) = 1 k max S⊆X:|S|≤k f W (S) with f W (S) := y∈Y max x∈S W (y|x).
One can consider non-signaling assistance shared between the sender and the receiver, which leads to the following maximum success probability: (2.7)

S NS (W, k) := maximize P 1 k i,x
A symmetrization can also be done to simplify the expression of the linear program de ning S NS (W, k):

B

Proposition 2.17 (Appendix A of [START_REF] Barman | Algorithmic aspects of optimal channel coding[END_REF]).

S NS (W, k) = maximize r,p 1 k x,y W (y|x)r x,y subject to x r x,y = 1 x p x = k 0 ≤ r x,y ≤ p x (2.8)
Finally, the main tool we will use from [START_REF] Barman | Algorithmic aspects of optimal channel coding[END_REF] is the following random coding technique, which describes how to nd a classical code with a success probability close to the nonsignaling assisted one:

Theorem 2.18 (Theorem 3.1 of [START_REF] Barman | Algorithmic aspects of optimal channel coding[END_REF]). Given a solution r, p of the program computing S NS (W, k), we have that: In this problem, we are given subsets T 1 , . . . , T m of a universe [n], along with a positive integer k, and the objective is to nd a size-k subset S ⊆ [m] that maximizes the coverage function C(S) := | i∈S T i |. A fundamental result in the eld of approximation algorithms establishes that an approximation ratio of 1 -e -1 can be achieved for this problem in polynomial time [Hoc97] and, in fact, this approximation guarantee is tight, under the assumption that P = NP [START_REF] Feige | A threshold of ln n for approximating set cover[END_REF].

E S f W (S) ≥ k 1 -1 - 1 k • 1 k x,y W (y|x)r x,
Note that in the maximum coverage problem, an element a ∈ [n] is counted at most once in the objective, even if a appears in several selected sets. However, if we think of elements a ∈ [n] as goods or resources, there are many settings wherein the utility indeed increases with the number of copies of a that get accumulated. Motivated, in part, by such settings, we consider a generalization of the maximum coverage problem where an element a can contribute by an amount that depends on the number of times it is covered.

Given a function ϕ : N → R + , an integer k ∈ N, a universe of elements [n], positive weights w a for each a ∈ [n], and subsets

T 1 , . . . , T m ⊆ [n], the ϕ-M C problem entails maximizing C ϕ (S) := a∈[n] w a ϕ(|S| a ) over subsets S ⊆ [m] of cardinality k; here |S| a = |{i ∈ S : a ∈ T i }|.
This chapter focuses on functions ϕ that are nondecreasing and concave (i.e., ϕ(i + 2)ϕ(i+1) ≤ ϕ(i+1)-ϕ(i) for i ∈ N). We will also assume that the function ϕ is normalized in the sense that ϕ(0) = 0 and ϕ(1) = 1. 1 Our approximation guarantees are in terms of the Poisson concavity ratio of ϕ, which we de ne as follows:

α ϕ := inf x∈N * E[ϕ(Poi(x))] ϕ(E[Poi(x)]) = inf x∈N * E[ϕ(Poi(x))] ϕ(x) . (3.1) 3. T A G C C P
Here Poi(x) denotes a Poisson-distributed random variable with parameter x. We will write α ϕ (x

) := E[ϕ(Poi(x))] ϕ(x)
, with α ϕ (0) = 1, and hence (see Proposition 3.18),

α ϕ = min x∈N * α ϕ (x) = inf x∈R + α ϕ (x). 2
Our main result is that the ϕ-M C problem admits an e cient α ϕ -approximation algorithm, when ϕ is normalized nondecreasing concave, and this approximation guarantee is tight when ϕ grows sublinearly. Formally, Theorem 3.1. For any normalized nondecreasing concave function ϕ, there exists a α ϕapproximation algorithm for the ϕ-M C problem running in polynomial time. Furthermore, for ϕ(n) = o(n), it is NP-hard to approximate the ϕ-M C problem within a factor better than α ϕ + ε, for any constant ε > 0, even restricted to instances with unit weights and regular subsets.

Before detailing the proof of the theorem, we provide a few remarks and connections to related work.

Applications and related work

We can directly reduce the standard maximum coverage problem to ϕ-M C by setting ϕ(j) = min(j, 1). In this case α ϕ = 1 -e -1 . One can also encapsulate, within our framework, the -M C problem studied in [START_REF] Barman | Tight approximation bounds for maximum multi-coverage[END_REF] by instantiating ϕ(j) = min(j, ). In this setting, we recover the approximation ratio α ϕ = 1 -e - ! , which matches the approximation guarantee obtained in [START_REF] Barman | Tight approximation bounds for maximum multi-coverage[END_REF] (see Proposition 3.31). Note that the hardness result in [START_REF] Barman | Tight approximation bounds for maximum multi-coverage[END_REF] was based on the unique games conjecture, whereas here we prove that this guarantee is tight under P = NP.

The initial motivation for studying ϕ-M C was to generalize the -M C problem studied in [START_REF] Barman | Tight approximation bounds for maximum multi-coverage[END_REF] as well as its -list-decoding interpretation. In the channel coding problem [START_REF] Barman | Algorithmic aspects of optimal channel coding[END_REF], we are given a noisy channel W (y|x), de ned as a conditional probability distribution, and the goal is to send a message chosen uniformly in [k] through this channel. In order to do so, the sender encodes the input message in X , and the receiver decodes back the output of the channel from Y to [k]. The objective is to nd a code that maximize the probability of successfully decoding the message. In the -list-decoding problem [START_REF] Elias | List decoding for noisy channels[END_REF][START_REF] John | List decoding[END_REF], instead of decoding the output into a single message, the decoder outputs a list of size of possible messages, where is some xed parameter common to all lists. We consider that the decoding is successful if the initial message belongs to that list. The generalization of that problem, which we call the ϕ-list-decoding problem, allows decoding lists of any size. However, the decoding will be successful only with probability ϕ( ) when the initial message belongs to that list, where is the size of that list. One can see that with a coverage function ϕ(j) = min(j, ), we recover the -list-decoding problem. When the channel is of the form W (y|x) = 1 t for y ∈ T x with |T x | = t and W (y|x) = 0 elsewhere, the success probability of the ϕ-list-decoding problem for a code S ⊆ X can be written as 1 kt y∈Y ϕ(|S| y ), with |S| y = |{x ∈ S : y ∈ T x }|, which is a particular instance of the ϕ-M C problem; see Section 3.3.1 for further details. Therefore, for that class of channels, a tight approximation ratio α ϕ follows from Theorem 3.1.

Another application of ϕ-M C is in the context of multiwinner elections that entail selecting k (out of m) candidates with the objective of maximizing the cumulative utility of n voters; here, the utility of each voter a ∈ [n] increases as more and more approved (by a) candidates get selected. One can reduce multiwinner elections to a coverage problem by considering subset T i ⊆ [n] as the set of voters that approve of candidate i ∈ [m] and ϕ(j) as the utility that an agent achieves from j approved selections. 3 Addressing multiwinner elections in this standard utilitarian model, Dudycz et al. [START_REF] Dudycz | Tight approximation for proportional approval voting[END_REF] obtain tight approximation guarantees for some well-studied classes of utilities. Speci cally, the result in [START_REF] Dudycz | Tight approximation for proportional approval voting[END_REF] applies to the classic proportional approval voting rule, which assigns a utility of j i=1 1 i for j approved selections. This voting rule corresponds to the coverage problem with ϕ(j) = j i=1 1 i , which we denote as the P A V problem (PAV for short). Section 3.3.2 shows that Theorem 3.1 holds for all the settings considered in [START_REF] Dudycz | Tight approximation for proportional approval voting[END_REF] and, in fact, applies more generally. In particular, the voting version of -M C (studied in [START_REF] Skowron | Finding a collective set of items: From proportional multirepresentation to group recommendation[END_REF]) can be addressed by Theorem 3.1, but not by the result in [START_REF] Dudycz | Tight approximation for proportional approval voting[END_REF]. Such a separation also arises when one truncates the proportional approval voting rule to, say, candidates, i.e., upon setting ϕ(j) = min(j, ) i=1

1 i .
Given that multiwinner elections model multiple real-world settings (e.g., committee selection [START_REF] Skowron | Finding a collective set of items: From proportional multirepresentation to group recommendation[END_REF] and parliamentary proceedings [BLS17]), instantiations of ϕ-M C in such social-choice contexts substantiate the applicability of our algorithmic result.

Coverage functions arise in numerous resource-allocation settings, such as sensor allocation [MW08], job scheduling, and plant location [START_REF] Cornuéjols | Exceptional paper-location of bank accounts to optimize oat: An analytic study of exact and approximate algorithms[END_REF]. The goal, broadly, in such setups is to select k subsets of resources (out of m pre-speci ed ones) such that the welfare generated by the selected resources is maximized-each resource's contribution to the welfare increases with the number of times it is selected. This problem can be cast as ϕ-M C by setting n to be the number of resources, {T i } i∈ [m] as the given collection of subsets, and ϕ(j) to be the welfare contribution of a resource when it is covered j times. 4 Here, we mention a speci c allocation problem to highlight the relevance of studying ϕ beyond the standard coverage and -coverage formulations (see Section 3.3.4 for details): in the V T A problem [Mur00, PM18] (VTA for short) the resources are n targets and covering a target j times contributes ϕ p (j) = 1-(1-p) j p to the welfare; here, p ∈ (0, 1) is a given parameter. Interestingly, we nd that for this problem, the approximation ratio α ϕ we obtain can outperform the Price of Anarchy (PoA), which corresponds to the approximation ratio of any method wherein the agents sel shly maximize their utilities (see Section 3.3.4 for further discussion of this point). By contrast, in the resource allocation problem with ϕ(j) = min(j, ), the price of anarchy is equal to α ϕ ; see [START_REF] Chandan | Optimal mechanisms for distributed resource-allocation[END_REF] for details. Another allocation problem studied in [START_REF] Paccagnan | Utility design for distributed resource allocationpart II: Applications to submodular, covering, and supermodular problems[END_REF] corresponds to ϕ-M C with ϕ(j) = j d , for a given parameter d ∈ (0, 1). We refer to this instantiation as the d-P function.

Theorem 3.1 gives us a tight approximation bound of α ϕ for all the above-mentioned applications of ϕ-M C . The values of α ϕ for these instantiations are listed in It is relevant to compare the approximation guarantee, α ϕ , obtained here with the approximation ratio based on the notion of curvature of submodular functions. Note that if ϕ is nondecreasing and concave, then C ϕ is submodular. One can show, via a direct calculation, that for such a submodular C ϕ the curvature (as de ned in [START_REF] Conforti | Submodular set functions, matroids and the greedy algorithm: Tight worst-case bounds and some generalizations of the Rado-Edmonds theorem[END_REF]) is given by c = 1 -(ϕ(m) -ϕ(m -1)) for instances with at most m cover sets; see Proposition 3.19. Therefore, the algorithm of Sviridenko et al. [START_REF] Sviridenko | Optimal approximation for submodular and supermodular optimization with bounded curvature[END_REF] provides an approximation ratio of 1 -ce -1 for the ϕ-M C problem. We note that the Poisson concavity ratio α ϕ is always greater than or equal to this curvature-dependent ratio (Proposition 3.22). Speci cally, for p-V T A , it is strictly better for all p / ∈ {0, 1} and for -M C , it is strictly better for all ≥ 2 as remarked in [START_REF] Barman | Tight approximation bounds for maximum multi-coverage[END_REF]. Therefore, for the setting at hand, we improve the approximation guarantee obtained in [START_REF] Sviridenko | Optimal approximation for submodular and supermodular optimization with bounded curvature[END_REF].

Remarks on the Poisson concavity ratio α ϕ . By Jensen's inequality along with the nonnegativity and concavity of ϕ, we have that α ϕ ∈ [0, 1]. We show that α ϕ can be computed numerically up to any precision ε > 0, in time that is polynomial in 1 ε . In fact, Proposition 3.17 shows that α ϕ (x) ≥ 1 -ε for all x ≥ N ε := 6 ε 4 . Thus, we can iterate over all x ∈ {1, 2, . . . , N ε } and nd min x∈[Nε] α ϕ (x) up to ε precision (under reasonable assumptions on ϕ). This gives us a method to overall compute α ϕ , up to an absolute error of 2ε:

if α ϕ ≤ 1 -ε, then computing min x∈[Nε] α ϕ (x) (up to ε precision) su ces. Otherwise, if α ϕ ≥ 1 -ε, then α ϕ (1)
≤ 1 provides the desired bound. Furthermore, we note that Proposition 3.16 shows that even if we consider α ϕ (x) over all x ∈ R + , an in mum (i.e., the value of α ϕ ) is achieved at an integer.

Further hardness under Gap-ETH Theorem 3.1 shows that, under the assumption P = NP, no polynomial-time algorithm can approximate ϕ-M C within a better ratio than α ϕ for sublinear ϕ. One natural question that arises is whether relaxing the running time constraint helps. More precisely, since there are m k = O(m k ) choices of k cover sets among the m available, a simple exhaustive search algorithm works in time O(m k ). We can ask if FPT algorithms with respect to k, running in time f (k) • m o(k) with f an arbitrary function, can do better. As in [START_REF] Dudycz | Tight approximation for proportional approval voting[END_REF], we use the result of [START_REF] Manurangsi | Tight running time lower bounds for strong inapproximability of maximum k-coverage, unique set cover and related problems (via t-wise agreement testing theorem[END_REF] to show in Theorem 3.14 that such algorithms cannot approximate ϕ-M C within a better ratio than α ϕ for sublinear ϕ, under the Gap-ETH hypothesis [CCK + 17]; see Section 3.2.3 for more details. This means that the brute-force strategy is essentially the best, if one wants to get a better approximation ratio than α ϕ .

Proof techniques and organization In Section 3.1, we present our approximation algorithm for the ϕ-M C . The algorithm is an application of pipage rounding, a technique introduced in [AS04], on a linear programming relaxation of ϕ-M C . We show that the multilinear extension F ϕ of C ϕ is e ciently computable and thus, we can compute an integer solution x int from the optimal fractional solution x * satisfying C ϕ (x int ) ≥ F ϕ (x * ). Using the notion of convex order between distributions, we show that

F ϕ (x * ) ≥ a∈[n] w a E[ϕ(Poi(|x * | a ))],
where |x| a = i∈[m]:a∈T i x i . Comparing this to the value a∈[n] w a ϕ(|x * | a ) taken by the linear program, we get a ratio given by the Poisson concavity ratio α ϕ . The concavity of ϕ is crucial at several steps of the proof: it guarantees that the natural relaxation can be written as a linear program, it is used to relate between sums of Bernouilli random variables and a Poisson random variable via the convex order, as well as for the fact that we can restrict the in mum in the de nition of α ϕ to integer values of x. The generalization to matroid constraints follows in a standard way and is presented in Section 3.1.1.

In Section 3.2, we present the hardness result for ϕ-M C

. For this, we de ne a generalization of the partitioning gadget of Feige [START_REF] Feige | A threshold of ln n for approximating set cover[END_REF], extending also [START_REF] Barman | Tight approximation bounds for maximum multi-coverage[END_REF]. Roughly speaking, for an integer x ϕ ∈ N, it is a collection of x ϕ -covers of the set [n] (an x-cover is a collection of subsets such that each element a ∈ [n] is covered x times, or in other words, its ϕ-coverage is ϕ(x)n) that are incompatible in the sense that if we take an element from each one of these x ϕ -covers, then the ϕ-coverage is bounded approximately by E[ϕ(Poi(x ϕ ))]n. Then, we construct an instance of ϕ-M C from an instance of the NP-hard problem Label Cover (as in [START_REF] Dudycz | Tight approximation for proportional approval voting[END_REF]) using such a gadget with x ϕ ∈ argmin x∈N α ϕ (x). Having set up the partitioning gadget, the analysis of the reduction can be obtained by carefully generalizing the reductions of [START_REF] Barman | Tight approximation bounds for maximum multi-coverage[END_REF] and [START_REF] Dudycz | Tight approximation for proportional approval voting[END_REF].

In Section 3.3, we present di erent domains of application of our result.

Approximation Algorithm for ϕ-M C

Fix a function ϕ : N → R + that is normalized, nondecreasing and concave. The ϕ-M C problem is de ned as follows. The input to the problem is given by positive integers n, m, t and m subsets T 1 , . . . , T m of the set [n] (described as characteristic vectors), the weights w a ∈ Q * + for a ∈ [n] (described as a pair of bitstring of length t), as well as an integer k ∈ {1, . . . , m}. The output is a subset S ⊆ [m] of size k that maximizes

C ϕ (S) = a∈[n] w a ϕ(|S| a ), where |S| a = |{i ∈ S : a ∈ T i }|.
Note that the input to this problem can be speci ed using n(m + 2t) + O(log nmt) bits. To reduce the number of parameters, we will assume that t is polynomial in n and m, so that a polynomial-time algorithm for this problem means an algorithm that runs in time polynomial in n and m. The counting function ϕ is xed and does not depend on the instance of the problem, but for a given instance the problem only depends on the values ϕ(0), ϕ(1), . . . , ϕ(m). We assume that we have black box access to ϕ and to ensure that all the algorithms run in polynomial time, we assume that ϕ(j) can be described with 3. T A G C C P a number of bits that is polynomial in j and that this description can be computed in polynomial time.

We now describe the approximation algorithm for ϕ-M C that we analyze. As described above, we follow the standard relax and round strategy, as in [START_REF] Barman | Tight approximation bounds for maximum multi-coverage[END_REF]. First, we de ne a natural convex relaxation.

De nition 3.1 (Relaxed program). maximize a∈[n] w a c a subject to c a ≤ ϕ(|x| a ), ∀a ∈ [n], with |x| a := i∈[m]:a∈T i x i 0 ≤ x i ≤ 1, ∀i ∈ [m] m i=1 x i = k . (3.2)
As previously mentioned, ϕ is de ned on R + by extending it in a piecewise linear fashion on non-integral points. As such, the constraint c a ≤ ϕ(|x| a ) is equivalent to m linear constraints. In fact, we can de ne ϕ j to be the linear function ϕ j (t) = (ϕ(j) -ϕ(j -1))t -(j -1)ϕ(j) + jϕ(j -1) for j ∈ [m]. Since ϕ is concave, we have that for all t ∈ [0, m], ϕ(t) = min j∈[m] ϕ j (t). Thus, the constraint c a ≤ ϕ(|x| a ) is equivalent to c a ≤ ϕ j (|x| a ) for all j ∈ [m] and so the program from De nition 3.1 can be stated as a linear program. Overall there are n + m variables and (n + 1)m + 1 linear constraints, and by assumption all the coe cients can be described using a number of bits that is polynomial in n and m. Hence, an optimal solution of this linear program can be found in polynomial time.

Observe that the program from De nition 3.1 is indeed a relaxation of the ϕ-M C problem. To see this, given a set S of size k, consider the characteristic vector x ∈ {0, 1} m de ned by x i = 1 if and only if i ∈ S. Then for all a ∈ [n], we can set c a = ϕ(|x| a ) = ϕ(|S| a ), and we get an objective value of a∈[n] w a ϕ(|S| a ) which is exactly C ϕ (S). When solving the program from De nition 3.1, we get an optimal x * ∈ [0, 1] m which is in general not integral. Next, we describe a method to round it to an integral vector x int ∈ {0, 1} m .

Rounding For a submodular function f : {0, 1} m → R , one can use pipage rounding [AS04, Von07, CCPV11] to transform, in polynomial time, any fractional solution

x ∈ [0, 1] m satisfying m i=1 x i = k into an integral vector x int ∈ {0, 1} m such that m i=1 x int i = k and F (x int ) ≥ F (x)
, where F corresponds to the multilinear extension of f , provided that F (x) is computable in polynomial time for a given x; see e.g., [START_REF] Vondrák | Submodularity in Combinatorial Optimization[END_REF]Lemma 3.4]. The multilinear extension F :

[0, 1] m → R of f is de ned by F (x 1 , . . . , x m ) := E[f (X 1 , . . . , X m )],
where X i are independent random variables with X i ∼ Ber(x i ), i.e., X i ∈ {0, 1} with P(X i = 1) = x i . Note that F (x) = f (x) for an integral vector x ∈ {0, 1} m . We apply this strategy to C ϕ , which is shown to be submodular in Proposition 3.19, and the solution x * of the linear programming relaxation from De nition 3.1. Note that overall the algorithm runs in polynomial time, since here F (x) is computable in polynomial time for a given x (see Proposition 3.23). We now analyze the value returned by the algorithm. Using the property of pipage rounding, with the notation X = (X 1 , . . . , X m ) and Ber(x) = (Ber(x 1 ), . . . , Ber(x m )), we get:

C ϕ (x int ) = E X∼Ber(x int ) [C ϕ (X)] ≥ E X∼Ber(x * ) [C ϕ (X)] .
Then, we only need to relate E X∼Ber(x * ) [C ϕ (X)] to the optimal value of the linear programming relaxation 3.1, which can only be larger than the optimal value of the ϕ-M C problem.

Theorem 3.2. Let x, c be a feasible solution of the program from De nition 3.1 and let X ∼ Ber(x). Recalling the de nition of α ϕ and α ϕ (j) from (3.1), we have

E X∼Ber(x) [C ϕ (X)] ≥ min j∈[m] α ϕ (j) a∈[n]
w a c a .

In particular, this implies that the described polynomial-time algorithm achieves an approximation ratio of α ϕ :

C ϕ (x int ) ≥ α ϕ a∈[n] w a c * a ≥ α ϕ max S⊆[m]:|S|=k C ϕ (S) .
Proof. By linearity of expectation and the fact that the weights w a are positive, it is su cient to show that for all a ∈ [n]:

E[C ϕ a (X)] ≥ min j∈[m] α ϕ (j) c a ,
where C ϕ a (S) := ϕ(|S| a ). Note that |X| a = i∈[m]:a∈T i X i , and thus:

E[C ϕ a (X)] = E   ϕ   i∈[m]:a∈T i X i     = E   ϕ   i∈[m]:a∈T i Ber(x i )     ≥ E   ϕ   Poi   i∈[m]:a∈T i x i       thanks to Proposition 2.2 = E[ϕ(Poi(|x| a ))] ≥ min(α ϕ ( |x| a ), α ϕ ( |x| a ))ϕ(|x| a ) thanks to Proposition 3.16 ≥ min j∈[m] α ϕ (j) ϕ(|x| a ) ≥ min j∈[m]
α ϕ (j) c a .

(3.3)

Generalization to Matroid Constraints

Instead of taking a cardinality constraint k on the size of the subset S, we look now at general matroid constraints on S. Speci cally, as input, instead of k, we take a matroid M de ned on [m] and given by a set of linear constraints describing its base polytope B(M).

T

A G C C P
The output is a set S ∈ M that maximizes C ϕ (S). Note that the cardinality constraint considered above is the special case where M is the uniform matroid (composed of all subsets of size at most k) with its base polytope B(M) = {x ∈ [0, 1] m : m i=1 x i = k}. We note that in order to establish Theorem 3.2, the cardinality constraint m i=1 x i = k is not used. Thus, since the pipage rounding strategy applies to matroid constraints M (see [START_REF] Vondrák | Submodularity in Combinatorial Optimization[END_REF]Lemma 3.4]), the strategy and the analysis of its e ciency generalize immediately when applied to the following linear program:

De nition 3.2 (Relaxed program for matroid constraints). maximize a∈[n] w a c a subject to c a ≤ ϕ(|x| a ), ∀a ∈ [n] 0 ≤ x i ≤ 1, ∀i ∈ [m]
x ∈ B(M) the base polytope of M .

(3.4) Theorem 3.3. Let x, c a feasible solution of the program from De nition 3.2 and X ∼ Ber(x). Then:

E X∼Ber(x) [C ϕ (X)] ≥ min j∈[m] α ϕ (j) a∈[n]
w a c a .

In particular, this implies that the described polynomial-time algorithm achieves an approximation ratio of α ϕ :

C ϕ (x int ) ≥ α ϕ a∈[n] w a c * a ≥ α ϕ max S∈M C ϕ (S) .

Hardness of Approximation for ϕ-M C

In this section, we establish an inapproximability result for the ϕ-M C problem with unit weights and regular subsets under cardinality constraints. Throughout this section, we use Γ to denote the universe of elements and, hence, an instance of the ϕ-M C problem consists of Γ, along with a collection of regular subsets F = {F i ⊆ Γ} m i=1 (i.e. ∀i ∈ [m], |F i | = t for some t) and an integer k. Recall that the objective of this problem is to nd a size-k subset S ⊆ [m] that maximizes C ϕ (S) = a∈Γ ϕ(|S| a ).

We establish the following theorem in this section: Theorem 3.4. It is NP-hard to approximate the ϕ-M C problem for ϕ(n) = o(n) within a factor greater that α ϕ + ε for any ε > 0, even restricted to instances with unit weights and regular subsets.

Our reduction is based on a problem called h-A L C

, which is equivalent to the more standard G L C problem as will be shown in Appendix 3.5.3.

De nition 3.3 (h-A L C

). An instance

G = (V, E, [L], [R], {π e,v } e∈E,v∈e ) of h-A L C is characterized by an h-uniform regular hypergraph (V, E) and con- straints π e,v : [L] → [R].
Here, each h-uniform hyperedge represents a h-ary constraint. Additionally, for any labeling σ : V → [L], we have the following notions of strongly and weakly satis ed constraints:

• An edge e = (v 1 , . . . , v h ) ∈ E is strongly satis ed by σ if: ∀x, y ∈ [h], π e,vx (σ(v x )) = π e,vy (σ(v y )) .
• An edge e = (v 1 , . . . , v h ) ∈ E is weakly satis ed by σ if:

∃x = y ∈ [h], π e,vx (σ(v x )) = π e,vy (σ(v y )) . Proposition 3.5 (δ, h-A G L C ).
For any xed integer h ≥ 2 and xed δ > 0, there exists an R 0 such that for any integer R ≥ R 0 , it is NP-hard for all instances

G = (V, E, [L], [R], {π e,v } e∈E,v∈e ) of h-A L C with right alphabet [R] to

distinguish between:

YES: There exists a labeling σ that strongly satis es all the edges. NO: No labeling weakly satis es more than δ fraction of the edges.

Partitioning System

In order to prove Theorem 3.4, we will need to generalize the partitioning system of Feige [START_REF] Feige | A threshold of ln n for approximating set cover[END_REF], a constant-size combinatorial object used to prove complexity hardness results. Note that it was already generalized in [START_REF] Barman | Tight approximation bounds for maximum multi-coverage[END_REF], over which we base our version. For any set [n] and Q ⊆ 2 [n] , we overload the de nition

C ϕ (Q) := a∈[n] ϕ(|Q| a ) with the notations |Q| a := |{P ∈ Q : a ∈ P }| and C ϕ a (Q) := ϕ(|Q| a ). Let us take x ϕ ∈ argmin x∈N * α ϕ (x), thus α ϕ = α ϕ (x ϕ ). We say that Q is an x-cover of x ∈ N if every element of [n] is covered x times, so C ϕ (Q) = nϕ(x). De nition 3.4. An ([n], h, R, ϕ, η)-partitioning system consists of R collections of subsets of [n], P 1 , . . . , P R ⊆ 2 [n] , that satisfy xϕn h ∈ N, x ϕ ≥ h and: 1. For every i ∈ [R], P i is a collection of h subsets P i,1 , . . . , P i,h ⊆ [n] each of size xϕn h
which is an x ϕ -cover.

For any

T ⊆ [R] and Q = {P i,j(i) : i ∈ T } for some function j : T → [h], we have C ϕ (Q) -ψ ϕ |T |,h n ≤ ηn where: ψ ϕ k,h := E ϕ Bin k, x ϕ h . (3.5) 29 3. T A G C C P Remark. In particular, for any Q = {Q 1 , . . . , Q k } with Q i of size xϕn h , we have that C ϕ (Q) ≤ nϕ(k xϕ h ). Indeed C ϕ (Q) = a∈[n] ϕ(|Q| a ) with a∈[n] |Q| a = i∈[k] |Q i | = k • xϕn h
. By concavity of ϕ and Jensen's inequality, this function is maximized when all |Q| a are equals, where we get nϕ(k

xϕ h ). Proposition 3.6. For every choice of R, h ∈ N with h ≥ x ϕ , η ∈ (0, 1), n ≥ η -2 Rϕ(R) 2 log(20(h + 1)) such that xϕn h ∈ N, there exists a ([n], h, R, ϕ, η)-partitioning system, which can be found in time exp(Rn log(n)) • poly(h).
The proof can be found in Appendix 3.5.4.

The Reduction

Proof of Theorem 3.4. Let ε > 0. Without loss of generality, we can assume that ε < 1. We show that it is NP-hard to reach an approximation ratio greater than

α ϕ + ε for the ϕ-M C problem, via a reduction from δ, h-A G L C . • η = ϕ(xϕ) 4xϕ ε, so 0 < η ≤ ε < 1, • h ≥ x ϕ such that ψ ϕ h,h -α ϕ ϕ(x ϕ ) ≤ η (see (3.5)
for the de nition of ψ ϕ ); such a choice exists thanks to Proposition 3.24,

• θ such that for all x ≥ θ, ϕ(x) x ≤ η, which exists since

ϕ(x) = o(x), • ξ = xϕ θ , • δ = η 2 ξ 3 h 2 ,
• R ≥ h large enough for Proposition 3.5 to hold.

Then, given an instance

G = (V, E, [L], [R], Σ, {π e,v } e∈E,v∈e ) of δ, h-A G L C , we construct an instance (Γ, F, k) of the ϕ-M C
problem with:

• n a large enough integer to have the existence of ([n], h, R, ϕ, η)-partitioning systems using Proposition 3.6. Note that the size of these partitioning systems is independent of the size of the instance G and that we can construct such a partitioning system in constant time (with relation to the size of the instance G) thanks to Proposition 3.6.

• Γ = [n] × E, • k = |V |,
• Consider a ([n], h, R, ϕ, η)-partitioning system, and call P = {P 1 , . . . , P R } the corresponding set of collections. De ne sets T e,v j β

:= P πe,v j (β),j × {e} where e = (v 1 , . . . , v h ) ∈ E, j ∈ [h], β ∈ [L]. Then, take F v β := e∈E:v∈e T e,v β
as cover sets and

F := {F v β , v ∈ V, β ∈ [L]}. Note that F v β = e∈E:v∈e T e,v β = e∈E:v∈e P πe,v j (β),j = |{e ∈ E : v ∈ e}| • xϕn h
which does not depend on v as the hypergraph (V, E) is regular. Therefore, those subsets are indeed regular.

We will now prove that if we are in a YES instance, we have that there exists T of size k such that C ϕ (T ) ≥ ϕ(x ϕ )|Γ| (completeness). Moreover, if we are in a NO instance, then we have that for all T of size k = |V |, C ϕ (T ) ≤ (α ϕ + ε)ϕ(x ϕ )|Γ| (soundness). Establishing these two properties would conclude the proof. In fact, an algorithm for ϕ-M C achieving a factor strictly greater than α ϕ + ε would allow us to decide whether we have YES or a NO instance of the NP-hard problem δ, h-A G L C .

In order to achieve this, let us de ne C ϕ,e := a∈[n]×{e} C ϕ a . In particular, we have

C ϕ = a∈Γ C ϕ a = e∈E C ϕ,e .
For T ⊆ F, we de ne the relevant part of T on e by:

T e := {T e,v β : v ∈ e, β ∈ [L], F v β ∈ T } = {F v β ∩ ([n] × {e}), F v β ∈ T } .
Note that C ϕ,e (T ) = C ϕ,e (T e ), and in particular C ϕ (T ) = e∈E C ϕ,e (T e ).

Completeness

Suppose the given h-A L C instance G is a YES instance. Then, there exists a labeling σ : V → [L] which strongly satis es all edges. Consider the collection of

|V | subsets T := {F v σ(v) : v ∈ V }. Fix e = (v 1 , . . . , v h ) ∈ E. Since e is strongly satis ed by σ, there exists r ∈ [R] such that π e,v i (σ(v i )) = r for all i ∈ [h]. Therefore, we get that T e = {T e,v i σ(v i ) } i∈[h] = {P r,i × {e}} i∈[h] is an x ϕ -cover of [n] × {e}, and so C ϕ,e (T e ) = nϕ(x ϕ ). Thus C ϕ (T ) = e∈E C ϕ,e (T e ) = |E|ϕ(x ϕ )n = ϕ(x ϕ )|Γ|.

Soundness

Suppose the given

h-A L C instance G is a NO instance.
Let us prove the contrapositive of the soundness: we suppose that there exists

T of size k = |V | such that C ϕ (T ) > (α ϕ + ε)ϕ(x ϕ )|Γ|.
Let us show that there exists a labeling σ that weakly satis es a strictly larger fraction of the edges than δ.

For every vertex v ∈ V , we de ne L(v) := {β ∈ [L] : F v β ∈ T } to be the candidate set of labels that can be associated with the vertex v. We extend this de nition to hyperedges e = (v 1 , . . . , v h ) where we de ne L(e) := i∈[h] L(v i ) to be the multiset of all labels associated with the edge. Note that |T e | = |L(e)|.

We say that

e = (v 1 , . . . , v h ) ∈ E is consistent if and only if there exists x = y ∈ [h] such that π e,vx (L(v x )) ∩ π e,vy (L(v y )) = ∅. We then decompose E in three parts: • B is the set of edges e ∈ E with |L(e)| ≥ h ξ .
• N is the set of consistent edges e ∈ E with |L(e)| < h ξ .

•

I = E -(B ∪ N ) is the set of inconsistent edges e ∈ E with |L(e)| < h ξ .
We want to show that the contribution of N is not too small, which we will use to construct a labeling weakly satisfying enough edges. This comes from the following lemmas:

|L(v)| = d v∈V |L(v)| = d|V | = |E|h .
Next, we bound the contribution of B:

Lemma 3.8. e∈B C ϕ,e (T e ) ≤ ε 4 ϕ(x ϕ )|Γ|.
Proof. We have:

e∈B C ϕ,
e∈B C ϕ,e (T e ) ≤ ηx ϕ |Γ| = ε 4 ϕ(x ϕ )|Γ| .
In order to bound the contribution of I, we will prove a property on inconsistent edges:

Proposition 3.9. Let e = (v 1 , . . . , v h ) ∈ E be an inconsistent hyperedge with respect to T . Then we have that C ϕ,e (T e ) -ψ ϕ |L(e)|,h n ≤ ηn. Proof. Since e is inconsistent, ∀x = y ∈ [h], π e,vx (L(v x )) ∩ π e,vy (L(v y )) = ∅. Therefore, for every i ∈ [R],
there is at most one v ∈ e such that i ∈ π e,v (L(v)), i.e., T e intersects with P i × {e} in at most one subset. This gives us a subset T ⊆ [R] and a function j

: T → [h] such that T e = {P i,j(i) × {e} : i ∈ T }. As a consequence, |T | = |T e | = |L(e)|
and by the second condition of the partitioning system, we get the expected result. Now, we can bound the contribution of I:

Lemma 3.10. e∈I C ϕ,e (T e ) ≤ (α ϕ + ε 2 )ϕ(x ϕ )|Γ|.
Proof. Thanks to Proposition 3.9, we have:

e∈I C ϕ,e (T e ) ≤ e∈I (ψ ϕ |L(e)|,h + η)n ≤ e∈E (ψ ϕ |L(e)|,h + η)n , since I ⊆ E and ψ ϕ |L(e)|,h ≥ 0. But e∈E |L(e)| =
|E|h by Lemma 3.7 and x → ψ ϕ x,h is concave thanks to Proposition 3.26, so we can use Jensen's inequality to get:

e∈E ψ ϕ |L(e)|,h ≤ |E|ψ ϕ e∈E |L(e)| |E| ,h = |E|ψ ϕ h,h ,
and thus:

e∈I C ϕ,e (T e ) ≤ (ψ ϕ h,h + η)n|E| ≤ (α ϕ ϕ(x ϕ ) + 2η)|Γ| ,
by de nition of h. This implies that the total contribution of inconsistent edges I is at most:

e∈I C ϕ,e (T e ) ≤ (α ϕ ϕ(x ϕ ) + 2η)|Γ| ≤ (α ϕ + ε 2 )ϕ(x ϕ )|Γ| , by de nition of η. Lemma 3.11. |N | |E| ≥ ξη.
Proof. Since we have supposed that e∈E C ϕ,e (T e ) = C ϕ (T ) > (α ϕ + ε)ϕ(x ϕ )|Γ|, and with the help of Lemmas 3.8 and 3.10, we have that the contribution of N is:

e∈N C ϕ,e (T e ) > ε 4 ϕ(x ϕ )|Γ| .
However, we have that for e ∈ N that:

C ϕ,e (T e ) ≤ nϕ |T e | x ϕ h = nϕ |L(e)| x ϕ h ≤ nϕ x ϕ ξ ≤ nx ϕ ξ ,
thanks to the remark on De nition 3.4 and the bound |L(e)| < h ξ . This implies that:

|N | |E| ≥ ξ x ϕ εϕ(x ϕ ) 4 = ξη .
Finally, we construct a randomized labeling σ :

V → [L] as follows: for v ∈ V , if L(v) = ∅, set σ(v) uniformly from L(v)
, otherwise set it arbitrarily. We claim that in expectation, this labeling must weakly satisfy δ fraction of the hyperedges.

To see this, x any e = (v 1 , . . . , v h ) ∈ N . Thus, there exists ∃x

= y ∈ [h] such that π e,vx (L(v x )) ∩ π e,vy (L(v y )) = ∅. Furthermore |L(v x )|, |L(v y )| ≤ h ξ . Thus, we have that π e,vx (L(v x )) = π e,vy (L(v y )) with probability at least 1 |L(vx)||L(vy)| ≥ ξ h 2
. Therefore:

E σ E e∼E [σ weakly satis es e] ≥ ξηE σ E e∼E [σ weakly satis es e|e ∈ N ] by Lemma 3.11 > η 2 ξ 3 h 2 = δ . (3.7)
In particular, there exists some labeling σ such that E e∼E [σ weakly satis es e] > δ, thus the soundness is proved.

T

A G C C P Furthermore, the previous proof implies an additive inapproximability result:

Corollary 3.12. There is no additive polynomial-time approximation scheme (PTAS) for the ϕ-M C problem for ϕ(n) = o(n) and α ϕ < 1 if P =NP; that is to say, a polynomialtime approximation that achieves an additive approximation error of ε • n, with n the size of the universe, for all ε > 0.

Remark. Contrary to fully polynomial-time approximation schemes (FPTAS), a PTAS does not necessarily run in polynomial time in 1 ε .

Proof. Let us x ε := 1-αϕ 3 ϕ(x ϕ ) > 0, which does not depend on the instance of the ϕ-M C problem. Assume for the sake of contradiction that we have a polynomialtime approximation that achieves an additive approximation error of ε • n, with n the size of the universe. In the previous proof (with the former ε de ned in Section 3.2.2 chosen equal to 1-αϕ 3 ), with such an additive approximation algorithm, one could distinguish between YES and NO instances of δ, h-A G L C . Indeed, a YES instance implies that the associated ϕ-M C instance has a solution of value ϕ(x ϕ )|Γ|, where Γ is the universe in that setting. On the other hand, a NO instance implies that all solutions of the associated ϕ-M C instance have a value smaller than or equal to 1+2αϕ 3 ϕ(x ϕ )|Γ|. Therefore, on a YES instance, our approximation algorithm would output a solution with a value larger than or equal to 2+αϕ 3 ϕ(x ϕ )|Γ|. This resulting inequality is enough to guarantee that it does not come from a NO instance, as it is strictly larger than 1+2αϕ 3

ϕ(x ϕ )|Γ| since α ϕ < 1. Thus, such an approximation algorithm would allow us to solve δ, h-A G L C in polynomial time, which is impossible if P =NP.

Further Hardness Under Gap-ETH

The Gap Exponential Time Hypothesis (Gap-ETH for short) states that, for some constant δ > 0, there is no 2 o(n) -time algorithm that, given n-variable 3-SAT formula, can distinguish whether the formula is fully satis able or that it is not even (1 -δ)-satis able. Gap-ETH is a standard assumption in proving FPT hardness of approximation (see e.g. [CCK + 17]). Under such hypothesis, Manurangsi showed the following theorem:

Theorem 3.13 ([Man20], adapted to (δ, h)-A G L C ). Assuming Gap-ETH, for every δ > 0, every h ∈ N, h ≥ 2 and any su ciently large R ∈ N (depending on δ, h), no f (k) • N o(k) -time algorithm can solve (δ, h)-A G L C with right alphabet [R],
where k denotes the number of vertices in h-A L C , N is the size of the instance, and f can be any function.

Such a statement can be made in terms of the (δ, h)-A G L

C problem, since it can be shown to be equivalent to δ-Gap-Label-Cover(t, R) (see Appendix 3.5.3 for more details).

Furthermore, in the previous reduction, the constructed instance (Γ, k, F) sizes satisfy |Γ| = n|E| (with n a constant independent of the size of the instance), k = |V |, and |F| = k • L. Therefore, plugin Theorem 3.13 in the previous reduction leads to the following hardness result: Theorem 3.14. Assuming Gap-ETH and ϕ(n) = o(n), we cannot achieve an (α ϕ + ε)approximation for the ϕ-M C problem, even in f (k) • m o(k) -time, for any function f , with m the number of cover sets and k the cardinality constraint.

Applications

In this section, we show how particular instances of ϕ-M C encapsule and generalize various problems across the elds of information theory [START_REF] Cover | Elements of Information Theory[END_REF], computational social choice [BCE + 16] and algorithmic game theory [START_REF] Nisan | Algorithmic Game Theory[END_REF].

Generalized List-Decoding

In the usual -list-decoding problem [START_REF] Elias | List decoding for noisy channels[END_REF][START_REF] John | List decoding[END_REF], we are given a noisy channel W (y|x), and the goal is to send a message chosen uniformly in [k] through this channel. In order to do so, the sender encodes the input message in X , and the receiver decodes a list of size of possible messages in [k] from the output of the channel in Y. The goal is to maximize the probability of successfully decoding the message, which happens when the initial message belongs to the decoded list. We consider the generalization of the -list-decoding problem that allows decoding lists of any size, with an associated cost function ϕ, which we call the ϕ-list-decoding problem. Now, the decoding will be successful only with probability ϕ( ) when the initial message belongs to a decoding list of size . Formally, the problem can be stated as:

max {Ly} y∈Y ,S⊆X :|S|=k 1 k x∈S y∈Y:x∈Ly ϕ(|L y |) |L y | W (y|x) ,
and can be directly reformulated as:

max S⊆X :|S|=k 1 k y∈Y max Ly⊆X ϕ(|L y |) |L y | x∈Ly 1 x∈S W (y|x) .
Let us show that with a coverage function ϕ(j) = min(j, ), we recover the -list-decoding problem. Indeed, let us consider an optimal solution S ⊆ X , {L y } y∈Y of the ϕ-list-decoding, with L y the list of accepted messages when the output of the channel is y. If all lists L y are of size , then as ϕ( ) = 1, we recover a solution of the -list-decoding problem with the same objective value. This shows in particular that the ϕ-list-decoding problem has an objective optimal value at least as large as the usual -list-decoding problem. Let us now consider an arbitrary set {L y } y∈Y , and let us look at a particular list L y with L y = . Let us show that we can e ciently build another list of size while keeping a solution with an objective value at least as good as with

L y . If |L y | ≤ , then ϕ(|L y |) |L y | = 1. Therefore, adding
elements in L y can only increase the objective value, as long as we keep ϕ(|Ly|) |Ly| = 1, i.e. that the list size is smaller than or equal to . Thus, we can transform all the lists L y with |L y | < into lists of size without decreasing the value of that solution. If |L y | > , then ϕ(|Ly|) |Ly| = |Ly| . We will keep only the most in uencing messages in L y and we will show that it does not decrease the objective value. Speci cally, let us sort the elements x ∈ L y in a nonincreasing order according to 1 x∈S W (y|x) and keep the rst elements of that list. We call L y the resulting list. We claim that replacing L y by L y will not decrease the 3. T A G C C P objective value. Indeed, we have:

x∈L y 1 x∈S W (y|x) ≥ |L y | x∈Ly 1 x∈S W (y|x) ,
as L y contains the largest elements of L y according to the weight 1 x∈S W (y|x). Therefore, since

ϕ(|L y |)
|L y | = 1, we have:

ϕ(|L y |) |L y | x∈L y 1 x∈S W (y|x) ≥ ϕ(|L y |) |L y | x∈Ly 1 x∈S W (y|x) ,
thus replacing L y by L y does not decrease the objective value. This concludes the equivalence of the -list-decoding problem with the ϕ-list-decoding with ϕ(j) = min(j, ).

When the channel is of the form W (y|x) = 1 t for y ∈ T x with |T x | = t and W (y|x) = 0 elsewhere, the success probability of the ϕ-list-decoding problem for a code S ⊆ X can be written as

1 kt y∈Y ϕ(|S y |), with S y := {x ∈ S : y ∈ T x }, which is a particular instance of the ϕ-M C
problem. Indeed, for a xed code S, the objective value becomes:

1 k y∈Y max Ly⊆X ϕ(|L y |) |L y | x∈Ly 1 x∈S W (y|x) = 1 kt y∈Y max Ly⊆X ϕ(|L y |) |L y | x∈Ly 1 x∈S 1 y∈Tx = 1 kt y∈Y max Ly⊆X ϕ(|L y |) |L y | |S y ∩ L y | .
(3.8)

We will show that max Ly⊆X ϕ(|Ly|)

|Ly| |S y ∩ L y | = ϕ(|S y |
), which will prove the claimed rewriting of the ϕ-list-decoding for those particular channels. As ϕ(0) = 0 and ϕ is concave, we have for all y ≥ x ≥ 0:

ϕ(y) -ϕ(0) y -0 ≤ ϕ(x) -ϕ(0) x -0 , i.e. x y ϕ(y) ≤ ϕ(x)
. Therefore, we have that:

ϕ(|L y |) |L y | |S y ∩ L y | ≤ ϕ(|S y ∩ L y |) ≤ ϕ(|S y |) , as ϕ is nondecreasing. Thus, max Ly⊆X ϕ(|Ly|) |Ly| |S y ∩ L y | ≤ ϕ(|S y |)
, with the equality obtained by choosing L y := S y .

Therefore, for that class of channels, a tight approximation ratio α ϕ follows from Theorem 3.1. As a consequence, the hardness part holds for the general ϕ-list-decoding problem, whereas nding an approximation algorithm achieving the ratio α ϕ for all channels is left as an open question.

Multiwinner Elections

As mentioned previously, multiwinner elections (with a utilitarian model for the voters) entail selection of k (out of m) candidates that maximize the utility across n voters. Here, the utility of each voter a ∈ [n] increases with the number of approved (by a) selections. The work of Dudycz et al. [START_REF] Dudycz | Tight approximation for proportional approval voting[END_REF] study the computational complexity of such elections and, in particular, address classic voting rules in which-for a speci ed sequence of nonnegative weights (w 1 , w 2 , . . .)-voter a's utility is equal to j i=1 w i , when he approves of j candidates among the selected ones. One can view this election exercise as a coverage problem by considering subset T i ⊆ [n] as the set of voters that approve of candidate i ∈ [m] and ϕ(j) = j i=1 w i . Indeed, for a subset of candidates S ⊆ [m], the utility of a voter a ∈

[n] is equal to ϕ(|S| a ), with |S| a = |{i ∈ S : a ∈ T i }|.
Dudycz et al. [START_REF] Dudycz | Tight approximation for proportional approval voting[END_REF] show that if the weights satisfy w 1 ≥ w 2 ≥ . . . (i.e., bear a diminishing returns property) along with geometric dominance (w i • w i+2 ≥ w 2 i+1 for all i ∈ N * ) and lim i→∞ w i = 0, then a tight approximation guarantee can be obtained for the election problem at hand. Note that the diminishing returns property implies that ϕ(j) = j i=1 w i is concave and lim i→∞ w i = 0 ensures that ϕ is sublinear (see Proposition 3.27). Hence, Theorem 3.1, together with Proposition 3.28, can be invoked to recover the result in [START_REF] Dudycz | Tight approximation for proportional approval voting[END_REF] where we get α ϕ = α ϕ (1). In fact, Theorem 3.1 does not require geometric dominance among the weights and, hence, applies to a broader class of voting rules. For instance, the geometric dominance property does not hold if one considers the voting weights induced by -M C , i.e., w i = 1, for 1 ≤ i ≤ , and w j = 0 for j > . However, using Theorem 3.1, we get that for this voting rule we can approximate the optimal utility within a factor of α ϕ = 1 -e - ! (see Proposition 3.31). Another example of such a separation arises if one truncates the proportional approval voting. The standard proportional approval voting corresponds to w i = 1 i , for all i ∈ N (equivalently, ϕ(j) = j i=1 1 i ) and falls within the purview of [START_REF] Dudycz | Tight approximation for proportional approval voting[END_REF]. While the truncated version with ϕ(j) = min(j, ) i=1 1 i , for a given threshold , does not satisfy geometric dominance, Theorem 3.1 continues to hold and provide a tight approximation ratio that can be computed numerically (see Proposition 3.21 and Table 3.1 for examples).

Resource Allocation in Multi-Agent Systems

A signi cant body of prior work in algorithmic game theory has addressed game-theoretic aspects of maximizing welfare among multiple (strategic) agents; see, e.g., [START_REF] Paccagnan | Utility design for distributed resource allocationpart II: Applications to submodular, covering, and supermodular problems[END_REF]. Complementing such results, this section shows that the optimization problem underlying multiple welfare-maximization games can be expressed in terms of ϕ-M C .

Speci cally, consider a setting with n resources, k agents, and a function ϕ

: N → R + .
Every agent i is endowed with a collection of resource subsets

A i = {T i 1 , . . . , T i m i } ⊆ 2 [n] (i.e., each T i j ⊆ [n]). The objective is to select a subset A i ∈ A i , for all i ∈ [k], so as to maximize W ϕ (A 1 , A 2 , . . . , A k ) := a∈[n] w a ϕ(|A| a ). Here, w a ∈ R + is a weight associated with a ∈ [n] and |A| a := |{i ∈ [k] : a ∈ A i }|.
We will refer to this problem as the ϕ-R A problem.

While ϕ-R A does not directly reduce to ϕ-M C , the next theorem shows that it corresponds to maximizing ϕ-coverage functions subject to a matroid constraint. Hence, invoking our result from Section 3.1.1, we obtain a tight α ϕ -approximation for ϕ-R A (see Appendix 3.5.5 for the proof):

Theorem 3.15. For any normalized nondecreasing concave function ϕ, there exists a α ϕapproximation algorithm for ϕ-R A running in polynomial time. Further-

3. T A G C C P more, for ϕ(n) = o(n), it is NP-hard to approximate ϕ-R
A within a factor better than α ϕ + ε, for any constant ε > 0.

Vehicle-Target Assignment

V T A [Mur00, PM18
] is another problem which highlights the applicability of coverage problems, with a concave ϕ. In particular, V T A can be directly expressed as ϕ-R A : the [n] resources correspond to targets, the agents correspond to vehicles i ∈ [k], each with a collection of covering choices A i ⊆ 2 [n] , and ϕ p (j) = 1-(1-p) j p , for a given parameter p ∈ (0, 1). As limit cases, we de ne ϕ 0 (j) := lim p→0 ϕ p (j) = j and ϕ 1 (j) := 1. Since ϕ p (j) is concave, by Proposition 3.32 and Theorem 3.15, we obtain a novel tight approximation ratio of α ϕ p = 1-e -p p for this problem. Also, one can look at the capped version of this problem, ϕ p (j) := ϕ p (min(j, )). In particular, we recover the -M C function when p = 0. In Figure 3.1, we have plotted several cases of the tight approximations α ϕ p in function of for several values of :

0 0.2 0.4 0.6 0.8 1 0.5 0.6 0.7 0.8 0.9 1 p α ϕ p ∞ α ϕ p 1 α ϕ p 2 α ϕ p 3 α ϕ p 4 α ϕ p 5 Figure 3.1 -Tight approximation ratios α ϕ p ,

where is the rank of the capped version of the p-V

T A problem. When p = 0, we recover the -coverage problem.

Paccagnan and Marden [START_REF] Paccagnan | Utility design for distributed resource allocationpart II: Applications to submodular, covering, and supermodular problems[END_REF] study the game-theoretic aspects of V T A . A key goal in [START_REF] Paccagnan | Utility design for distributed resource allocationpart II: Applications to submodular, covering, and supermodular problems[END_REF] is to bound the welfare loss incurred due to strategic selection by the k vehicles, i.e., the selection of each

A i ∈ A i by a self-interested vehi- cle/agent i ∈ [k].
The loss is quanti ed in terms of the Price of Anarchy (PoA). Formally, this performance metric is de ned as ratio between the welfare of the worst-possible equilibria and the optimal welfare. Paccagnan and Marden [START_REF] Paccagnan | Utility design for distributed resource allocationpart II: Applications to submodular, covering, and supermodular problems[END_REF] show that, for computationally tractable equilibrium concepts (in particular, for coarse correlated equilibria), tight price of anarchy bounds can be obtained via linear programs.

Note that our hardness result (Theorem 3.1) provides upper bounds on PoA of tractable equilibrium concepts-this follows from the observation that computing an equilibrium provides a speci c method for nding a coverage solution. In [START_REF] Chandan | Optimal mechanisms for distributed resource-allocation[END_REF] and in the partic-ular case of the -M C problem, it is shown that this in fact an equality, i.e., PoA = α ϕ if ϕ(j) = min(j, ) for all values of . However, numerically comparing the approximation ratio for V T A , α ϕ p = 1-e -p p , with the optimal PoA bound, we note that α ϕ p can in fact be strictly greater than the PoA guarantee; see Figure 3.2.

Another form of the current problem, considered in [START_REF] Paccagnan | Utility design for distributed resource allocationpart II: Applications to submodular, covering, and supermodular problems[END_REF], corresponds to ϕ d (j) = j d , for a given parameter d ∈ (0, 1). We refer to this instantiation as the d-P function and for it obtain the approximation ratio [START_REF] Paccagnan | Utility design for distributed resource allocationpart II: Applications to submodular, covering, and supermodular problems[END_REF], we were able to compute the blue curve PoA 20 , the Price of Anarchy of this problem for m = 20 players. Since the PoA only decreases when the number of players grows, this means that PoA < α ϕ in that case. As a comparison, the red curve Curv depicts the general approximation ratio (see [START_REF] Sviridenko | Optimal approximation for submodular and supermodular optimization with bounded curvature[END_REF]) obtained for submodular function with curvature c, with c = 1 -ϕ p (m) + ϕ p (m -1) here.

α ϕ d = e -1 +∞

Welfare Maximization for ϕ-Coverage

Maximizing (social) welfare by partitioning items among agents is a key problem in algorithmic game theory; see, e.g., the extensive work on combinatorial auctions [START_REF] Nisan | Algorithmic Game Theory[END_REF]. The goal here is to partition t items among a set of k agents such that the sum of values achieved by the agents-referred to as the social welfare-is maximized. That is, one needs to partition

[t] into k pairwise disjoint subsets A 1 , A 2 , . . . , A k with the objective of max- imizing k i=1 v i (A i ).
Here, v i (S) denotes the valuation that agent i has for a subset of items S ⊆ [t].

When each agent's valuation v i is submodular, a tight (1 -e -1 )-approximation ratio is known for social welfare maximization [START_REF] Vondrák | Submodularity in Combinatorial Optimization[END_REF]. This section shows that improved approximation guarantees can be achieved if, in particular, the agents' valuations are ϕcoverage functions. Towards a stylized application of such valuations, consider a setting in which each "item" b ∈ [t] represents a bundle (subset) of goods T b ⊆ [n] and the value of an agent increases with the number of copies of any good a ∈ [n] that get accumulated. Indeed, if each agent's value for j copies of a good is ϕ(j), then we have a ϕ-coverage function and the overall optimization problem is nd a k

-partition, A 1 , A 2 , . . . , A k , of [t] that maximizes k i=1 a∈[n] ϕ (|A i | a ) , where |A i | a := {b ∈ A i : a ∈ T b }.
In the current setup, one can obtain an α ϕ approximation ratio for social-welfare maximization by reducing this problem to ϕ-coverage with a matroid constraint, and applying the result from Section 3.1.1. Speci cally, we can consider a partition matroid over the universe [t] × [k]: for a bundle/item b ∈ [t] and an agent i ∈ [k], the element (b, i) in the universe represents that bundle b is assigned to agent i, i.e., b ∈ A i . The partition-matroid constraint is imposed to ensure that each bundle b is assigned to at most one agent. Furthermore, we can create k copies of the underlying set of goods [n] and set T (b,i) := {(a, i) : a ∈ T b } to map the ϕ-coverage over the universe to the social-welfare objective. This, overall, gives us the desired α ϕ approximation guarantee.

Conclusion

We have introduced the ϕ-M C problem where having c copies of element a gives a value ϕ(c). We have shown that when ϕ is normalized, nondecreasing and concave, we can obtain an approximation guarantee given by the Poisson concavity ratio

α ϕ := min x∈N * E[ϕ(Poi(x))] ϕ(E[Poi(x)]
) and we showed it is tight for sublinear functions ϕ. The Poisson concavity ratio strictly beats the bound one gets using the notion of curvature of submodular functions, except in limit cases such as M C where the bounds are equal.

An interesting open question is whether there exists combinatorial algorithms that achieve this approximation ratio. As mentioned in [START_REF] Barman | Tight approximation bounds for maximum multi-coverage[END_REF], for the -M C with ≥ 2, which is the special case where ϕ(x) = min(x, ), the simple greedy algorithm only gives a 1 -e -1 approximation ratio, which is strictly less than the ratio α ϕ = 1 -e - ! in that case. Also, for any geometrically dominant vector w = (ϕ(i + 1) -ϕ(i)) i∈N which is not p-geometric, such as P A V , the greedy algorithm achieves an approximation ratio which is strictly less than α ϕ (see Theorem 18 of [START_REF] Dudycz | Tight approximation for proportional approval voting[END_REF]).

Another open question is whether the hardness result remains true when ϕ(n) = o(n). A good example is given by ϕ(0) = 0 and ϕ(1 + t) = 1 + (1 -c)t with c ∈ (0, 1). We know that the problem is hard for c = 1 but easy for c = 0. One can show that the approximation ratio achieved by our algorithm is α ϕ = 1-c e in that case (which is the same approximation ratio obtained from the curvature in [START_REF] Sviridenko | Optimal approximation for submodular and supermodular optimization with bounded curvature[END_REF]), but the tightness of this approximation ratio remains open.

T

A G C C P

Appendix

General Properties

In this section, we will assume that ϕ is speci ed over nonnegative integers (i.e. ϕ :

N → R + )
and that it is nondecreasing, concave, and normalized (i.e. ϕ(0) = 0 and ϕ(1) = 1). We will consider its piecewise linear extension on R + by de ning ϕ(x)

:= λϕ( x ) + (1 -λ)ϕ( x ); here, parameter λ ∈ [0, 1] satis es x = λ x + (1 -λ) x .
Note that the piecewise linear extension is also nondecreasing and concave.

Proposition 3.16. For all x ∈ R + , we have

α ϕ (x) ≥ min(α ϕ ( x ), α ϕ ( x )); here, α ϕ (0) := lim x→0 α ϕ (x) = 1. Proof. For any x ≥ 1, consider parameter λ ∈ [0, 1] such that x = λ x + (1 -λ) x . Since x → E[ϕ(Poi(x))
] is concave (Proposition 3.25), the following bound holds for all x ≥ 1:

E[ϕ(Poi(x))] ≥ λE[ϕ(Poi( x ))] + (1 -λ)E[ϕ(Poi( x ))] = λα ϕ ( x )ϕ( x ) + (1 -λ)α ϕ ( x )ϕ( x ) by de nition of α ϕ (x) ≥ min(α ϕ ( x ), α ϕ ( x )) (λϕ( x ) + (1 -λ)ϕ( x )) = min(α ϕ ( x ), α ϕ ( x ))ϕ(x) , (3.9) since ϕ linear between integers. Therefore, α ϕ (x) = E[ϕ(Poi(x))] ϕ(x) ≥ min(α ϕ ( x ), α ϕ ( x )).
We will show next that α ϕ (x) is nonincreasing from 0 to 1, which implies that for x ∈ [0, 1), we have α ϕ (x) ≥ min(α ϕ ( x ), α ϕ ( x )). Since ϕ is linear between integers, ϕ(0) = 0 and ϕ(1) = 1, we have that ϕ(x) = x for all x ∈ [0, 1]. Therefore:

α ϕ (x) = E[ϕ(Poi(x))] x = e -x +∞ k=1 ϕ(k) k x k-1 (k -1)! = e -x +∞ k=0 ϕ(k + 1) k + 1 x k k! .
In particular, α ϕ (x) is well-de ned at 0 and α ϕ (0) = e -0 +∞ k=0 ϕ(k+1) k+1

0 k k! = 1. Now, consider the derivative: α ϕ (x) = e -x - +∞ k=0 ϕ(k + 1) k + 1 x k k! + +∞ k=1 ϕ(k + 1) k + 1 x k-1 (k -1)! = e -x +∞ k=0 ϕ(k + 2) k + 2 - ϕ(k + 1) k + 1 x k k! . (3.10) Note that ϕ(k+2) k+2 -ϕ(k+1) k+1 = ϕ(k+2)-ϕ(0) (k+2)-0 -ϕ(k+1)-ϕ(0) (k+1)-0
≤ 0; the last inequality follows from the concavity of ϕ. Hence, α ϕ (x) ≤ 0. That is, α ϕ (x) is nonincreasing from 0 to 1.

Proposition 3.17. For any ε > 0, the bound

1 -α ϕ (x) ≤ ε holds for all x ≥ 6 ε 4 . Proof. Write X ∼ Poi(x) and note that P(X ≤ x(1 -δ(x))) ≤ exp -xδ(x) 2 2(1+δ(x))
, for any positive function δ(•) which satis es δ(x) < 1, for all x > 1; see for instance [Can17].

Appendix

Therefore:

E[ϕ(X)] ≥ e -x +∞ k= x(1-δ(x)) ϕ(k) x k k! since ϕ nonnegative ≥ ϕ(x(1 -δ(x))) +∞ k= x(1-δ(x)) e -x x k k! since ϕ nondecreasing ≥ ϕ(x(1 -δ(x)))(1 -P(X ≤ x(1 -δ(x)))) ≥ ϕ(x(1 -δ(x))) 1 -exp - xδ(x) 2 2(1 + δ(x))
.

(3.11)

Next, we will show that ϕ(x(1-δ(x)))

ϕ(x) ≥ 1 - δ(x)+ 1 x 1-δ(x)
. Towards this end, we will rst bound ϕ(x+y)-ϕ(x) in terms of w x k = ϕ(x+k)-ϕ(x+k-1), which constitutes a nonincreasing sequence (since ϕ is concave):

ϕ(x + y) -ϕ(x) ≤ ϕ(x + y + 1) -ϕ(x + y ) + y k=1 w x k ≤ ( y + 1)w x 1 .
Applying this bound to x(1 -δ(x)) and xδ(x) gives us:

1 - ϕ(x(1 -δ(x))) ϕ(x) = ϕ(x) -ϕ(x(1 -δ(x))) ϕ(x) ≤ ( xδ(x) + 1)w x(1-δ(x)) 1 ϕ(x) ≤ xδ(x) + 1 ϕ(x) ϕ(x(1 -δ(x))) x(1 -δ(x)) ≤ xδ(x) + 1 x(1 -δ(x)) = δ(x) + 1 x 1 -δ(x) . (3.12)
Here, w

x(1-δ(x)) 1 = ϕ(x(1-δ(x))+1)-ϕ(x(1-δ(x))) (x(1-δ(x))+1)-(x(1-δ(x))) ≤ ϕ(x(1-δ(x)))-ϕ(0) x(1-δ(x))-0 = ϕ(x(1-δ(x)))
x(1-δ(x)) follows from the concavity of ϕ and ϕ(x(1-δ(x))) ϕ(x) ≤ 1 from the fact that ϕ is nondecreasing. Inequalities (3.11) and (3.12) lead to following upper bound on 1 -α ϕ (x) in terms of δ(x):

1 -α ϕ (x) = 1 - E[ϕ(Poi(x))] ϕ(x) ≤ 1 -1 - δ(x) + 1 x 1 -δ(x) 1 -exp - xδ(x) 2 2(1 + δ(x)) ≤ δ(x) + 1 x 1 -δ(x) + exp - xδ(x) 2 2(1 + δ(x)) . (3.13) Speci cally setting δ(x) = x -1 4 , we have (for all x ≥ 16): δ(x) ≤ 1 2 , 1 x ≤ x -1 4 , and exp -xδ(x) 2 2(1+δ(x)) ≤ exp - √ x 4 ≤ 2x -1 4 . Hence, inequality (3.13) reduces to 1 -α ϕ (x) ≤ 2x -1 4 1 -1 2 + 2x -1 4 ≤ 6x -1 4 for all x ≥ 16. If ε ≥ 1, we have 1 -α ϕ (x) ≤ 1 ≤ ε. Otherwise, we have that 6 ε 4 ≥ 6 4 ≥ 16. Therefore, given any ε > 0, for all x ≥ 6 ε 4 we have 1 -α ϕ (x) ≤ ε. 3. T A G C C P Proposition 3.18. We have that α ϕ = inf x∈R + α ϕ (x) = min x∈N * α ϕ (x).
Proof. Thanks to Proposition 3.16, we have that inf x∈R + α ϕ (x) = inf x∈N * α ϕ (x), and thanks to Proposition 3.17, since α ϕ (x) ≤ 1, we have that

inf x∈N * α ϕ (x) = min x∈N * α ϕ (x). Proposition 3.19. C ϕ is submodular, its curvature is at most c = 1 -(ϕ(m) -ϕ(m -1))
and it cannot be improved for a general instance with m cover sets.

Proof. We will use the following straightforward lemma:

Lemma 3.20 (Properties of |S| a = |{i ∈ S : a ∈ T i }|.
). We have:

1. |S| a ≤ |S|, 2. |S ∪ S | a ≤ |S| a + |S | a . If S ⊆ T then |S| a ≤ |T | a and |S ∪ {x}| a ≤ |S| a + 1, 3. If S ⊆ T , x ∈ T then |S| a = |T | a ⇒ |S ∪ {x}| a = |T ∪ {x}| a .
Let us show rst the submodularity of C ϕ . Let S ⊆ T ⊆ [m] and x ∈ T :

C ϕ (S ∪ {x}) -C ϕ (S) -(C ϕ (T ∪ {x}) -C ϕ (T )) = = a∈[n]
|T | a > |S| a so |T | a ≥ k + 1. i. If |T ∪ {x}| a = |T | a , then g(a) = ϕ(k + 1) -ϕ(k) ≥ 0 since ϕ is nondecreasing. ii. Else |T ∪ {x}| a = |T | a so with |T | a = with ≥ k + 1, we get that
|T | a = + 1. So we have that:

g(a) = ϕ(k + 1) -ϕ(k) -(ϕ( + 1) -ϕ( )) = ϕ(k + 1) -ϕ(k) (k + 1) -k - ϕ( + 1) -ϕ( ) ( + 1) - ≥ 0 , (3.15)
by concavity of ϕ: its slopes are nonincreasing.

Appendix

So in all cases, we have g(a) ≥ 0 so

C ϕ (S ∪ {x}) -C ϕ (S) -(C ϕ (T ∪ {x}) -C ϕ (T )) ≥ 0: C ϕ is submodular.
Let us now compute its curvature:

c = 1 -min i∈[m] C ϕ ([m]) -C ϕ ([m] -{i}) C ϕ ({i}) -C ϕ (∅) . Let i ∈ [m] xed: C ϕ ([m]) -C ϕ ([m] -{i}) C ϕ ({i}) -C ϕ (∅) = a∈[n] w a [ϕ(|[m]| a ) -ϕ(|[m] -{i}| a )] a∈[n] w a [ϕ(|{i}| a ) -ϕ(|∅| a )] = a∈T i w a [ϕ(|[m]| a ) -ϕ(|[m] -{i}| a )] a∈T i w a = a∈T i w a [ϕ(|[m]| a ) -ϕ(|[m]| a -1)] a∈T i w a , (3.16) since a ∈ T i . But |[m]| a ≤ m and ϕ concave, so ϕ(|[m]| a )) -ϕ(|[m]| a -1) ≥ ϕ(m) - ϕ(m -1) for all a ∈ [n].
As a consequence we have that:

C ϕ ([m]) -C ϕ ([m] -{i}) C ϕ ({i}) -C ϕ (∅) ≥ ϕ(m) -ϕ(m -1) ,
and this lower bound is true for its minimum over i. Thus we get that c ≤ 1 -(ϕ(m)ϕ(m-1)). Also one can nd instances for all m such that this bound is tight: take T 1 = {a} and ∀j ∈ [m], a ∈ T j for instance.

Proposition 3.21. Let ∈ N * . if ∀x ≥ , ϕ(x) = ϕ( ) + a(x -) for some 0 ≤ a ≤ ϕ( ) -ϕ( -1), then α ϕ (x) is nondecreasing from to +∞ and:

α ϕ (x) = ϕ( ) + a(x -) ϕ(x) - e -x ϕ(x) k=0 (ϕ( ) + a(x -) -ϕ(k)) x k k! -a x +1 ! .
In particular, α ϕ = min x∈[ ] α ϕ (x), and the argmin can be computed numerically.

Proof. One can compute a closed form value for α ϕ (x) using the fact that ϕ is linear from :

α ϕ (x) = e -x ϕ(x) +∞ k=0 ϕ(k) x k k! = e -x ϕ(x)   k=0 ϕ(k) x k k! + +∞ k= +1 (ϕ( ) + a(k -)) x k k!   = ϕ( ) -a ϕ(x) + e -x ϕ(x)   k=0 (ϕ(k) -ϕ( ) + a ) x k k! + ax +∞ k= +1 x k-1 (k -1)!   = ϕ( ) + a(x -) ϕ(x) + e -x ϕ(x) k=0 (ϕ(k) -ϕ( ) + a( -x)) x k k! + ax x ! ,
(3.17) and thus we get:

α ϕ (x) = ϕ( ) + a(x -) ϕ(x) - e -x ϕ(x) k=0 (ϕ( ) + a(x -) -ϕ(k)) x k k! -a x +1 ! . 3. T A G C C P
Let us show that it is nondecreasing from to +∞ by computing its derivative. Indeed, for x ≥ , we have that ϕ(x) = ϕ( ) + a(x -) and ϕ (x) = a, so:

α ϕ (x) = 1 -e -x k=0 x k k! - 1 ϕ(x) k=0 ϕ(k) x k k! + a x +1 ! .
Thus:

α ϕ (x) = e -x k=0 x k k! - 1 ϕ(x) k=0 ϕ(k) x k k! + a x +1 ! -e -x -1 k=0 x k k! - 1 ϕ(x) -1 k=0 ϕ(k + 1) x k k! + a( + 1) x ! + e -x a ϕ(x) 2 k=0 ϕ(k) x k k! + a x +1 ! = e -x x ! - 1 ϕ(x) -1 k=0 (ϕ(k) -ϕ(k + 1)) x k k! + ϕ( ) x ! + (a(x -) -a) x ! -e -x a ϕ(x) 2 k=0 ϕ(k) x k k! + a x +1 ! = e -x x ! + 1 ϕ(x) -1 k=0 (ϕ(k + 1) -ϕ(k)) x k k! - ϕ(x) -a ϕ(x) x ! -e -x a ϕ(x) 2 k=0 ϕ(k) x k k! + a x +1 ! = e -x 1 ϕ(x) -1 k=0 (ϕ(k + 1) -ϕ(k)) x k k! + e -x a ϕ(x) 2 ϕ(x) x ! - k=0 ϕ(k) x k k! -a x +1 ! .
(3.18)

If a = 0, then it is nonnegative since ϕ nondecreasing and nonnegative. Otherwise, suppose that a > 0. Then:

α ϕ (x) = ae -x ϕ(x) 2 -1 k=0 ϕ(x) ϕ(k + 1) -ϕ(k) a -ϕ(k) x k k! + (ϕ(x) -ϕ( ) -ax) x ! ≥ ae -x ϕ(x) 2 -1 k=0 (ϕ(x) -ϕ(k)) x k k! -a x ( -1)! , (3.19) since ϕ(k+1)-ϕ(k) a ≥ ϕ(k+1)-ϕ(k)
ϕ( )-ϕ( -1) ≥ 1 by concavity of ϕ. Thus:

α ϕ (x) ≥ ae -x ϕ(x) 2 -1 k=0 (ϕ( ) -ϕ(k)) x k k! + a (x -) -1 k=0 x k k! - x ( -1)! , but: (x -) -1 k=0 x k k! - x ( -1)! = k=1 x k k! - -1 k=0 x k k! - x ( -1)! = -, 3.5. Appendix so: α ϕ (x) ≥ ae -x ϕ(x) 2 -1 k=0 (ϕ( ) -ϕ(k)) x k k! -a ≥ ae -x ϕ(x) 2 -1 k=0 (ϕ( ) -ϕ(k)) x k k! -(ϕ( ) -ϕ( -1)) ≥ ae -x ϕ(x) 2 -1 k=0 (ϕ( ) -ϕ(k)) x k k! - -1 k=0 (ϕ( ) -ϕ( -1)) x k k! since x k k! ≥ k k! ≥ 1 for k ≤ = ae -x ϕ(x) 2 -1 k=0 (ϕ( -1) -ϕ(k)) x k k! ≥ 0 since ϕ nondecreasing .
(3.20) Thus, α ϕ (x) is nondecreasing from to +∞, and we get that

α ϕ = min x∈[ ] α ϕ (x).
Proposition 3.22. The Poisson concavity ratio α ϕ is always greater than or equal to the curvature-dependent ratio de ned in

[SVW17]: if ϕ is linear from m with slope 1 -c = ϕ(m) -ϕ(m -1), then we have α ϕ ≥ 1 -ce -1 .
Proof. Note that by Proposition 3.19, the curvature of C ϕ is equal to c, so the e ciency of the algorithm described in [START_REF] Sviridenko | Optimal approximation for submodular and supermodular optimization with bounded curvature[END_REF] is indeed 1 -ce -1 . Thanks to Proposition 3.21, we have that α ϕ = min x∈[ ] α ϕ (x), so we only have to show that:

min ∈[m] α ϕ ( ) ≥ 1 -ce -1 .
Let us denote by ϕ ( ,a) the function which is equal to ϕ for k ≤ and linear from with nonnegative coe cient a: ∀k ≥ , ϕ ( ,a) (k) = ϕ( ) + a(k -). Note that we ask that 0 ≤ a ≤ ϕ( ) -ϕ( -1) in order to ϕ ( ,a) to be nondecreasing concave, and ≥ 1.

This is done in two steps:

1. Let 1 ≤ ≤ m, then:

α ϕ ( ) = E[ϕ(Poi( ))] ϕ( ) = E[ϕ(Poi( ))] ϕ ( ,1-c) ( ) ≥ E[ϕ ( ,1-c) (Poi( ))] ϕ ( ,1-c) ( ) = α ϕ ( ,1-c) ( ) , since ϕ ( ,1-c) (x) ≤ ϕ(x) for all x.
Note that we have:

ϕ( ) -ϕ( -1) ≥ ϕ(m) -ϕ(m -1) = 1 -c ,
by concavity of ϕ. So, we only have to show that for all 1 ≤ ≤ m, we have

α ( ,1-c) := α ϕ ( ,1-c) ( ) ≥ 1 -ce -1 . 2. Let us show that α ( ,1-c) := α ϕ ( ,1-c) ( ) ≥ 1 -ce -1 for 1 ≤ ≤ m. 3. T A G C C P
Using the closed-form expression of Proposition 3.21 on ϕ ( ,1-c) evaluated at , one gets:

α ( ,1-c) = α ϕ ( ,1-c) ( ) = 1 -e - -1 k=0 ϕ( ) -ϕ(k) ϕ( ) k k! - 1 -c ϕ( ) +1 ! .
The worst case occurs when ϕ ( ,1-c) is linear between 1 and , which we call ϕ

( ,1-c) lin . Indeed, with b := ϕ( )-1 -1 , for 1 ≤ k ≤ , we have that ϕ ( ,1-c) lin (k) = 1 + b(k -1). But: -1 k=0 ϕ( ) -ϕ(k) ϕ( ) k k! ≤ 1 + -1 k=1 ϕ( ) -(1 + b(k -1)) ϕ( ) k k! , since ϕ(k) ≥ 1 + b(k -1), because ϕ(k)-ϕ(1) k-1 ≥ ϕ( )-ϕ(1) -1
= b by concavity of ϕ. In that case, the expression can be simpli ed:

α ( ,1-c) ≥ α ϕ ( ,1-c) lin ( ) = 1 -e -1 + -1 k=1 b( -k) ϕ( ) k k! - 1 -c ϕ( ) +1 ! = 1 -e -1 + b ϕ( ) -1 k=1 k k! - b ϕ( ) -1 k=1 k-1 (k -1)! - 1 -c ϕ( ) +1 ! = 1 - e - ϕ( ) ϕ( ) + b -1 ( -1)! -1 -(1 -c) +1 ! = 1 - e - ϕ( ) 1 + b( -1) + b ( -1)! - -(1 -c) +1 ! = 1 -e -1 -b + (b -(1 -c)) +1 ! ϕ( ) .
(3.21)

We have also that b ≥ ϕ( ) -ϕ( -1) ≥ 1 -c since ϕ concave. As a function of

(b -(1 -c)) for c xed, we get g(x) := 1 -e - c+x +1 ! -1 1+(x+(1-c))( -1) . In particular, we have that α ϕ ( ,(1-c)) lin ( ) = g(b -(1 -c)), since ϕ( ) = 1 + b( -1). We have that g (x) = -e - ! -1 +(1-c) +1 ! ( -1) (1+(x+(1-c))( -1)) 2 ≤ 0, so g is nonincreasing: it is thus enough to show that g(c) ≥ 1 -ce -1 to get the result, since α ( ,1-c) ≥ g(b -(1 -c)) ≥ g(c) ≥ 1 -ce -1 . But: g(c) = 1 - c +1 ! 1 + -1 e -= 1 -c ! e -≥ 1 -ce -1 ,
since ! e -is a decreasing sequence.

Proposition 3.23. Let F (x) := E X∼x [C ϕ (X)] for x ∈ {0, 1} m .
We have an explicit formula for F :

F (x) = n a=1 m k=0   1 m + 1 m =0 ω -k m+1 j∈[m]:a∈T j (1 + (ω m+1 -1)x j )   ϕ(k) , 48 3.5. Appendix
with ω m+1 := exp 2iπ m+1 . Thus, F is computable in polynomial time in n and m.

Proof. Recall that C ϕ (S) = n a=1 C ϕ a (S), so by linearity of expectation we focus on

E X∼x [C ϕ a (X)]. But C ϕ a (X) = ϕ(|X| a ) where |X| a = |{i ∈ [m] : X i = 1 and a ∈ T i }|. Thus: E X∼x [C ϕ a (X)] = m k=0 P X∼x (|X| a = k)ϕ(k) .
Let us compute the distribution of |X| a . We have

|X| a = i∈[m]:a∈T i X i and X i ∼ Ber(x i ). Thus, |X| a ∼ Poi Bin((x i ) i∈[m]:a∈T i ),
which is known as the Poisson binomial law. Thanks to [START_REF] Fernández | Closed-form expression for the Poissonbinomial probability density function[END_REF], we have that:

P X∼x (|X| a = k) = 1 m + 1 m =0 ω -k m+1 j∈[m]:a∈T j (1 + (ω m+1 -1)x j ) ,
where ω m+1 := exp 2iπ m+1 , and the result is proved.

Proposition 3.24. We have that

|E[ϕ(Bin(n, x/n))] -E[ϕ(Poi(x))]| ≤ xϕ(n) 2n + x n+1 n! .
In particular when ϕ(n) = o(n):

lim n→∞ E[ϕ(Bin(n, x ϕ /n))] = E[ϕ(Poi(x ϕ ))] = α ϕ ϕ(x ϕ ) .
Proof. Thanks to [BH84, TT19], the total variation distance between Bin(n, x/n) and Poi(x) is bounded in the following way:

∆(Bin(n, x/n), Poi(x)) ≤ 1 -e -x 2x n • x n 2 ≤ x 2n .
Thus with B ∼ Bin(n, x/n) and P ∼ Poi(x):

|E[ϕ(B)] -E[ϕ(P )]| = +∞ k=0 ϕ(k)P(B = k) - +∞ k=0 ϕ(k)P(P = k) = +∞ k=0 ϕ(k)(P(B = k) -P(P = k)) ≤ +∞ k=0 ϕ(k)|P(B = k) -P(P = k)| ≤ ϕ(n)∆(Bin(n, x/n), Poi(x)) + +∞ k=n+1 ϕ(k)P(P = k) ≤ xϕ(n) 2n + e -x +∞ k=n+1 k x k k! since ϕ(k) ≤ k = xϕ(n) 2n + xe -x +∞ k=n x k k! ≤ xϕ(n) 2n + x n+1 n! → n→∞ 0 when ϕ(n) = o(n) , (3.22) 3. T A G C C P
by a standard upper bound on the remainder of the exponential series.

Proposition 3.25. The function g :

x → E[ϕ(Poi(x))] on R + is C ∞ nondecreasing concave.
Proof. Since we have that

0 ≤ ϕ(k) ≤ k for k ∈ N, in particular g(x) = e -x +∞ k=0 ϕ(k) x k k! is C ∞ .
It is thus enough to compute its rst and second derivatives:

g (x) = -e -x +∞ k=0 ϕ(k) x k k! + e -x +∞ k=1 ϕ(k)k x k-1 k! = -e -x +∞ k=0 ϕ(k) x k k! + e -x +∞ k=0 ϕ(k + 1) x k k! = e -x +∞ k=0 (ϕ(k + 1) -ϕ(k)) x k k! . (3.23) But ϕ(k + 1) -ϕ(k) ≥ 0 since ϕ nondecreasing, so g (x) ≥ 0 and g is nondecreasing.
The calculus of g is the same where we replace ϕ by ψ(k) := ϕ(k + 1) -ϕ(k) which is a nonincreasing function by concavity of ϕ. Thus:

g (x) = e -x +∞ k=0 (ψ(k + 1) -ψ(k)) x k k! ≤ 0 .
since ψ(k + 1) -ψ(k) ≤ 0, and so g is concave.

Proposition 3.26. The function g q : n → E[ϕ(Bin(n, q))] de ned on N is nondecreasing concave. As a consequence, one can uses Jensen's inequality on the piecewise linear extension of g q which is also continuous.

Proof. We have Bin(n, q) ≤ st Bin(n + 1, q) and E[ϕ(Bin(n, q))] ≤ E[ϕ(Bin(n + 1, q))] since ϕ is nondecreasing, ie g q (n + 1) -g q (n) ≥ 0: g q is nondecreasing.

We show then the concavity, i.e. g q (n + 2) -g q (n + 1) ≤ g q (n + 1) -g q (n). Let us de ne ψ(x) = ϕ(x + 1) -ϕ(x), which is nonincreasing since ϕ concave. Let us take X k,q ∼ Bin(k, q). Then:

g q (n + 1) = E[ϕ(X n+1,q )] = n i=0 E[ϕ(X n,q + X 1,q )|X n,q = i]P(X n,q = i) = n i=0 E[ϕ(i + X 1,q ) -ϕ(i)]P(X n,q = i) + n i=0 ϕ(i)P(X n,q = i) = n i=0 E[ϕ(i + X 1,q ) -ϕ(i)]P(X n,q = i) + g q (n) .
(3.24) 50 3.5. Appendix Thus:

g q (n + 1) -g q (n) = n i=0 E[ϕ(i + X 1,q ) -ϕ(i)]P(X n,q = i) = n i=0 q(ϕ(i + 1) -ϕ(i))P(X n,q = i) = qE[ψ(Bin(n, q))] .
(3.25)

Then thanks to the fact that Bin(n, q) ≤ st Bin(n+1, q) and ψ is nonincreasing, we have that

E[ψ(Bin(n, q))] ≥ E[ψ(Bin(n + 1, q))],
i.e. g q (n + 2) -g q (n + 1) ≤ g q (n + 1) -g q (n).

Proposition 3.27. With w i := ϕ(i) -ϕ(i -1), we have:

lim i→+∞ w i = 0 ⇐⇒ ϕ(n) = o(n) . Proof. (⇒) Let > 0, let us nd a rank N such that for n ≥ N , ϕ(n) n ≤ . Let N 0 the rank from which w i ≤ 2 and N 1 the rank from which 1 n N 0 -1 i=1 w i ≤ 2 . We have ϕ(n) n = 1 n n i=1 w i ≤ 1 n N 0 -1 i=1 w i + 1 n n-1 i=N 0 2 ≤ 2 + 2 = for n ≥ max(N 0 , N 1 ) =: N . (3.26) (⇐) Since w i = ϕ(i) -ϕ(i -1
) is nonnegative and nonincreasing (respectively because ϕ is nondecreasing and concave), then the sequence w has a limit L ≥ 0. But

ϕ(n) n = 1 n n i=1 w i ≥ L .
Since the left hand side tends to 0 by hypothesis, this means that L = 0.

Proposition 3.28.

If w i := ϕ(i) -ϕ(i -1) is geometrically dominant, i.e. it satis es ∀i ∈ N * , w i w i+1 ≥ w i+1 w i+2 , then α ϕ = α ϕ (1).
Remark. Proposition 3.28 and in particular its proof uses similar ideas to the sketch provided in [START_REF] Dudycz | Tight approximation for proportional approval voting[END_REF].

Proof. Let g(k) := E[ϕ(Poi(k))],
and thus α ϕ (k) = g(k) ϕ(k) . Let us show that for k ∈ N * , α ϕ (k) ≥ α ϕ (1), which will be enough to conclude. In order to show this, we will need the following lemmas:

Lemma 3.29. ∀k < i ∈ N, w i ≥ w k+1 w i-k and thus ∀k, j ∈ N, ϕ(k + j) -ϕ(k) ≥ w k+1 ϕ(j). 3. T A G C C P
Proof. We have that:

w i = w i w i-1 w i-1 w i-2 . . . w i-k+1 w i-k w i-k . But for j ∈ [k]: w i-j+1 w i-j ≥ w (i-1)-j+1 w (i-1)-j ≥ . . . ≥ w (k+1)-j+1 w (k+1)-j ,
since w is geometrically dominant and k + 1 ≤ i. Thus applying this bound on each term of the previous product, we get:

w i ≥ w k+1 w k w k w k-1 . . . w 2 w 1 w i-k = w k+1 w 1 w i-k = w k+1 w i-k .
In particular, ∀k, j ∈ N, we get:

ϕ(k + j) -ϕ(k) = k+j i=k+1 w i ≥ w k+1 j i=1 w i = w k+1 ϕ(j) .
Lemma 3.30. The piecewise linear extension on [1, +∞[ of w, de ned on integers with w(k) := w k , is convex.

Proof. We will show that ∀k ∈ N * , w k+2 -w k+1 ≥ w k+1 -w k which implies the convexity of its piecewise linear extension on [1, +∞[. For k ∈ N * we have:

w k+1 w k+2 -1 ≤ w k+1 w k+2 w k+1 w k+2 -1 ≤ w k+1 w k+2 w k w k+1 -1 = w k -w k+1 w k+2 ,
since w is nonnegative nonincreasing (respectively ϕ nondecreasing concave) and

w k+1 w k+2 ≤ w k
w k+1 since w is geometrically dominant. Then, multiplying by -w k+2 ≤ 0 gives the expected result w k+2 -w k+1 ≥ w k+1 -w k .

We have

g(k + 1) = E[ϕ(Poi(k + 1))] = E[ϕ(Poi(k) + Poi(1))] since Poi(k + 1) ∼ Poi(k) + Poi(1)
. Thus:

g(k + 1) -g(k) = E X,X ∼Poi(k),Y ∼Poi(1) [ϕ(X + Y ) -ϕ(X )] = E X∼Poi(k),Y ∼Poi(1) [ϕ(X + Y ) -ϕ(X)] ≥ E X∼Poi(k),Y ∼Poi(1) [w X+1 ϕ(Y )] by Lemma 3.29 = E X∼Poi(k) [w(X + 1)]E Y ∼Poi(1) [ϕ(Y )] ,
(3.27) by independence of w(X + 1) and ϕ(Y ). Since w is convex on [1, +∞[ by Lemma 3.30 and Poi(k

) + 1 ∈ [1, +∞[, we have that E[w(Poi(k) + 1)] ≥ w(E[Poi(k) + 1]) = w(k + 1) = w k+1 thanks to Jensen's inequality. Note that g(0) = E[ϕ(Poi(0))] = ϕ(0) = 0. Then: g(k) = k-1 i=0 g(i + 1) -g(i) ≥ k-1 i=0 w i+1 E[ϕ(Poi(1))] = ϕ(k)g(1)
.

Therefore:

α ϕ (k) = g(k) ϕ(k) ≥ g(1) = g(1) ϕ(1) = α ϕ (1) .

Calculations of α ϕ

Proposition 3.31. For ∈ N * and ϕ(j) = min(j, ), we have that

α ϕ = 1 -e - ! .
Proof. Thanks to Proposition 3.22, we have that

α ϕ = min x∈N * α ϕ (x). Let us compute E[ϕ(Poi(x))]: E[ϕ(Poi(x))] = e -x +∞ k=0 ϕ(k) x k k! = e -x k=0 k x k k! + e -x +∞ k= +1 x k k! = e -x x -1 k=0 x k k! + e -x +∞ k= +1 x k k! = e -x (x -) -1 k=0 x k k! - x ! + e -x +∞ k=0 x k k! = -e -x x ( -1)! -(x -) -1 k=0 x k k! .
(3.28)

Let us show that x → α ϕ (x) takes its minimum in , where we have indeed:

α ϕ ( ) = 1 -e - ( -1)! -( -) -1 k=0 k k! = 1 -e - ! .
Thanks to proposition 3.21, x → α ϕ (x) is nondecreasing from to +∞. Suppose now that ≥ 2 (otherwise the result is already proved). Since x → α ϕ (x) is di erentiable, we have 53

3. T A G C C P for 1 ≤ x ≤ : α ϕ (x) = - x 2 + e -x x -1 ( -1)! - -1 k=0 x k k! + -2 k=0 x k (k + 1)! + x -e -x x -2 ( -2)! - -2 k=0 x k k! + -3 k=0 x k (k + 2)k! - x 2 = x e -x 1 + 1 x - 1 x + e -x -1 -1 x -2 ( -2)! + -3 k=0 x k (k + 1)! - x k (k + 2)k! = x e -x 1 + 1 x - 1 x + e -x x -2 ( -1)! + -3 k=0 x k k! 1 k + 1 - 1 k + 2 = x e -x 1 + 1 x + x -1 ! + x -3 k=0 x k k! 1 (k + 1)(k + 2) - 1 x = e -x x 2 1 + x + x ! + -3 k=0 x k+2 (k + 2)! -e x = e -x x 2 k=0 x k k! -e x ≤ 0 .
(3.29) since the partial sum of the exponential series is bounded by its total sum. Thus α ϕ (x) is nonincreasing from 1 to , and nondecreasing after, so it takes indeed its minimum in and the proposition is proved.

Proposition 3.32. For p ∈ (0, 1) and ϕ(j) = 1-(1-p) j p , we have that α ϕ = 1-e -p p .

Proof. By de nition:

α ϕ (x) = E[ϕ(Poi(x))] ϕ(x) = +∞ k=0 ϕ(k)e -x x k k! ϕ(x) = 1 -e -x +∞ k=0 (1 -p) k x k k! pϕ(x) = 1 -e -x e (1-p)x pϕ(x) = 1 -e -px pϕ(x) .
(3.30)

If x ≥ 1, α ϕ (x) = 1-e -px 1-(1-p) x = 1-e -px
1-e -qx with q = ln 1 1-p > 0 and:

α ϕ (x) = pe -px (1 -e -qx ) -qe -qx (1 -e -px ) (1 -e -qx ) 2 = pe -px -qe -qx + (q -p)e -(p+q)x (1 -e -qx ) 2 .
Let us take t = p q ∈ (0, 1), since q = ln 1 1-p > p > 0, x 1 = -px and x 2 = -(p + q)x. Then by strict convexity of the exponential function, we have:

e tx 1 +(1-t)x 2 < te x 1 + (1 -t)e x 2 = pe -px + (q -p)e -(p+q)x q . 54 3.5. Appendix But tx 1 + (1 -t)x 2 = -p 2 x q + -(q-p)(p+q)x q = -p 2 x q + -(q 2 x-p 2 x) q
= -qx, so we get pe -px -qe -qx + (q -p)e -(p+q)x > 0, and α ϕ (x) > 0. Thus, α ϕ (x) increases from 1 to in nity and takes its minimum in 1:

α ϕ = α ϕ (1) =
1 -e -p p .

Proposition 3.33. For d ∈ (0, 1) and ϕ(j) = j d , we have that

α ϕ = e -1 +∞ k=1 k d k! .
Proof. We have for x ≥ 1:

α ϕ (x) = E[Poi(x) d ] ϕ(x) = e -x +∞ k=0 k d x k k! ϕ(x) = e -x +∞ k=0 k d x k-d k! .
Then:

α ϕ (x) = -α ϕ (x) + e -x +∞ k=1 (k -d)k d x k-d-1 k! = -α ϕ (x) + e -x +∞ k=0 (k + 1 -d)(k + 1) d x k-d (k + 1)! = -α ϕ (x) + e -x (1 -d)x -d + +∞ k=1 (k + 1 -d)(k + 1) d-1 x k-d k! = e -x x -d 1 -d + +∞ k=1 k + 1 -d k + 1 (k + 1) d -k d x k k! . (3.31) But the function f (k) = k+1-d k+1 (k + 1) d -k d is positive on R * + , so we get that α ϕ (x) > 0 for x ≥ 1, thus α ϕ (x) is increasing from 1 to +∞, so α ϕ = α ϕ (1) = e -1 +∞ k=1 k d k! .

NP-hardness of δ, h-A G L C

Proof of Proposition 3.5. We reduce from the Label Cover problem described in [START_REF] Dudycz | Tight approximation for proportional approval voting[END_REF] which is known to be an NP-hard problem. The main idea of this reduction is the usual equivalence between bipartite graphs and hypergraphs. Note that both problems are in fact linearly equivalent since we could do the same reduction backwards.

Proof of Existence of Partitioning Systems

Proof of Proposition 3.6. The existential proof is based on the probabilistic method. We take P i an h-equi-sized uniform random x ϕ -cover of [n]. Hence in the collection P i = (P i,1 , . . . , P i,h ), each of the h subsets is of cardinality xϕn h . Write P = (P 1 , . . . , P R ). We have that for any a ∈ [n], P(a ∈ P i,j ) = xϕ h . Note that these events are independent for di erent is.

By construction, the rst condition is ful lled. Let us prove the second one.

Fix T ⊆ [R] and Q := {P i,j(i) : i ∈ T } for some function j : T → [h]. We have for a ∈ [n]:

E[C ϕ a (Q)] = E[ϕ(|Q| a )] = E ϕ {i ∈ T : a ∈ P i,j(i) } .
But the random variables {X a i := 1 a∈P i,j(i) } i∈T are independent and

X a i ∼ Ber( xϕ h ), so X a := {i ∈ T : a ∈ P i,j(i) } = i∈T X a i ∼ Bin(|T |, xϕ h
), and thus:

E[C ϕ a (Q)] = E ϕ Bin |T |, x ϕ h = ψ ϕ |T |,h . Since |Q| a ≤ |Q| ≤ R and ϕ nondecreasing, we have 0 ≤ C ϕ a (Q) ≤ ϕ(R). The random variables {C ϕ a (Q)} a∈[n]
are not independent in general. However, we will show that they are negatively associated (see De nition 2.2), thus we can apply the version of the Cherno -Hoe ding bound described in Proposition 2.9 to get: Since there are at most (h + 1) R choices of T and Q, a union bound gives:

P   1 n a∈[n] C ϕ a (Q) -ψ ϕ |T |,h > η   ≤ 2 exp -2 η ϕ(R) 2 n . Note that C ϕ a (Q) = ϕ(|Q| a ) = ϕ(X a ) is a nondecreasing function of {X a i } i∈[R] , since ϕ is nondecreasing and X a = i∈T X a i .
P ∃C, Q : C ϕ (Q) -ψ ϕ |T |,h n > ηn ≤ 2(h + 1) R exp -2 η ϕ(R) 2 n .
Thus with probability at least 9/10, we have that

C ϕ (Q) -ψ ϕ
|T |,h n ≤ ηn, since we have taken n ≥ η -2 Rϕ(R) 2 log(20(h + 1)). So there must exists some choice of P that satis es the rst and second constraints of partitioning systems. Thus, we can enumerate over all choices of P in time exp(Rn log(n)) • poly(h) to nd such a partitioning system.

Proof of Theorem 3.15

Proof. We show that ϕ-R

A corresponds to ϕ-M C under a matroid constraint. Given an instance of ϕ-R A , consider the partition matroid M on [ i∈[k] m i ] := [m 1 ]+. . .+[m k ], where (B i ) i∈[k] := ([m i ]) i∈[k]
is a partition of the ground set and the cardinality constraint for each i is to d i = 1.

Here, I ⊆ [ i∈[k] m i ] is an independent set of the matroid if and only if |I ∩ B i | ≤ d i = 1, for all i ∈ [k]
. This corresponds to each agent i ∈ [k] selecting at most one element from the available m i choices. In other words, we have a bijection f between tuples (A 1 , . . . , A k ) ∈ A 1 × . . . × A k and maximal independent sets (bases) of M such that W ϕ (A) = C ϕ (f (A)). Therefore, Theorem 3.3 leads to a polynomial-time α ϕ -approximation algorithm for ϕ-R A .

For the hardness part of the theorem, the proof is exactly the same as in Theorem 3.4, but instead of

F := {F v β , v ∈ V, β ∈ [L]} and k = |V |, we take k = |V | to

be the number of agents and A

i := {F v i β , β ∈ [L]} where V = {v 1 , . . . , v k }.
Hence, instead of subsets of F of size k, we only consider one set F v β ∈ F, for each v ∈ V . The function we maximize in the reduction remains unchanged.

To establish completeness, we note that the subset described is already of the right form and, hence, the arguments continue to hold. For proving soundness, the constraint on the shape of the subset of F only helps us, since it gives more constraints on the given subset from which we want to construct a labeling. Therefore, the NP-hardness follows. 57 C 4

Multiple-Access Channel Coding With Non-Signaling Correlations

Multiple-access channels (MACs for short) are one of the simplest models of network communication settings, where two senders aim to transmit individual messages to one receiver. The capacity of such channels has been entirely characterized by the seminal works by Liao [START_REF] Herng | Multiple access channels[END_REF] and Ahlswede [START_REF] Ahlswede | Multi-way communication channels[END_REF] in terms of a simple single-letter formula. From the point of view of quantum information, it is natural to ask whether additional resources, such as quantum entanglement or more generally non-signaling correlations between the parties, change the capacity region. A non-signaling correlation is a multipartite inputoutput box shared between parties that, as the name suggests, cannot by itself be used to send information between parties. However, non-signaling correlations such as the ones generated by measurements of entangled quantum particles, can provide an advantage for various information processing tasks and nonlocal games. The study of such correlations has given rise to the quantum information area known as nonlocality [BCP + 14]. For example, in the context of channel coding, there exists classical point-to-point channels for which quantum entanglement between the sender and the receiver can increase the optimal success probability for sending one bit of information with a single use of the channel [PLM + 11, BF18]. However, for classical point-to-point channels, entanglement [BBC + 93, BSST99] and even more generally non-signaling correlations [START_REF] Matthews | A linear program for the nite block length converse of Polyanskiy-Poor-Verdú via nonsignaling codes[END_REF] do not change the capacity of the channel.

In the network setting, behavior is di erent. Quek and Shor showed in [START_REF] Quek | Quantum and superquantum enhancements to twosender, two-receiver channels[END_REF] the existence of two-sender two-receiver interference channels with gaps between their classical, quantum-entanglement assisted and non-signaling assisted capacity regions. Following this result, Leditzky et al. [START_REF] Leditzky | Playing games with multiple access channels[END_REF][START_REF] Seshadri | On the separation of correlation-assisted sum capacities of multiple access channels[END_REF] showed that quantum entanglement shared between the two senders of a MAC can strictly enlarge the capacity region. This has been demonstrated through channels that are constructed from two-player nonlocal games, such as the Magic Square game [Mer90, Per90, Ara02, BBT05], by translating known gaps between classical and quantum values of games into MAC capacity gaps. Other instances of network channels for which entanglement increases the capacity region were studied in [Noe20, ND20, YRB20, YRB23]. This raises the following natural question: Can nonsignaling correlations lead to signi cant gains in capacity for natural MACs? Can we nd a characterization of the capacity region of the MAC when non-signaling resources between the parties are allowed?

59 4. M A C C W N S C
Contributions We focus here on the MAC with two senders and we allow arbitrary tripartite non-signaling correlations between the two senders and the receiver. This is the most optimistic setting, in the sense that we only enforce the non-signaling constraints between the parties, and also the mathematically simplest setting. Even if not all nonsignaling correlations are feasible within quantum theory, the setting we study here can be seen as a tractable and physically motivated outer approximation of what can be achieved with quantum theory. In fact, the quantum set is notoriously complicated and deciding membership in this set is not computable [JNV + 20]. We note that very recently, Pereg et al.

[PDB23] found a regularized characterization for the capacity of MACs with quantum entanglement shared between the two senders. Unfortunately, even the associated singleletter inner bound is very di cult to evaluate for any xed channel.

We rst show that the multiple-access channel coding problem, which entails maximizing the success probability of sending a xed number of messages through a MAC, cannot be approximated in polynomial time within any constant ratio, under a complexity hypothesis on random k-SAT formulas; see Theorem 4.5.

We denote by S NS (W, k 1 , k 2 ) the success probability of the best non-signaling assisted (k 1 , k 2 )-code for the MAC W . Contrary to the unassisted value that we denote S(W, k 1 , k 2 ), S NS (W, k 1 , k 2 ) can be formulated as a linear program; see Proposition 4.7. Furthermore, using symmetries, we have developed a linear program computing S NS for a nite number of copies of a MAC W with a size growing polynomially in the number of copies; see Theorem 4.12 and Corollary 4.13. Using this result, we describe a method to derive inner bounds on the non-signaling assisted capacity region achievable with zero error; see Proposition 4.17. Applied to the binary adder channel, which maps (x 1 , x 2 ) ∈ {0, 1} 2 to x 1 + x 2 ∈ {0, 1, 2}, we show that the sum-rate log 2 (72) 4

1.5425 can be reached with zero error, which beats the maximum classical sum-rate capacity of 3 2 ; see Theorem 4.18. For noisy channels, where the zero-error non-signaling assisted capacity region is trivial, we can use concatenated codes to obtain achievable points in the capacity region; see Proposition 4.20. Applied to a noisy version of the binary adder channel, we show that non-signaling assistance still improves the sum-rate capacity.

In order to nd outer bounds, we de ne a relaxed notion of non-signaling assistance and characterize its capacity region by a single-letter expression, which is the same as the well-known expression for the capacity of the MAC (see Theorem 4.1) except that the inputs X 1 and X 2 are not required to be independent; see Theorem 4.25. This gives in particular an outer bound on the non-signaling assisted capacity region; see Corollary 4.32. The main open problem that we leave is whether this outer bound on the non-signaling capacity region is tight. We give an example of a channel for which the relaxed notion of non-signaling assistance gives a strictly larger success probability than non-signaling assistance but we do not know if such a gap can persist for the capacity region.

We also study the case where non-signaling assistance is shared only between each sender and the receiver independently. Note that no assistance is shared between the senders. We show that this capacity region is the same as the capacity region without any assistance; see Theorem 4.34 and Corollary 4.35. We note that a similar setting with independent entangled states between each sender and the receiver was studied by Hsieh et al. [START_REF] Hsieh | Entanglement-assisted capacity of quantum multiple-access channels[END_REF]: a regularized characterization of the capacity region is obtained for any quantum MAC in this setting. It is possible to show using their result that for a classical MAC, this type of entanglement does not change the capacity region given in Theorem 4.1.

Organization In Section 4.1, we de ne precisely the di erent notions of MAC capacities: the classical capacity (i.e. without any assistance) as well as the non-signaling assisted capacity. In Section 4.2, we show that no polynomial-time constant approximation for multiple-access channel coding exists if a complexity hypothesis on random k-SAT formulas is true. In Section 4.3, we address computational complexity questions concerning the probability of success of the best classical coding strategy and the best non-signaling strategy for a MAC. In Section 4.4, we develop numerical methods to nd inner bounds on non-signaling assisted capacity regions, and apply those to the binary adder channel and a noisy variant. In Section 4.5, we de ne our relaxation of non-signaling assistance, we characterize its capacity region by a single-letter formula, and apply those to the binary adder channel. Finally, in Section 4.6, we show that the capacity region with non-signaling assistance shared only between each sender and the receiver independently is the same as without assistance.

Multiple-Access Channels Capacity Regions

Classical Capacities

The coding problem for a MAC W : X 1 × X 2 → Y can be stated in the following way. We want to encode messages from [k 1 ] into X 1 and messages from [k 2 ] into X 2 independently. They are given as inputs to W , which results in a random output in Y. From this output, we want to decode back the original messages in [k 1 ] and [k 2 ]. We will call e 1 : [k 1 ] → X 1 the rst encoder, e 2 : [k 2 ] → X 2 the second encoder and We want to maximize over all encoders e 1 , e 2 and decoders d the probability of successfully encoding and decoding the messages through W , i.e. the probability that j 1 = i 1 and j 2 = i 2 , given that the input messages are taken uniformly in

d : Y → [k 1 ] × [k 2 ] the decoder. The scenario is depicted in Figure 4.1. e 1 W e 2 d x 1 x 2 y i 1 i 2 (j 1 , j 2 )
[k 1 ] and [k 2 ]. We call this 61 4. M A C C W N S C
quantity S(W, k 1 , k 2 ), which is characterized by the following optimization program:

S(W, k 1 , k 2 ) := maximize e 1 ,e 2 ,d 1 k 1 k 2 i 1 ,i 2 ,x 1 ,x 2 ,y W (y|x 1 x 2 )e 1 (x 1 |i 1 )e 2 (x 2 |i 2 )d(i 1 i 2 |y) subject to x 1 ∈X 1 e 1 (x 1 |i 1 ) = 1, ∀i 1 ∈ [k 1 ] x 2 ∈X 2 e 2 (x 2 |i 2 ) = 1, ∀i 2 ∈ [k 2 ] j 1 ∈[k 1 ],j 2 ∈[k 2 ] d(j 1 j 2 |y) = 1, ∀y ∈ Y e 1 (x 1 |i 1 ), e 2 (x 2 |i 2 ), d(j 1 j 2 |y) ≥ 0 (4.1)
Proof. One should note that we allow randomized encoders and decoders for generality reasons, although the value of the program is not changed as it is convex. Besides that remark, let us name I 1 , I 2 , J 1 , J 2 , X 1 , X 2 , Y the random variables corresponding to i 1 , i 2 , j 1 , j 2 , x 1 , x 2 , y in the coding and decoding process. Then, for given e 1 , e 2 , d and W , the objective value of the previous program is:

P (J 1 = I 1 , J 2 = I 2 ) = 1 k 1 k 2 i 1 ,i 2 P (J 1 = I 1 , J 2 = I 2 |I 1 = i 1 , I 2 = i 2 ) = 1 k 1 k 2 i 1 ,i 2 ,x 1 ,x 2 e 1 (x 1 |i 1 )e 2 (x 2 |i 2 )P (J 1 = i 1 , J 2 = i 2 |i 1 , i 2 , X 1 = x 1 , X 2 = x 2 ) = 1 k 1 k 2 i 1 ,i 2 ,x 1 ,x 2 ,y W (y|x 1 x 2 )e 1 (x 1 |i 1 )e 2 (x 2 |i 2 )P (J 1 = i 1 , J 2 = i 2 |i 1 , i 2 , x 1 , x 2 , Y = y) = 1 k 1 k 2 i 1 ,i 2 ,x 1 ,x 2 ,y W (y|x 1 x 2 )e 1 (x 1 |i 1 )e 2 (x 2 |i 2 )d(i 1 , i 2 |y) . (4.2)
The (classical) capacity region of a MAC W , as de ned for example in [START_REF] Cover | Elements of Information Theory[END_REF], can be reformulated in terms of the previous success probability S using De nition 2.12 and by C(W ) := C[S](W ), and its zero-error equivalent is de ned by C 0 (W ) := C 0 [S](W ).

The capacity region C(W ) is characterized by a single-letter formula:

Theorem 4.1 (Liao [START_REF] Herng | Multiple access channels[END_REF] and Ahlswede [START_REF] Ahlswede | Multi-way communication channels[END_REF]). C(W ) is the closure of the convex hull of all rate pairs (R 1 , R 2 ) satisfying:

R 1 < I(X 1 : Y |X 2 ) , R 2 < I(X 2 : Y |X 1 ) , R 1 + R 2 < I((X 1 , X 2 ) : Y ) , for (X 1 , X 2 ) ∈ X 1 × X 2 following a product law P X 1 × P X 2 , and Y ∈ Y the outcome of W on inputs X 1 , X 2 .
We will also consider what we call the sum success probability S sum (W, k 1 , k 2 ), de ned using P(J 1 =I 1 )+P(J2 =I 2 )

to the following optimization program:

S sum (W, k 1 , k 2 ) := maximize e 1 ,e 2 ,d 1 ,d 2 1 2k 1 k 2 i 1 ,i 2 ,x 1 ,x 2 ,y W (y|x 1 x 2 )e 1 (x 1 |i 1 )e 2 (x 2 |i 2 )d 1 (i 1 |y) + 1 2k 1 k 2 i 1 ,i 2 ,x 1 ,x 2 ,y W (y|x 1 x 2 )e 1 (x 1 |i 1 )e 2 (x 2 |i 2 )d 2 (i 2 |y)
subject to

x 1 ∈X 1 e 1 (x 1 |i 1 ) = 1, ∀i 1 ∈ [k 1 ] x 2 ∈X 2 e 2 (x 2 |i 2 ) = 1, ∀i 2 ∈ [k 2 ] j 1 ∈[k 1 ] d 1 (j 1 |y) = 1, ∀y ∈ Y j 2 ∈[k 2 ] d 2 (j 2 |y) = 1, ∀y ∈ Y e 1 (x 1 |i 1 ), e 2 (x 2 |i 2 ), d 1 (j 1 |y), d 2 (j 2 |y) ≥ 0 (4.3)
Note that we used independent decoders d 1 (j 1 |y), d 2 (j 2 |y) rather than a global d(j 1 j 2 |y) here. This does not change the value of the optimization program. Indeed, since the program is convex, an optimal solution can be found on the extremal points of the search space. Thus, if we had used the variable d(j 1 j 2 |y), we could always take it to be a function

d from Y to [k 1 ] × [k 2 ].
Taking d 1 , d 2 as the rst and second coordinates of that function satis es the equality d(j 1 j 2 |y) = d 1 (j 1 |y)d 2 (j 2 |y), and therefore, the value of the program is the same in both cases. Note that it is also true for the program computing S(W, k 1 , k 2 ).

As for the usual (joint) success probability, we can de ne the sum-capacity region using De nition 2.12 by C sum (W ) := C[S sum ](W ). It turns out those two notions of success de ne the same capacity region:

Proposition 4.2. C(W ) = C sum (W )
Proof. Let us focus on error probabilities rather than success ones. Call them respectively

E(W, k 1 , k 2 ) := 1 -S(W, k 1 , k 2 ) and E sum (W, k 1 , k 2 ) := 1 -S sum (W, k 1 , k 2 ). Let us x a solution e 1 , d 1 , e 2 , d 2 of the optimization program computing S(W, k 1 , k 2 ).
Let us remark rst that:

i 1 ,i 2 ,x 1 ,x 2 ,y W (y|x 1 x 2 )e 1 (x 1 |i 1 )e 2 (x 2 |i 2 ) = k 1 k 2 ,
thus, the value of the maximum error for those encoders and decoders, which we call

63 4. M A C C W N S C E(W, k 1 , k 2 , e 1 , d 1 , e 2 , d 2 ), is: 1 - 1 k 1 k 2 i 1 ,i 2 ,x 1 ,x 2 ,y W (y|x 1 x 2 )e 1 (x 1 |i 1 )e 2 (x 2 |i 2 )d 1 (i 1 |y)d 2 (i 2 |y) = 1 k 1 k 2 i 1 ,i 2 ,x 1 ,x 2 ,y W (y|x 1 x 2 )e 1 (x 1 |i 1 )e 2 (x 2 |i 2 ) - 1 k 1 k 2 i 1 ,i 2 ,x 1 ,x 2 ,y W (y|x 1 x 2 )e 1 (x 1 |i 1 )e 2 (x 2 |i 2 )d 1 (i 1 |y)d 2 (i 2 |y) = 1 k 1 k 2 i 1 ,i 2 ,x 1 ,x 2 ,y W (y|x 1 x 2 )e 1 (x 1 |i 1 )e 2 (x 2 |i 2 ) [1 -d 1 (i 1 |y)d 2 (i 2 |y)] . (4.4)
Similarly, the value of the sum error for those encoder and decoders, which we call

E sum (W, k 1 , k 2 , e 1 , d 1 , e 2 , d 2 ), is: 1 - 1 k 1 k 2 i 1 ,i 2 ,x 1 ,x 2 ,y W (y|x 1 x 2 )e 1 (x 1 |i 1 )e 2 (x 2 |i 2 ) d 1 (i 1 |y) + d 2 (i 2 |y) 2 = 1 k 1 k 2 i 1 ,i 2 ,x 1 ,x 2 ,y W (y|x 1 x 2 )e 1 (x 1 |i 1 )e 2 (x 2 |i 2 ) 1 - d 1 (i 1 |y) + d 2 (i 2 |y) 2 .
(4.5)

However, for x, y ∈ [0, 1], we have that:

1 -xy ≥ max (1 -x, 1 -y) ≥ 1 - x + y 2 ,
and:

1 -xy ≤ (1 -x) + (1 -y) = 2 1 - x + y 2 .
This means that, for all e 1 , d 1 , e 2 , d 2 , we have:

E sum (W, k 1 , k 2 , e 1 , d 1 , e 2 , d 2 ) ≤ E(W, k 1 , k 2 , e 1 , d 1 , e 2 , d 2 ) ,
and:

E(W, k 1 , k 2 , e 1 , d 1 , e 2 , d 2 ) ≤ 2E sum (W, k 1 , k 2 , e 1 , d 1 , e 2 , d 2 ) ,
so, maximizing over all e 1 , d 1 , e 2 , d 2 , we get:

E sum (W, k 1 , k 2 ) ≤ E(W, k 1 , k 2 ) ≤ 2E sum (W, k 1 , k 2 ) .
Thus, up to a multiplicative factor 2, the error is the same. In particular, when one of those errors tends to zero, the other one tends to zero as well. This implies that the capacity regions are the same.

Non-Signaling Assisted Capacities

Three-party non-signaling assistance We now consider the case where the senders and the receiver are given non-signaling assistance. This resource, which is a theoretical but easier to manipulate generalization of quantum entanglement, can be characterized as follows. A tripartite non-signaling box is described by a joint conditional probability distribution P (x 1 x 2 (j 1 j 2 )|i 1 i 2 y) such that the marginal from any two parties is independent of the removed party's input, i.e., we have:

∀x 2 , j 1 , j 2 , i 1 , i 2 , y, i 1 , x 1 P (x 1 x 2 (j 1 j 2 )|i 1 i 2 y) = x 1 P (x 1 x 2 (j 1 j 2 )|i 1 i 2 y) , ∀x 1 , j 1 , j 2 , i 1 , i 2 , y, i 2 , x 2 P (x 1 x 2 (j 1 j 2 )|i 1 i 2 y) = x 2 P (x 1 x 2 (j 1 j 2 )|i 1 i 2 y) , ∀x 1 , x 2 , i 1 , i 2 , y, y , j 1 ,j 2 P (x 1 x 2 (j 1 j 2 )|i 1 i 2 y) = j 1 ,j 2 P (x 1 x 2 (j 1 j 2 )|i 1 i 2 y ) . (4.6)
This implies that one can consider for example P (x 1 x 2 |i 1 i 2 ) since it does not depend on y, or even P (x 1 |i 1 ) since it does not depend on i 2 , y. Then, in our coding scenario, when the senders and the receiver are given non-signaling assistance, it means that they share a tripartite non-signaling box, the behavior of which is described by P . In this case, the expression e 1 (x 1 |i 1 )e 2 (x 2 |i 2 )d(j 1 j 2 |y) in (4.1) is replaced by P (x 1 x 2 (j 1 j 2 )|i 1 i 2 y), as depicted in Figure 4.2. The cyclicity of Figure 4.2 is at rst sight counter-intuitive. Note rst that P being a non-signaling box is completely independent of W : in particular, the variable y does not need to follow any law in the de nition of P being a non-signaling box. Therefore, the remaining ambiguity is the apparent need to encode and decode at the same time. However, since P is a non-signaling box, we do not need to do both at the same time. Indeed, ∀y, P (x 1 x 2 |i 1 i 2 ) = P (x 1 x 2 |i 1 i 2 y) by the non-signaling property of P . Thus, one can get the outputs x 1 , x 2 on inputs i 1 , i 2 without access to y, as that knowledge won't a ect the laws of x 1 , x 2 . Then y follows the law given by W given those x 1 , x 2 . Finally, given access to y, the decoding process is described by:

e 1 e 2 d i 1 i 2 (j 1 , j 2 ) W x 1 x 2 y P (x 1 x 2 (j 1 j 2 )|i 1 i 2 y) i 1 i 2 (j 1 , j 2 ) W x 1 x 2 y
P ((j 1 j 2 )|i 1 i 2 yx 1 x 2 ) = P (x 1 x 2 (j 1 j 2 )|i 1 i 2 y) P (x 1 x 2 |i 1 i 2 y) = P (x 1 x 2 (j 1 j 2 )|i 1 i 2 y) P (x 1 x 2 |i 1 i 2 ) ,
so we recover globally P ((j

1 j 2 )|i 1 i 2 yx 1 x 2 ) × P (x 1 x 2 |i 1 i 2 ) = P (x 1 x 2 (j 1 j 2 )|i 1 i 2 y)
the prescribed conditional probability. The non-signaling condition ensures that it is possible 65

4. M A C C W N S C
to consider the conditional probabilities of each party independently. This clari es how one can e ectively encode and then decode messages through a non-signaling box.

As in the unassisted case, we want to maximize over all non-signaling boxes P the probability of successfully encoding and decoding the messages through W , i.e. the probability that j 1 = i 1 and j 2 = i 2 , given that the input messages are taken uniformly in

[k 1 ] and [k 2 ].
We call this quantity S NS (W, k 1 , k 2 ), which is characterized by the following optimization program:

S NS (W, k 1 , k 2 ) := maximize P 1 k 1 k 2 i 1 ,i 2 ,x 1 ,x 2 ,y W (y|x 1 x 2 )P (x 1 x 2 (i 1 i 2 )|i 1 i 2 y)
subject to

x 1 P (x 1 x 2 (j 1 j 2 )|i 1 i 2 y) = x 1 P (x 1 x 2 (j 1 j 2 )|i 1 i 2 y) x 2 P (x 1 x 2 (j 1 j 2 )|i 1 i 2 y) = x 2 P (x 1 x 2 (j 1 j 2 )|i 1 i 2 y) j 1 ,j 2 P (x 1 x 2 (j 1 j 2 )|i 1 i 2 y) = j 1 ,j 2 P (x 1 x 2 (j 1 j 2 )|i 1 i 2 y ) x 1 ,x 2 ,j 1 ,j 2 P (x 1 x 2 (j 1 j 2 )|i 1 i 2 y) = 1 P (x 1 x 2 (j 1 j 2 )|i 1 i 2 y) ≥ 0 (4.7)
Since it is given as a linear program, the complexity of computing S NS (W, k 1 , k 2 ) is polynomial in the number of variables and constraints (see for instance Section 7.1 of [START_REF] Gärtner | Understanding and using linear programming[END_REF]), which is a polynomial in |X 1 |, |X 2 |, |Y|, k 1 and k 2 . Also, as it is easy to check that a classical strategy is a particular case of a non-signaling assisted strategy, we have that

S NS (W, k 1 , k 2 ) ≥ S(W, k 1 , k 2 ).
As for the unassisted case, we de ne the non-signaling assisted capacity region using De nition 2.12 and S NS by C NS (W ) := C[S NS ](W ), and its zero-error equivalent is de ned by

C NS 0 (W ) := C 0 [S NS ](W ).
Independent non-signaling assistance One can also consider the case where nonsignaling assistance is shared independently between the rst sender and the receiver as well as between the second encoder and the receiver, which we call independent nonsignaling assistance. The precise scenario is depicted in Figure 4.3: This leads to the following de nition of the success probability S NS SR (W, k 1 , k 2 ):

e 1 d 1 e 2 d 2 i 1 i 2 j 1 j 2 W x 1 x 2 y P 1 (x 1 j 1 |i 1 y) P 2 (x 2 j 2 |i 2 y) i 1 i 2 j 1 j 2 W x 1 x 2 y
maximize

P 1 ,P 2 1 k 1 k 2 i 1 ,i 2 ,x 1 ,x 2 ,y W (y|x 1 x 2 )P 1 (x 1 i 1 |i 1 y)P 2 (x 2 i 2 |i 2 y)
subject to

x 1 P 1 (x 1 j 1 |i 1 y) = x 1 P 1 (x 1 j 1 |i 1 y) j 1 P 1 (x 1 j 1 |i 1 y) = j 1 P 1 (x 1 j 1 |i 1 y ) x 1 ,j 1 P 1 (x 1 j 1 |i 1 y) = 1 x 2 P 2 (x 2 j 2 |i 2 y) = x 2 P 2 (x 2 j 2 |i 2 y) j 2 P 2 (x 2 j 2 |i 2 y) = j 2 P 2 (x 2 j 2 |i 2 y ) x 2 ,j 2 P 2 (x 2 j 2 |i 2 y) = 1 P 1 (x 1 j 1 |i 1 y), P 2 (x 2 j 2 |i 2 y) ≥ 0 (4.8) 67 4. M A C C W N S C
As before, one can also consider the sum-success probability S NS SR sum (W, k 1 , k 2 ):

maximize

P 1 ,P 2 1 2k 1 k 2 i 1 ,i 2 ,x 1 ,x 2 ,y W (y|x 1 x 2 )P 1 (x 1 i 1 |i 1 y) j 2 P 2 (x 2 j 2 |i 2 y) + 1 2k 1 k 2 i 1 ,i 2 ,x 1 ,x 2 ,y W (y|x 1 x 2 )P 2 (x 2 i 2 |i 2 y) j 1 P 1 (x 1 j 1 |i 1 y)
subject to

x 1 P 1 (x 1 j 1 |i 1 y) = x 1 P 1 (x 1 j 1 |i 1 y) j 1 P 1 (x 1 j 1 |i 1 y) = j 1 P 1 (x 1 j 1 |i 1 y ) x 1 ,j 1 P 1 (x 1 j 1 |i 1 y) = 1 x 2 P 2 (x 2 j 2 |i 2 y) = x 2 P 2 (x 2 j 2 |i 2 y) j 2 P 2 (x 2 j 2 |i 2 y) = j 2 P 2 (x 2 j 2 |i 2 y ) x 2 ,j 2 P 2 (x 2 j 2 |i 2 y) = 1 P 1 (x 1 j 1 |i 1 y), P 2 (x 2 j 2 |i 2 y) ≥ 0 (4.9)
We de ne the independent non-signaling assisted capacity (resp. sum-capacity) region using De nition 2.12 by

C NS SR (W ) := C[S NS SR ](W ) (resp. C NS SR sum (W ) := C[S NS SR sum ](W )
). It turns out those two notions of success de ne the same capacity region:

Proposition 4.3. C NS SR (W ) = C NS SR sum (W )
Proof. Given non-signaling boxes P 1 , P 2 , the maximum success probability of encoding and decoding correctly with those is given by:

S NS SR (W, k 1 , k 2 , P 1 , P 2 ) := 1 k 1 k 2 i 1 ,i 2 ,x 1 ,x 2 ,y W (y|x 1 x 2 )P 1 (x 1 i 1 |i 1 y)P 2 (x 2 i 2 |i 2 y) .
This should be compared to the sum success probability of encoding and decoding correctly with those, which we call S NS SR sum (W, k 1 , k 2 , P 1 , P 2 ) and is equal to:

1 2k 1 k 2 i 1 ,i 2 ,x 1 ,x 2 ,y W (y|x 1 x 2 )P 1 (x 1 i 1 |i 1 y) j 2 P 2 (x 2 j 2 |i 2 y) + 1 2k 1 k 2 i 1 ,i 2 ,x 1 ,x 2 ,y W (y|x 1 x 2 )P 2 (x 2 i 2 |i 2 y) j 1 P 1 (x 1 j 1 |i 1 y) .
(4.10)

Similarly to what was done in Proposition 4.2, we focus on error probabilities rather than success probabilities. We have that:

1 k 1 k 2 i 1 ,i 2 ,x 1 ,x 2 ,y W (y|x 1 x 2 ) j 1 ,j 2 P 1 (x 1 j 1 |i 1 y)P 2 (x 2 j 2 |i 2 y) = 1 , so we get that E NS SR (W, k 1 , k 2 , P 1 , P 2 ) is equal to: 1 k 1 k 2 i 1 ,i 2 ,x 1 ,x 2 ,y W (y|x 1 x 2 ) j 1 ,j 2 P 1 (x 1 j 1 |i 1 y)P 2 (x 2 j 2 |i 2 y) - 1 k 1 k 2 i 1 ,i 2 ,x 1 ,x 2 ,y W (y|x 1 x 2 )P 1 (x 1 i 1 |i 1 y)P 2 (x 2 i 2 |i 2 y) = i 1 ,i 2 ,x 1 ,x 2 ,y W (y|x 1 x 2 ) (j 1 ,j 2 ) =(i 1 ,i 2 ) P 1 (x 1 j 1 |i 1 y)P 2 (x 2 j 2 |i 2 y) .
(4.11)

On the other hand, since:

j 1 ,j 2 P 1 (x 1 j 1 |i 1 y)P 2 (x 2 j 2 |i 2 y) -P 1 (x 1 i 1 |i 1 y) j 2 P 2 (x 2 j 2 |i 2 y) = j 1 =i 1 ,j 2 P 1 (x 1 j 1 |i 1 y)P 2 (x 2 j 2 |i 2 y) , (4.12)
and symetrically, we get that

E NS SR sum (W, k 1 , k 2 , P 1 , P 2 ) is equal to: 1 2k 1 k 2 i 1 ,i 2 ,x 1 ,x 2 ,y W (y|x 1 x 2 ) j 1 =i 1 ,j 2 P 1 (x 1 j 1 |i 1 y)P 2 (x 2 j 2 |i 2 y) + 1 2k 1 k 2 i 1 ,i 2 ,x 1 ,x 2 ,y W (y|x 1 x 2 ) j 1 ,j 2 =i 2 P 1 (x 1 j 1 |i 1 y)P 2 (x 2 j 2 |i 2 y) = 1 k 1 k 2 i 1 ,i 2 ,x 1 ,x 2 ,y W (y|x 1 x 2 ) j 1 =i 1 ,j 2 =i 2 P 1 (x 1 j 1 |i 1 y)P 2 (x 2 j 2 |i 2 y) + 1 k 1 k 2 i 1 ,i 2 ,x 1 ,x 2 ,y W (y|x 1 x 2 ) (j 1 ,j 2 )∈S P 1 (x 1 j 1 |i 1 y)P 2 (x 2 j 2 |i 2 y) 2 , ( 4.13)
with S := {(j 1 , i 2 ) :

j 1 ∈ [k 1 ] -{i 1 }} ∪ {(i 1 , j 2 ) : j 2 ∈ [k 2 ]
-{i 2 }}. However, we have that:

j 1 =i 1 ,j 2 =i 2 P 1 (x 1 j 1 |i 1 y)P 2 (x 2 j 2 |i 2 y) + (j 1 ,j 2 )∈S P 1 (x 1 j 1 |i 1 y)P 2 (x 2 j 2 |i 2 y) 2 ≤ j 1 =i 1 ,j 2 =i 2 P 1 (x 1 j 1 |i 1 y)P 2 (x 2 j 2 |i 2 y) + (j 1 ,j 2 )∈S P 1 (x 1 j 1 |i 1 y)P 2 (x 2 j 2 |i 2 y) = (j 1 ,j 2 ) =(i 1 ,i 2 ) P 1 (x 1 j 1 |i 1 y)P 2 (x 2 j 2 |i 2 y) ≤ 2   j 1 =i 1 ,j 2 =i 2 P 1 (x 1 j 1 |i 1 y)P 2 (x 2 j 2 |i 2 y) + (j 1 ,j 2 )∈S P 1 (x 1 j 1 |i 1 y)P 2 (x 2 j 2 |i 2 y) 2   . (4.14)
This implies that:

E NS SR sum (W, k 1 , k 2 , P 1 , P 2 ) ≤ E NS SR (W, k 1 , k 2 , P 1 , P 2 ) ≤ 2E NS SR sum (W, k 1 , k 2 , P 1 , P 2
) , and by maximizing over all P 1 and P 2 :

E NS SR sum (W, k 1 , k 2 ) ≤ E NS SR (W, k 1 , k 2 ) ≤ 2E NS SR sum (W, k 1 , k 2 )
. Thus, as before, the capacity regions are the same.

Hardness of Approximation for Multiple-Access Channel Coding

Since MACs are more general than point-to-point channels (by de ning W (y|x 1 x 2 ) := Ŵ (y|x 1 ) for Ŵ a point-to-point channel and looking only at its rst input), computing S(W, k 1 , k 2 ) is NP-hard, and it is even NP-hard to approximate within a better ratio than 1 -e -1 , as a consequence of the hardness result for point-to-point channels from [START_REF] Barman | Algorithmic aspects of optimal channel coding[END_REF].

On the other hand, it has been shown in [START_REF] Leditzky | Playing games with multiple access channels[END_REF] that it is NP-hard to decide whether a rate pair belongs to the classical capacity region of a MAC.

The goal of this section is to show that, under some reasonable hardness assumptions, the multiple-access channel coding problem cannot be approximated in polynomial time within a Ω(1) factor. We will achieve this through a ). For some constant c > 0, for every k, for ∆ a su ciently large constant independent of n, there is no polynomial-time algorithm that on most random k-SAT formulas with n variables and m = ∆n clauses outputs typical, but never outputs typical on k-SAT formulas with at least m2 -c √ k satis able clauses.

We will consider the following reformulation of the multiple-access channel coding problem:

Proposition 4.4. Given a MAC W and integers k 1 , k 2 , the multiple-access channel coding problem entails maximizing:

f W (S, T ) := 1 |S||T | y∈Y max x 1 ∈S,x 2 ∈T W (y|x 1 x 2 ) , over all sets S ⊆ X 1 , T ⊆ X 2 with |S| = k 1 , |T | = k 2 .
Proof. As the multiple-access channel coding problem is a convex program, its optimal solution can be found on integer solutions, i.e. when e 1 , e 2 and d are functions. Note that this restriction does not a ect the hardness of the problem as a deterministic code with a better or equal value can be retrieved easily from any code.

Therefore, given encoders e 1 , e 2 , the objective value becomes: is the same as optimizing over sets S, T of size k 1 , k 2 (corresponding to their images). Thus, we have that the multiple-access channel coding can be reformulated as:

max d 1 k 1 k 2 i 1 ,i 2 ,y W (y|e 1 (i 1 )e 2 (i 2 ))1 d(y)=(i 1 ,i 2 ) = 1 k 1 k 2 y max i 1 ,i 2 W (y|e 1 (i 1 )
max S⊆X 1 ,T ⊆X 2 :|S|=k 1 ,|T |=k 2 1 |S||T | y∈Y max x 1 ∈S,x 2 ∈T W (y|x 1 x 2 ) .
The previously mentioned hardness of approximation for the bipartite densest subgraph problem from [AAM + 11] will imply the following hardness of approximation for the multiple-access channel coding problem:

Theorem 4.5. Under Hypothesis 4.1, there is no polynomial-time Ω(1)-approximation algorithm for the multiple-access channel coding problem.

Proof. Given a bipartite graph G = (A B, E), we de ne its associated MAC W G on X 1 := A, X 2 := B and Y := A × B in the following way:

W ((a , b )|ab) =      1 a=a 1 b=b if (a, b) ∈ E , 1 |A||B| otherwise . ( 4
f W G (S, T ) -1 |S||T | 1 -1 |A||B| = |(S × T ) ∩ E| |S||T | .
Proof of Lemma 4.6. We have that: Let us now assume that a polynomial-time constant approximation of the multiple-access channel coding problem exists. Let us call λ ∈ (0, 1) the ratio of that algorithm. Then, for any G = (A B, E), k 1 , k 2 , we can apply our algorithm to W G , k 1 , k 2 and get S approx , T approx of size k 1 , k 2 such that:

f W G (S, T ) = 1 |S||T | a ,b max a∈S,b∈T W ((a , b )|ab) = 1 |S||T |   (a ,b )∈(S×T )∩E max a∈S,b∈T W ((a , b )|ab) + (a ,b ) ∈(S×T )∩E max a∈S,b∈T W ((a , b )|ab)   = 1 |S||T |   (a ,b )∈(S×T )∩E 1 + (a ,b ) ∈(S×T )∩E 1 |A||B|   = 1 |S||T | |(S × T ) ∩ E| + |A||B| -|(S × T ) ∩ E| |A||B| = 1 |S||T | + |(S × T ) ∩ E| |S||T | 1 - 1 |A||B| , ( 4
f W G (S approx , T approx ) ≥ λ • max S,T :|S|=k 1 ,|T |=k 2 f W G (S, T ) .
But, thanks to Lemma 4.6, we have that:

max S,T :|S|=k 1 ,|T |=k 2 f W G (S, T ) -1 k 1 k 2 1 -1 |A||B| = max S,T :|S|=k 1 ,|T |=k 2 |(S × T ) ∩ E| k 1 k 2 .
So in all, we have:

|(S approx × T approx ) ∩ E| k 1 k 2 = f W G (S approx , T approx ) -1 k 1 k 2 1 -1 |A||B| ≥ λ • max S,T :|S|=k 1 ,|T |=k 2 f W G (S, T ) -1 k 1 k 2 1 -1 |A||B| = λ • max S,T :|S|=k 1 ,|T |=k 2 f W G (S, T ) -1 k 1 k 2 1 -1 |A||B| - 1-λ k 1 k 2 1 -1 |A||B| = λ • max S,T :|S|=k 1 ,|T |=k 2 |(S × T ) ∩ E| k 1 k 2 - 1-λ k 1 k 2 1 -1 |A||B| , ( 4.17)
and thus:

|(S approx × T approx ) ∩ E| ≥ λ • max S,T :|S|=k 1 ,|T |=k 2 |(S × T ) ∩ E| - 1 -λ 1 -1 |A||B| .
Let us consider instances of the bipartite densest subgraph problem with |E| ≥ min(k 1 , k 2 ), which implies directly that max S,T :|S|=k 1 ,|T |=k 2 |(S × T ) ∩ E| ≥ min(k 1 , k 2 ) by picking min(k 1 , k 2 ) edges and taking their endpoints to construct S and T . As |A||B| ≥ 2, we have that

1-λ 1-1 |A||B| ≤ 2. Then: |(S approx × T approx ) ∩ E| ≥ λ • max S,T :|S|=k 1 ,|T |=k 2 |(S × T ) ∩ E| -2 ≥ (1 -c)λ • max S,T :|S|=k 1 ,|T |=k 2 |(S × T ) ∩ E| + cλ min(k 1 , k 2 ) -2 for any c ∈ (0, 1) = 1 - 2 λ min(k 1 , k 2 ) λ • max S,T :|S|=k 1 ,|T |=k 2 |(S × T ) ∩ E| , (4.18) by choosing c = 2 λ min(k 1 ,k 2 ) . Finally, for instances such that k 1 , k 2 ≥ 4
λ , we have that:

|(S approx × T approx ) ∩ E| ≥ λ 2 • max S,T :|S|=k 1 ,|T |=k 2 |(S × T ) ∩ E| .
Therefore, if a polynomial-time approximation of the multiple-access channel coding problem of constant ratio λ exists, then we have a polynomial-time λ 2 -approximation of the bipartite densest subgraph problem for instances satisfying |E| ≥ min(k 1 , k 2 ) and k 1 , k 2 ≥ 4 λ . Note that the bipartite densest subgraph problem can be solved in polynomial time for instances which do not satisfy these constraints (as 1 λ is constant). Thus, we have in all a constant polynomial-time approximation for the bipartite densest subgraph problem, which is in contradiction with Hypothesis 4.1 as proved in [AAM + 11]. Hence, under Hypothesis 4.1, there is no polynomial-time Ω(1)-approximation algorithm for the multiple-access channel coding problem.

Properties of Non-Signaling Assisted Codes

Symmetrization

One can prove an equivalent formulation of the linear program computing S NS (W, k 1 , k 2 ) with a number of variables and constraints polynomial in only |X 1 |, |X 2 | and |Y| and independent of k 1 and k 2 : Proposition 4.7. For a MAC W : X 1 × X 2 → Y and k 1 , k 2 ∈ N * , we have:

S NS (W, k 1 , k 2 ) = maximize r,r 1 ,r 2 ,p 1 k 1 k 2 x 1 ,x 2 ,y W (y|x 1 x 2 )r x 1 ,x 2 ,y subject to x 1 ,x 2 r x 1 ,x 2 ,y = 1 x 1 r 1 x 1 ,x 2 ,y = k 1 x 1 r x 1 ,x 2 ,y x 2 r 2 x 1 ,x 2 ,y = k 2 x 2 r x 1 ,x 2 ,y x 1 p x 1 ,x 2 = k 1 x 1 r 2 x 1 ,x 2 ,y x 2 p x 1 ,x 2 = k 2 x 2 r 1 x 1 ,x 2 ,y 0 ≤ r x 1 ,x 2 ,y ≤ r 1 x 1 ,x 2 ,y , r 2 x 1 ,x 2 ,y ≤ p x 1 ,x 2 p x 1 ,x 2 -r 1 x 1 ,x 2 ,y -r 2 x 1 ,x 2 ,y + r x 1 ,x 2 ,y ≥ 0 (4.19)
Proof. One can check that given a solution of the original program, the following choice of variables is a valid solution of the second program achieving the same objective value:

r x 1 ,x 2 ,y := i 1 ,i 2 P (x 1 x 2 (i 1 i 2 )|i 1 i 2 y) , r 1 x 1 ,x 2 ,y := j 1 ,i 1 ,i 2 P (x 1 x 2 (j 1 i 2 )|i 1 i 2 y) , r 2 x 1 ,x 2 ,y := j 2 ,i 1 ,i 2 P (x 1 x 2 (i 1 j 2 )|i 1 i 2 y) , p x 1 ,x 2 := j 1 ,j 2 ,i 1 ,i 2 P (x 1 x 2 (j 1 j 2 )|i 1 i 2 y) .
(4.20) Note that p x 1 ,x 2 is well-de ned since j 1 ,j 2 ,i 1 ,i 2 P (x 1 x 2 (j 1 j 2 )|i 1 i 2 y) is independent of y as P is a non-signaling box.

For the other direction, given those variables, a non-signaling probability distribution P (x 1 x 2 (j 1 j 2 )|i 1 i 2 y) achieving the same objective value is given by, for j 1 = i 1 and 73

4. M A C C W N S C j 2 = i 2 : P (x 1 x 2 (i 1 i 2 )|i 1 i 2 y) := r x 1 ,x 2 ,y k 1 k 2 , P (x 1 x 2 (j 1 i 2 )|i 1 i 2 y) := r 1 x 1 ,x 2 ,y -r x 1 ,x 2 ,y k 1 k 2 (k 1 -1) , P (x 1 x 2 (i 1 j 2 )|i 1 i 2 y) := r 2 x 1 ,x 2 ,y -r x 1 ,x 2 ,y k 1 k 2 (k 2 -1) , P (x 1 x 2 (j 1 j 2 )|i 1 i 2 y) := p x 1 ,x 2 -r 1 x 1 ,x 2 ,y -r 2 x 1 ,x 2 ,y + r x 1 ,x 2 ,y k 1 k 2 (k 1 -1)(k 2 -1) . (4.21)
This symmetrization can also be done for the program computing S NS SR sum (W, k 1 , k 2 ):

Proposition 4.8.

S NS SR sum (W, k 1 , k 2 ) = maximize r 1 ,r 2 ,p 1 ,p 2 1 2k 1 k 2 x 1 ,x 2 ,y W (y|x 1 x 2 ) p 2 x 2 r 1 x 1 ,y + p 1 x 1 r 2 x 2 ,y = 1 2k 1 x 1 ,y W 1 p 2 ,k 2 (y|x 1 )r 1 x 1 ,y + 1 2k 2 x 2 ,y W 2 p 1 ,k 1 (y|x 2 )r 2 x 2 ,y with W 1 p 2 ,k 2 (y|x 1 ) := 1 k 2 x 2 W (y|x 1 x 2 )p 2 x 2 and W 2 p 1 ,k 1 (y|x 2 ) := 1 k 1 x 1 W (y|x 1 x 2 )p 1 x 1 subject to x 1 r 1 x 1 ,y = 1, x 2 r 2 x 2 ,y = 1 x 1 p 1 x 1 = k 1 , x 2 p 2 x 2 = k 2 0 ≤ r 1 x 1 ,y ≤ p 1 x 1 , 0 ≤ r 2 x 2 ,y ≤ p 2 x 2 (4.22)
Proof. One can check that given a solution of the original program, the following choice of variables is a valid solution of the second program achieving the same objective value:

r 1 x 1 ,y := i 1 P 1 (x 1 i 1 |i 1 y) , p 1 x 1 := j 1 ,i 1 P 1 (x 1 j 1 |i 1 y) , r 2 x 2 ,y := i 2 P 2 (x 2 i 2 |i 2 y) , p 2 x 2 := j 2 ,i 2 P 2 (x 2 j 2 |i 2 y) . (4.23)
Note that p 1 x 1 and p 2 x 2 are well-de ned since j 1 ,i 1 P 1 (x 1 j 1 |i 1 y) and j 2 ,i 2 P 2 (x 2 j 2 |i 2 y) are independent of y as P 1 and P 2 are non-signaling boxes.

For the other direction, given those variables, non-signaling probability distributions P 1 (x 1 j 1 |i 1 y) and P 2 (x 2 j 2 |i 2 y) achieving the same objective value are given by, for j 1 = i 1 and j 2 = i 2 :

P 1 (x 1 i 1 |i 1 y) := r 1 x 1 ,y k 1 , P 1 (x 1 j 1 |i 1 y) := p 1 x 1 ,y -r 1 x 1 ,y k 1 (k 1 -1) , P 2 (x 2 i 2 |i 2 y) := r 2 x 2 ,y k 2 , P 2 (x 2 j 2 |i 2 y) := p 2 x 2 ,y -r 2 x 2 ,y k 2 (k 2 -1) . (4.24) 4.3.2 Properties of S NS (W, k 1 , k 2 ), C NS (W ) and C NS 0 (W )
Proposition 4.9. For a MAC W : X 1 × X 2 → Y and k 1 , k 2 ∈ N * , we have:

1. 1 k 1 k 2 ≤ S NS (W, k 1 , k 2 ) ≤ 1. 2. S NS (W, k 1 , k 2 ) ≤ min |X 1 | k 1 , |X 2 | k 2 , |Y| k 1 k 2 . 3. If k 1 ≤ k 1 and k 2 ≤ k 2 , then S NS (W, k 1 , k 2 ) ≥ S NS (W, k 1 , k 2 ). 4. For W : X 1 × X 2 → Y and k 1 , k 2 ∈ N * , we have S NS (W ⊗ W , k 1 k 1 , k 2 k 2 ) ≥ S NS (W, k 1 , k 2 ) • S NS (W , k 1 , k 2 ).
In particular, for any positive integer n, we have

S NS (W ⊗n , k n 1 , k n 2 ) ≥ S NS (W, k 1 , k 2 ) n and S NS (W ⊗W , k 1 , k 2 ) ≥ S NS (W, k 1 , k 2 ).
Proof.

1. We will start by proving that S NS (W,

k 1 , k 2 ) ≥ 1 k 1 k 2 . Take p x 1 ,x 2 := k 1 k 2 |X 1 ||X 2 | , r 1 x 1 ,x 2 ,y := px 1 ,x 2 k 2 , r 2 x 1 ,x 2 ,y := px 1 ,x 2 k 1 and r x 1 ,x 2 ,y := px 1 ,x 2 k 1 k 2 = 1 |X 1 ||X 2 | .
One can easily check that it is indeed a valid solution of the linear program computing S NS (W, k 1 , k 2 ). Thus we have:

S NS (W, k 1 , k 2 ) ≥ 1 k 1 k 2 x 1 ,x 2 ,y W (y|x 1 x 2 )r x 1 ,x 2 ,y = 1 k 1 k 2 x 1 ,x 2 1 |X 1 ||X 2 | y W (y|x 1 x 2 ) = 1 k 1 k 2 x 1 ,x 2 1 |X 1 ||X 2 | = 1 k 1 k 2 . (4.25)
Furthermore, in order to show that it is at most 1, let us consider an optimal solution of S NS (W, k 1 , k 2 ). We have:

S NS (W, k 1 , k 2 ) = 1 k 1 k 2 x 1 ,x 2 ,y W (y|x 1 x 2 )r x 1 ,x 2 ,y ≤ 1 k 1 k 2 x 1 ,x 2 ,y W (y|x 1 x 2 )p x 1 ,x 2 = 1 k 1 k 2 x 1 ,x 2 p x 1 ,x 2 y W (y|x 1 x 2 ) = 1 k 1 k 2 x 1 ,x 2 p x 1 ,x 2 = 1 , (4.26) since x 1 ,x 2 p x 1 ,x 2 = k 1 x 1 ,x 2 r 2 x 1 ,x 2 ,y = k 1 k 2 x 1 ,x 2 r x 1 ,x 2 ,y = k 1 k 2 . 75 4. M A C C W N S C 2. First let us show that S NS (W, k 1 , k 2 ) ≤ |X 1 | k 1 (the case S NS (W, k 1 , k 2 ) ≤ |X 2 | k 2 is symmetric): S NS (W, k 1 , k 2 ) = 1 k 1 k 2 x 1 ,x 2 ,y W (y|x 1 x 2 )r x 1 ,x 2 ,y ≤ 1 k 1 k 2 x 1 ,x 2 ,y W (y|x 1 x 2 )r 2 x 1 ,x 2 ,y ≤ 1 k 1 k 2 x 2 ,y   x 1 W (y|x 1 x 2 )   • x 1 r 2 x 1 ,x 2 ,y = 1 k 1 k 2 x 2 ,y   x 1 W (y|x 1 x 2 )   • 1 k 1 x 1 p x 1 ,x 2 = 1 k 2 1 k 2 x 1 ,x 2 p x 1 ,x 2 x 1 y W (y|x 1 x 2 ) = |X 1 | k 2 1 k 2 x 1 ,x 2 p x 1 ,x 2 = |X 1 | k 1 . (4.27) Let us show now that S NS (W, k 1 , k 2 ) ≤ |Y| k 1 k 2 : S NS (W, k 1 , k 2 ) = 1 k 1 k 2 x 1 ,x 2 ,y W (y|x 1 x 2 )r x 1 ,x 2 ,y ≤ 1 k 1 k 2 y max x 1 ,x 2 W (y|x 1 x 2 ) x 1 ,x 2 r x 1 ,x 2 ,y ≤ 1 k 1 k 2 y x 1 ,x 2 r x 1 ,x 2 ,y = |Y| k 1 k 2 .
(4.28)

3. Let us assume that k 1 ≤ k 1 and that k 2 = k 2 , since this latter case will follow by symmetry. Consider an optimal solution of S NS (W,

k 1 , k 2 ) = 1 k 1 i 1 ∈[k 1 ] f (i 1 ) with: f (i 1 ) := 1 k 2 x 1 ,x 2 ,y,i 2 W (y|x 1 x 2 )P (x 1 x 2 (i 1 i 2 )|i 1 i 2 y) ,
and P non-signaling. Let us consider S ∈ argmax

S ⊆[k 1 ]:|S |=k 1 i 1 ∈S f (i 1 )
. Then, by construction, we have that 1

k 1 i 1 ∈S f (i 1 ) ≥ 1 k 1 i 1 ∈[k 1 ] f (i 1 ) = S NS (W, k 1 , k 2
), since we have taken the average of the k 1 largest values of the sum.

Let us de ne the strategy P on the smallest set

X 1 × X 2 × (S × [k 2 ]) × S × [k 2 ] × Y: P (x 1 x 2 (j 1 j 2 )|i 1 i 2 y) := P (x 1 x 2 (j 1 j 2 )|i 1 i 2 y) + C(x 1 x 2 j 2 |i 1 i 2 y) , with C(x 1 x 2 j 2 |i 1 i 2 y) := 1 k 1 j 1 ∈[k 1 ]-S P (x 1 x 2 (j 1 j 2 )|i 1 i 2 y) . (4.29)
P is a correct conditional probability distribution. Indeed, it is nonnegative by construction, and we have that x 1 ,x 2 ,j 1 ∈S,j 2 P (x 1 x 2 (j 1 j 2 )|i 1 i 2 y) is equal to:

x 1 ,x 2 ,j 1 ∈S,j 2 P (x 1 x 2 (j 1 j 2 )|i 1 i 2 y) + x 1 ,x 2 ,j 1 ∈S,j 2 C(x 1 x 2 j 2 |i 1 i 2 y) = x 1 ,x 2 j 2 j 1 ∈S P (x 1 x 2 (j 1 j 2 )|i 1 i 2 y) + x 1 ,x 2 ,j 2 j 1 ∈S 1 k 1 j 1 ∈[k 1 ]-S P (x 1 x 2 (j 1 j 2 )|i 1 i 2 y) = x 1 ,x 2 ,j 2 j 1 ∈S P (x 1 x 2 (j 1 j 2 )|i 1 i 2 y) + x 1 ,x 2 ,j 2 j 1 ∈[k 1 ]-S P (x 1 x 2 (j 1 j 2 )|i 1 i 2 y) = x 1 ,x 2 ,j 1 ,j 2 P (x 1 x 2 (j 1 j 2 )|i 1 i 2 y) = 1 .
(4.30)

Let us show that P is non-signaling: a) First with x 1 :

x 1 P (x 1 x 2 (j 1 j 2 )|i 1 i 2 y) = x 1 P (x 1 x 2 (j 1 j 2 )|i 1 i 2 y) + x 1 C(x 1 x 2 j 2 |i 1 i 2 y) = x 1 P (x 1 x 2 (j 1 j 2 )|i 1 i 2 y) + 1 k 1 j 1 ∈[k 1 ]-S x 1 P (x 1 x 2 (j 1 j 2 )|i 1 i 2 y) = x 1 P (x 1 x 2 (j 1 j 2 )|i 1 i 2 y) + 1 k 1 j 1 ∈[k 1 ]-S x 1 P (x 1 x 2 (j 1 j 2 )|i 1 i 2 y)
since P is non-signaling.

=

x 1 P (x 1 x 2 (j 1 j 2 )|i 1 i 2 y) .

(4.31) b) Then with x 2 :

x 2 P (x 1 x 2 (j 1 j 2 )|i 1 i 2 y) = x 2 P (x 1 x 2 (j 1 j 2 )|i 1 i 2 y) + x 2 C(x 1 x 2 j 2 |i 1 i 2 y) = x 2 P (x 1 x 2 (j 1 j 2 )|i 1 i 2 y) + 1 k 1 j 1 ∈[k 1 ]-S x 2 P (x 1 x 2 (j 1 j 2 )|i 1 i 2 y) = x 2 P (x 1 x 2 (j 1 j 2 )|i 1 i 2 y) + 1 k 1 j 1 ∈[k 1 ]-S x 2 P (x 1 x 2 (j 1 j 2 )|i 1 i 2 y)
since P is non-signaling.

=

x 2 P (x 1 x 2 (j 1 j 2 )|i 1 i 2 y) .

(4.32) 77 4. M A C C W N S C
c) Finally with (j 1 j 2 ), we have that j 1 ∈S,j 2 P (x 1 x 2 (j 1 j 2 )|i 1 i 2 y) is equal to:

j 2 j 1 ∈S P (x 1 x 2 (j 1 j 2 )|i 1 i 2 y) + j 2 j 1 ∈S C(x 1 x 2 j 2 |i 1 i 2 y) = j 2 j 1 ∈S P (x 1 x 2 (j 1 j 2 )|i 1 i 2 y) + j 2 j 1 ∈S 1 k 1 j 1 ∈[k 1 ]-S P (x 1 x 2 (j j 2 )|i 1 i 2 y) = j 2 j 1 ∈S P (x 1 x 2 (j 1 j 2 )|i 1 i 2 y) + j 2 j 1 ∈[k 1 ]-S P (x 1 x 2 (j 1 j 2 )|i 1 i 2 y) = j 1 ,j 2 P (x 1 x 2 (j 1 j 2 )|i 1 i 2 y) = j 1 ,j 2 P (x 1 x 2 (j 1 j 2 )|i 1 i 2 y ) since P is non-signaling. = j 1 ∈S,j 2 P (x 1 x 2 (j 1 j 2 )|i 1 i 2 y ) .
(4.33) Thus P is a correct solution of the program computing S NS (W, k , k 2 ), and it leads to the value:

S NS (W, k 1 , k 2 ) ≥ 1 k 1 k 2 x 1 ,x 2 ,y,i 1 ∈S,i 2 W (y|x 1 x 2 )P (x 1 x 2 (i 1 i 2 )|i 1 i 2 y) ≥ 1 k 1 k 2 x 1 ,x 2 ,y,i 1 ∈S,i 2 W (y|x 1 x 2 )P (x 1 x 2 (i 1 i 2 )|i 1 i 2 y) = 1 k 1 i 1 ∈S f (i 1 ) ≥ 1 k 1 i 1 ∈[k 1 ] f (i 1 ) = S NS (W, k 1 , k 2 ) .
(4.34) 4. Consider optimal non-signaling probability distributions P and P reaching respectively the values S NS (W, k 1 , k 2 ) and S NS (W , k 1 , k 2 ). Then by Proposition 2.11,

P ⊗ P is a non-signaling probability distribution on (X 1 × X 1 ) × (X 2 × X 2 ) × (([k 1 ] × [k 1 ]) × ([k 2 ] × [k 2 ])) × ([k 1 ] × [k 1 ]) × ([k 2 ] × [k 2 ]) × (Y × Y ), which is trivially in bijection with (X 1 × X 1 ) × (X 2 × X 2 ) × ([k 1 k 1 ] × [k 2 k 2 ]) × [k 1 k 1 ] × [k 2 k 2 ] × (Y × Y
). This will be a feasible solution of the program computing

S NS (W ⊗ W , k 1 k 1 , k 2 k 2 ). Thus, we get that S NS (W ⊗ W , k 1 k 1 , k 2 k 2 )
is larger than or equal to (where the rst sums are over all indices x 1 x 1 , x 2 x 2 , yy , i 1 i 1 , i 2 i 2 ):

W ⊗ W (yy |x 1 x 1 x 2 x 2 ) P ⊗ P (x 1 x 1 x 2 x 2 (i 1 i 1 i 2 i 2 )|i 1 i 1 , i i 2 yy ) = W (y|x 1 x 2 ) • W (y |x 1 x 2 ) P (x 1 x 2 (i 1 i 2 )|i 1 i 2 y) • P (x 1 x 2 (i 1 i 2 )|i 1 i 2 y ) =   i 1 ,i 2 ,x 1 ,x 2 ,y W (y|x 1 x 2 )P (x 1 x 2 (i 1 i 2 )|i 1 i 2 y)   ×   x 1 ,x 2 ,y ,i 1 ,i 2 W (y |x 1 x 2 )P (x 1 x 2 (i 1 i 2 )|i 1 i 2 y )   =S NS (W, k 1 , k 2 ) • S NS (W , k 1 , k 2 ) . (4.35)
In particular, applying this n times on the same MAC W gives the rst corollary, and the second one comes from the fact that

S NS (W ⊗ W , k 1 , k 2 ) ≥ S NS (W, k 1 , k 2 ) • 4. M A C C W N S C S NS (W ⊗n , 2 R 1 n , 2 R 2 n ) m = 1, so S NS ((W ⊗n ) ⊗m , 2 R 1 n m , 2 R 2 n m ) = 1 since S NS (W, k 1 , k 2 )
≤ 1 by the rst property of Proposition 4.9. But (W ⊗n ) ⊗m = W ⊗nm , and

2 R 1 n m ≥ 2 R 1 nm , 2 R 2 n m ≥ 2 R 2 nm
, so by the third property of Proposition 4.9, we have

S NS (W ⊗nm , 2 R 1 nm , 2 R 2 nm ) ≥ 1, so S NS (W ⊗nm , 2 R 1 nm , 2 R 2 nm ) = 1.
Then, consider some r ∈ {0, . . . , n -1}. By the fourth property of Proposition 4.9, we have that:

S NS (W ⊗(nm+r) , 2 R 1 nm , 2 R 2 nm ) = S NS (W ⊗nm ⊗ W ⊗r , 2 R 1 nm , 2 R 2 nm ) ≥ S NS (W ⊗nm , 2 R 1 nm , 2 R 2 nm ) = 1 , (4.36) so S NS (W ⊗(nm+r) , 2 R 1 nm , 2 R 2 nm ) = 1. But 2 R 1 nm = 2 R 1 nm nm+r (nm+r) = 2 R 1 1+δ (nm+r) with δ = r nm ≤ 1 m , and symmetrically 2 R 1 nm = 2 R 1 1+δ (nm+r) . Thus in particular, for all R 1 ≤ R 1 1+ 1 m and R 2 ≤ R 2 1+ 1 m , we have that for all n ≥ nm, S NS (W ⊗n , 2 R 1 n , 2 R 2 n ) = 1. So for any (R 1 , R 2 ) such that R 1 < R 1 and R 2 < R 2 , there is large enough m such that R 1 ≤ R 1 1+ 1 m and R 2 ≤ R 2 1+ 1 m ,
and thus we get the expected property on (R 1 , R 2 ) for n 0 := nm.

Linear Program With Reduced Size for Structured Channels

Although S NS (W, k 1 , k 2 ) can be computed in polynomial time in W , k 1 and k 2 , a channel of the form W ⊗n has exponential size in n. Thus, the linear program for S NS (W ⊗n , k 1 , k 2 ) grows exponentially with n. However, using the invariance of W ⊗n under permutations, one can nd a much smaller linear program computing S NS (W ⊗n , k 1 , k 2 ).

De nition 4.2.

Let G a group acting on X 1 , X 2 , Y. We say that a MAC W :

X 1 × X 2 → Y is invariant under the action of G if: ∀g ∈ G, W (g • y|g • x 1 g • x 2 ) = W (y|x 1 x 2 ) .
In particular, for W ⊗n : X n 1 × X n 2 → Y n , the symmetric group G := S n acts in a natural way in any set A raised to power n. So for σ ∈ S n , we have that:

W ⊗n (σ • y n |σ • x n 1 σ • x n 2 ) = n i=1 W (y σ(i) |x 1,σ(i) x 2,σ(i) ) = n i=1 W (y i |x 1,i x 2,i ) = W ⊗n (y n |x n 1 x n 2 ) , (4.37)
and so W ⊗n is invariant under the action of S n .

Let

Z := {X 1 , X 2 , Y, X 1 × Y, X 2 × Y, X 1 × X 2 , X 1 × X 2 × Y}. Let us call O G (A)
the set of orbits of A under the action of G. Then, one can nd an equivalent smaller linear program for S NS (W, k 1 , k 2 ):

Theorem 4.12. Let W :

X 1 × X 2 → Y a MAC invariant under the action of G. Let us name systematically w ∈ O G (X 1 × X 2 × Y), u ∈ O G (X 1 × X 2 ), u 1 ∈ O G (X 1 ), u 2 ∈ O G (X 2 ), v 1 ∈ O G (X 1 × Y), v 2 ∈ O G (X 2 × Y), v ∈ O G (Y).
We will also call z A the projection of z ∈ O G (B) on A, for A, B ∈ Z and A projection of B; note that z A ∈ O G (A), since by de nition of the action, the projection of an orbit is an orbit. Let us nally call W (w) := W (y|x 1 x 2 ) for any (x 1 , x 2 , y) ∈ w, which is well-de ned since W is invariant under G. We have that S NS (W, k 1 , k 2 ) is the solution of the following linear program: maximize r,r 1 ,r 2 ,p Proof. We use the linear program obtained in Theorem 4.12 with G := S n acting on W ⊗n as described before. The number of variables and constraints is linear in the number of orbits of the action of S n on the di erent sets A ∈ Z, where here

1 k 1 k 2 w∈O G (X 1 ×X 2 ×Y) W (w)r w subject to w:w Y =v r w = |v|, ∀v ∈ O G (Y) w:w X 2 Y =v 2 r 1 w = k 1 w:w X 2 Y =v 2 r w , ∀v 2 ∈ O G (X 2 × Y) w:w X 1 Y =v 1 r 2 w = k 2 w:w X 1 Y =v 1 r w , ∀v 1 ∈ O G (X 1 × Y) u:u X 2 =v 2 X 2 p u = |v 2 X 2 | |v 2 | k 1 w:w X 2 Y =v 2 r 2 w , ∀v 2 ∈ O G (X 2 × Y) u:u X 1 =v 1 X 1 p u = |v 1 X 1 | |v 1 | k 2 w:w X 1 Y =v 1 r 1 w , ∀v 1 ∈ O G (X 1 × Y) 0 ≤ r w ≤ r 1 w , r 2 w ≤ |w| |w X 1 X 2 | p w X 1 X 2 , ∀w ∈ O G (X 1 × X 2 × Y) |w| |w X 1 X 2 | p w X 1 X 2 -r 1 w -r 2 w + r w ≥ 0, ∀w ∈ O G (X 1 × X 2 × Y) .
Z := {X n 1 , X n 2 , Y n , X n 1 × Y n , X n 2 × Y n , X n 1 × X n 2 , X n 1 × X n 2 × Y n }. For example, for A ∈ X n 1 × X n 2 × Y n , we have that: |O Sn (X n 1 × X n 2 × Y n )| = n + |X 1 ||X 2 ||Y| -1 |X 1 ||X 2 ||Y| -1 ≤ (n + |X 1 ||X 2 ||Y| -1) |X 1 ||X 2 ||Y|-1 .

So the number of variables and constraints is

O(n |X 1 |•|X 2 |•|Y|-1
(n |X 1 |•|X 2 |•|Y|-1
), and thus S NS (W ⊗n , k 1 , k 2 ) can be computed in polynomial time in n; see for instance Section 7.1 of [START_REF] Gärtner | Understanding and using linear programming[END_REF].

In order to prove Theorem 4.12, we will need several lemmas. For all of them, A and B will denote nite sets on which a group G is acting, and x G will denote the orbit of x under G: Proof. Let x, x ∈ ν. Thus there exists g ∈ G such that x = g • x. Let:

f : B x τ → B x τ y → g • y .
First, f is well de ned. Indeed, if y ∈ B x τ = {y : (x, y) ∈ τ }, then we have that g andv x,y variable indexed by elements of A × B, let us de ne v τ := (x,y)∈τ v x,y . We have:

• y ∈ {y : (g • x, y) ∈ τ } = B x τ , since τ ∈ O G (A × B). Let us show that f is injective. If g • y = g • y , then g -1 • (g • y) = (g -1 g) • y = y, g -1 • (g • y ) = y ,
:= v (x,y) G |(x,y) G | . We have: x∈A v x,y = 1 |y G | τ ∈O G (A×B):τ B =y G v τ , ∀y ∈ B . Proof. x∈A v x,y = τ ∈O G (A×B):τ B =y G x∈A:(x,y)∈τ v x,y = τ ∈O G (A×B):τ B =y G x∈A:(x,y)∈τ v τ |τ | since (x, y) G = τ = τ ∈O G (A×B):τ B =y G c y G τ v τ |τ | by Lemma 4.14, since y ∈ τ B = τ ∈O G (A×B):τ B =y G |τ | |y G | v τ |τ | = 1 |y G | τ ∈O G (A×B):τ B =y G v τ . (4.39) Lemma 4.16. For any τ ∈ O G (A × B), µ ∈ O G (B)
τ ∈O G (A×B):τ B =µ v τ = y∈µ x∈A v x,y . Proof. τ ∈O G (A×B):τ B =µ v τ = τ ∈O G (A×B):τ B =µ (x,y)∈τ v x,y = y∈µ x∈A v x,y .
Proof of Theorem 4.12. Let r x 1 ,x 2 ,y , r 1

x 1 ,x 2 ,y , r 2 x 1 ,x 2 ,y , p x 1 ,x 2 a feasible solution of the program de ned in Proposition 4.7, and call S := 1

k 1 k 2 x 1 ,x 2 ,y W (y|x 1 x 2 )r x 1 ,x 2 ,y its value. De ne: r w := (x 1 ,x 2 ,y)∈w r x 1 ,x 2 ,y , r 1 w := (x 1 ,x 2 ,y)∈w r 1 x 1 ,x 2 ,y , r 2 w := (x 1 ,x 2 ,y)∈w r 2 x 1 ,x 2 ,y , p u := (x 1 ,x 2 )∈u p x 1 ,x 2 . (4.40)
Let us show that r w , r 1 w , r 2 w , p u is a feasible solution of the program de ned in Theorem 4.12, and that its value

S * := 1 k 1 k 2 w W (w)r w = S.
First, we have S * = S. Indeed:

S * = 1 k 1 k 2 w W (w)r w = 1 k 1 k 2 w W (w) (x 1 ,x 2 ,y)∈w r x 1 ,x 2 ,y = 1 k 1 k 2 w (x 1 ,x 2 ,y)∈w W (y|x 1 x 2 )r x 1 ,x 2 ,y = 1 k 1 k 2 x 1 ,x 2 ,y W (y|x 1 x 2 )r x 1 ,x 2 ,y = S .
(4.41)

Then, all the constraints are satis ed. Indeed, thanks to Lemma 4.16, we have for the rst constraint:

w:w Y =v r w = y∈v x 1 ,x 2 r x 1 ,x 2 ,y = y∈v 1 = |v| .
For the second constraint (and symmetrically for the third constraint), we have:

w:w X 2 Y =v 2 r 1 w = (x 2 ,y)∈v 2 x 1 r 1 x 1 ,x 2 ,y = (x 2 ,y)∈v 2 k 1 x 1 r x 1 ,x 2 ,y = k 1 w:w X 2 Y =v 2 r w .
For the fourth (and symmetrically for the fth), we have:

w:w X 2 Y =v 2 r 2 w = (x 2 ,y)∈v 2 x 1 r 2 x 1 ,x 2 ,y = (x 2 ,y)∈v 2 1 k 1 x 1 p x 1 ,x 2 = 1 k 1 x 2 ∈v 2 X 2 y:(x 2 ,y)∈v 2 x 1 p x 1 ,x 2 = 1 k 1 x 2 ∈v 2 X 2 |v 2 | |v 2 X 2 | x 1 p x 1 ,x 2 thanks to Lemma 4.14 = 1 k 1 |v 2 | |v 2 X 2 | u:u X 2 =v 2 X 2 p u .
(4.42)

Finally for the last constraints, we only need to compute:

(x 1 ,x 2 ,y)∈w p x 1 ,x 2 = (x 1 ,x 2 )∈w X 1 X 2 y:(x 1 ,x 2 ,y)∈w p x 1 ,x 2 = (x 1 ,x 2 )∈w X 1 X 2 |w| |w X 1 X 2 | p x 1 ,x 2 = |w| |w X 1 X 2 | p w X 1 X 2 , (4.43) 83 4. M A C C W N S C
which implies that the linear inequalities on p x 1 ,x 2 , r x 1 ,x 2 ,y , r 1 x 1 ,x 2 ,y , r 2 x 1 ,x 2 ,y get transposed respectively to the values |w| |w X 1 X 2 | p w X 1 X 2 , r w , r 1 w , r 2 w . Indeed, for instance, one has for any x 1 , x 2 , y that p x 1 ,x 2 -r 1

x 1 ,x 2 ,y -r 2 x 1 ,x 2 ,y + r x 1 ,x 2 ,y ≥ 0. Thus for some orbit w:

(x 1 ,x 2 ,y)∈w p x 1 ,x 2 -r 1 x 1 ,x 2 ,y -r 2 x 1 ,x 2 ,y + r x 1 ,x 2 ,y ≥ 0 , and then |w| |w X 1 X 2 | p w X 1 X 2 -r 1 w -r 2 w + r w ≥ 0,
which was what we wanted to show. Now let us consider a feasible solution r w , r 1 w , r 2 w , p u of the program de ned in Theorem 4.12, with a value S * := 1

k 1 k 2 w W (w)r w . De ne: r x 1 ,x 2 ,y := r (x 1 ,x 2 ,y) G |(x 1 , x 2 , y) G | , r 1 x 1 ,x 2 ,y := r 1 (x 1 ,x 2 ,y) G |(x 1 , x 2 , y) G | , r 2 x 1 ,x 2 ,y := r 2 (x 1 ,x 2 ,y) G |(x 1 , x 2 , y) G | , p x 1 ,x 2 := p (x 1 ,x 2 ) G |(x 1 , x 2 ) G | . (4.44)
Let us show that r x 1 ,x 2 ,y , r 1 x 1 ,x 2 ,y , r 2 x 1 ,x 2 ,y , p x 1 ,x 2 is a feasible solution of the program de ned in Proposition 4.7, and that its value S := 1 k 1 k 2 x 1 ,x 2 ,y W (y|x 1 x 2 )r x 1 ,x 2 ,y = S * . First we have S = S * . Indeed:

S = 1 k 1 k 2 x 1 ,x 2 ,y W (y|x 1 x 2 )r x 1 ,x 2 ,y = 1 k 1 k 2 x 1 ,x 2 ,y W (y|x 1 x 2 ) r (x 1 ,x 2 ,y) G |r (x 1 ,x 2 ,y) G | = 1 k 1 k 2 w (x 1 ,x 2 ,y)∈w W (y|x 1 x 2 ) r w |w| = 1 k 1 k 2 w (x 1 ,x 2 ,y)∈w W (w) r w |w| = 1 k 1 k 2 w |w|W (w) r w |w| = 1 k 1 k 2 w W (w)r w = S * . (4.45)
Then, all the constraints are satis ed. Indeed, thanks to Lemma 4.15, we have for the rst constraint:

x 1 ,x 2 r x 1 ,x 2 ,y = 1 |y G | w:w Y =y G r w = |y G | |y G | = 1 .
For the second constraint (and symmetrically for the third constraint), we have:

x 1 r 1 x 1 ,x 2 ,y = 1 |(x 2 , y) G | w:w X 2 Y =(x 2 ,y) G r 1 w = k 1 |(x 2 , y) G | w:w X 2 Y =(x 2 ,y) G r w = k 1 x 1 r x 1 ,x 2 ,y . (4.46)
For the fourth (and symmetrically for the fth), we have:

x 1 r 2 x 1 ,x 2 ,y = 1 |(x 2 , y) G | w:w X 2 Y =(x 2 ,y) G r 2 w = 1 |(x 2 , y) G | 1 k 1 |(x 2 , y) G | |(x 2 , y) G X 2 | u:u X 2 =(x 2 ,y) G X 2 p u = 1 k 1 1 |(x 2 , y) G X 2 | u:u X 2 =(x 2 ,y) G X 2 p u = 1 k 1 1 |x G 2 | u:u X 2 =x G 2 p u since (x 2 , y) G X 2 = x G 2 = 1 k 1 x 1 p x 1 ,x 2 .
(4.47)

Finally, to conclude with the last constraints, one has only to see that for any x 1 , x 2 , y:

|(x 1 , x 2 , y) G | |(x 1 , x 2 , y) G X 1 X 2 | p (x 1 ,x 2 ,y) G X 1 X 2 = |(x 1 , x 2 , y) G | |(x 1 , x 2 ) G | p (x 1 ,x 2 ) G = |(x 1 , x 2 , y) G |p x 1 ,x 2 ,
which implies that the linear inequalities on |w| |w X 1 X 2 | p w X 1 X 2 , r w , r 1 w , r 2 w get transposed respectively to the values p x 1 ,x 2 , r x 1 ,x 2 ,y , r 1

x 1 ,x 2 ,y , r 2 x 1 ,x 2 ,y . Indeed, for instance, one has for any w that

|w| |w X 1 X 2 | p w X 1 X 2 -r 1 w -r 2 w + r w ≥ 0. But for any (x 1 , x 2 , y) ∈ w, one has that r x 1 ,x 2 ,y = rw |w| , r 1 x 1 ,x 2 ,y = r 1 w |w| , r 2 x 1 ,x 2 ,y = r 2 w |w| .
Thanks to the previous inequality, we have that p x 1 ,x 2 = pw X 1 X 2 |w X 1 X 2 | , and thus:

p x 1 ,x 2 -r 1 x 1 ,x 2 ,y -r 2 x 1 ,x 2 ,y + r x 1 ,x 2 ,y = p w X 1 X 2 |w X 1 X 2 | - r 1 w |w| - r 2 w |w| + r w |w| ≥ 0 ,
which was what we wanted to show.

Non-Signaling Achievability Bounds

Zero-Error Non-Signaling Assisted Achievable Rate Pairs

We will now present a numerical method to nd e ciently inner bounds on C NS 0 (W ). Thanks to Corollary 4.13, we know how to decide in polynomial time in n, k 1 , k 2 whether

S NS (W ⊗n , k 1 , k 2 ) = 1. However, by Proposition 4.11, if S NS (W ⊗n , k 1 , k 2 ) = 1, then we have that log(k 1 ) n , log(k 2 ) n ∈ C NS 0 (W )
, which describes a way of computing achievable points for that capacity region. More precisely, this leads to the following result: Proposition 4.17 (Inner Bounds on C NS 0 (W )). Let us de ne the zero-error non-signaling assisted n-shots capacity region C NS 0,≤n (W ) in the following way:

C NS 0,≤n (W ) := log(k 1 ) n , log(k 2 ) n : S NS (W ⊗n , k 1 , k 2 ) = 1 .
Then, we have that ∀n ∈ N, C NS 0,≤n (W ) ⊆ C NS 0 (W ), and that one can decide in polynomial time

in n, k 1 , k 2 if log(k 1 ) n , log(k 2 ) n ∈ C NS 0,≤n (W ).
This implies that we can nd e ciently achievable rate pairs for MACs.

85 4. M A C C W N S C
Application to the binary adder channel The binary adder channel W BAC is de ned by:

∀x 1 , x 2 ∈ {0, 1}, ∀y ∈ {0, 1, 2}, W BAC (y|x 1 x 2 ) := 1 y=x 1 +x 2 .
Its classical capacity region C(W BAC ) is well known and consists of all The rst noticeable result coming from these curves is that the zero-error non-signaling assisted sum-rate capacity beats with only 7 copies the classical sum-rate capacity of 3 2 , even without a zero-error constraint, with a value of 2 log 2 (42) 7 1.5406, coming from the fact that S NS (W ⊗7 BAC , 42, 42) = 1 and Proposition 4.11. This implies that C NS 0 (W BAC ) has larger sum-rate pairs than C(W BAC ), and that C NS (W BAC ) is strictly larger than C(W BAC ). This sum-rate can even be increased up to log 2 (72) 1.5425, since we have computed S NS (W ⊗8 BAC , 72, 72) = 1, which is the largest number of copies we have been able to manage with our e cient version of the linear program from Theorem 4.12. This should be compared with the upper bound on the non-signaling assisted sum-rate capacity coming from Proposition 4.26, which is log 2 (3) 1.5850 for R 1 = R 2 .

(R 1 , R 2 ) such that R 1 ≤ 1, R 2 ≤ 1, R 1 + R 2 ≤ 3 2 ,
Another surprising property is the speed at which one obtains e cient zero-error nonsignaling assisted codes compared to classical zero-error codes. Indeed, with only three copies of the binary adder channel, one gets that S NS (W ⊗3 BAC , 4, 5) = 1, which corresponds to a sum-rate of 2+log 2 (5) 3 1.4406, which already largely beats the best known zero-error achieved sum-rate of log 2 (240/6) 1.3178 [START_REF] Mattas | A new bound for the zero-error capacity region of the two-user binary adder channel[END_REF]. These results are summarized in the following theorem: Theorem 4.18. We have that ; see [START_REF] Mattas | A new bound for the zero-error capacity region of the two-user binary adder channel[END_REF] for a description of this border.

R 1 R 2 C(W BAC ) Best Inner Bounds on C 0 (W BAC ) C NS 0,≤2 (W BAC ) C NS 0,≤3 (W BAC ) C NS 0,≤7 (W BAC )
On the other hand, the continuous curves depict the best zero-error non-signaling assisted achievable rate pairs for respectively 2, 3 and 7 copies of the binary adder channel.

Non-Signaling Assisted Achievable Rate Pairs With Non-Zero Error

We have analyzed the non-signaling assisted capacity region through zero-error strategies and applied it to the BAC. However, if some noise is added to that channel, its zero-error non-signaling assisted capacity region becomes trivial (see Proposition 4.21). Thus, the previous method fails to nd signi cant inner bounds on the non-signaling assisted capacity region of noisy MACs.

In this section, we use concatenated codes to obtain achievable rate pairs, and apply it to a noisy version of the BAC:

De nition 4.3 (Concatenated Codes). Given a MAC W and a non-signaling assisted code P , let

W [P ] : [k 1 ]×[k 2 ] → [ ] with W [P ](j|i 1 i 2 ) := x 1 ,x 2 ,y W (y|x 1 x 2 )P (x 1 x 2 j|i 1 i 2 y): 87 4. M A C C W N S C P (x 1 x 2 j|i 1 i 2 y) i 1 i 2 j W x 1 x 2 y := W [P ] i 1 i 2 j Note that W [P ] is a MAC since W [P ](j|i 1 i 2 ) ≥ 0 and: j W [P ](j|i 1 i 2 ) = x 1 ,x 2 ,y W (y|x 1 x 2 ) j P (x 1 x 2 j|i 1 i 2 y) = x 1 ,x 2 y W (y|x 1 x 2 ) P (x 1 x 2 |i 1 i 2 ) since P is non-signaling = x 1 ,x 2 P (x 1 x 2 |i 1 i 2 ) = 1 .
(4.48)

The following proposition states that combining a classical code to a non-signaling strategy leads to inner bounds on the non-signaling assisted capacity region of a MAC:

Proposition 4.19. If P is a non-signaling assisted code for the MAC W , we have that

C(W [P ]) ⊆ C NS (W ). Proof. Let (R 1 , R 2 ) ∈ C(W [P ]
). Then, by de nition, we have that:

lim n→+∞ S(W [P ] ⊗n , 2 R 1 n , 2 R 2 n ) = 1 .
Let us x ε > 0. For a large enough N , we have S(W

[P ] ⊗N , 2 R 1 N , 2 R 2 N ) ≥ 1 -ε. Let us call 1 := 2 R 1 N and 2 := 2 R 2 N . Thus, there exists encoders e 1 : [ 1 ] → [k 1 ], e 2 : [ 2 ] → [k 2 ] and a decoder d : [ ] → [ 1 ] × [ 2 ] such that:
1

1 2 i 1 ,i 2 ,j W [P ](j|i 1 i 2 ) a 1 ∈[ 1 ],a 2 ∈[ 2 ] e 1 (i 1 |a 1 )e 2 (i 2 |a 2 )d(a 1 a 2 |j) ≥ 1 -ε .
In particular, we have:

1 1 2 x 1 ,x 2 ,y W (y|x 1 x 2 )   i 1 ,i 2 ,j,a 1 ,a 2 P (x 1 x 2 j|i 1 i 2 y)e 1 (i 1 |a 1 )e 2 (i 2 |a 2 )d(a 1 a 2 |j)   ≥ 1-ε . De ne P (x 1 x 2 (b 1 b 2 )|a 1 a 2 y) := i 1 ,i 2 ,j P (x 1 x 2 j|i 1 i 2 y)e 1 (i 1 |a 1 )e 2 (i 2 |a 2 )d(b 1 b 2 |j).
One can easily check that P is non-signaling, and thus:

S NS (W ⊗N , 1 , 2 ) ≥ 1 1 2 x 1 ,x 2 ,y W (y|x 1 x 2 ) a 1 ,a 2 P (x 1 x 2 (a 1 , a 2 )|a 1 a 2 y) ≥ 1 -ε . This implies that lim n→+∞ S NS (W ⊗n , 2 R 1 n , 2 R 2 n ) = 1, i.e. (R 1 , R 2 ) ∈ C NS (W ).
Thanks to Proposition 4.19, we have for any non-signaling assisted code P , C(W

⊗n [P ]) ⊆ C NS (W ⊗n ). But if (R 1 , R 2 ) ∈ C NS (W ⊗n ), we have that ( R 1 n , R 2 n ) ∈ C NS (W ).
Thus, applying Theorem 4.1 on W ⊗n [P ] leads to inner bounds on C NS (W ): Proposition 4.20 (Inner Bounds on C NS (W )). For any number of copies n, number of inputs

k 1 ∈ [|X 1 | n ] and k 2 ∈ [|X 2 | n ], non-signaling assisted codes P on inputs in [k 1 ], [k 2 ]
for W ⊗n , and distributions q 1 , q 2 on [k 1 ], [k 2 ], we have that the rate pairs (R 1 , R 2 ) satisfying the following constraints are in C NS (W ):

R 1 ≤ I(I 1 : J|I 2 ) n , R 2 ≤ I(I 2 : J|I 1 ) n , R 1 + R 2 ≤ I((I 1 , I 2 ) : J) n , for (I 1 , I 2 ) ∈ [k 1 ] × [k 2 ]
following the product law q 1 × q 2 , and J ∈ [ ] the outcome of W ⊗n [P ] on inputs I 1 , I 2 . In particular, the corner points of this capacity region are given by:

I(I 1 : J|I 2 ) n , I(I 2 : J) n and I(I 1 : J) n , I(I 2 : J|I 1 ) n .
Proof. The achievable region comes from the previous discussion. We just need to prove that the corner points are of the given form. If R 1 = I(I 1 :J|I 2 ) n , constraints on R 2 and R 1 + R 2 leads to a maximum R 2 = min I(I 2 :J|I 1 ) n , I((I 1 ,I 2 ):J) n -I(I 1 :J|I 2 ) n . However, I((I 1 , I 2 ) : J) -I(I 1 : J|I 2 ) = I(I 2 : J) by the chain rule. We only need to show that I(I 2 : J) ≤ I(I 2 : J|I 1 ) and the proof will be complete, since the other corner point is symmetric. We have:

I(I 2 : J) = H(I 2 ) -H(I 2 |J) = H(I 2 |I 1 ) -H(I 2 |J) ≤ H(I 2 |I 1 ) -H(I 2 |JI 1 ) = I(I 2 : J|I 1 ) , (4.49)
the second equality coming from the fact that I 1 and I 2 are independent, and the inequality coming from the fact that H(A|BC) ≤ H(A|B) for any A, B, C.

Application to the Noisy Binary Adder Channel

We will now apply this strategy to a noisy version of the BAC. We will consider ip errors ε 1 , ε 2 of inputs x 1 , x 2 on W BAC , which leads to the following de nition of W BAC,ε 1 ,ε 2 :

∀y, x 1 , x 2 , W BAC,ε 1 ,ε 2 (y|x 1 x 2 ) := (1 -ε 1 )(1 -ε 2 )W BAC (y|x 1 x 2 ) + ε 1 (1 -ε 2 )W BAC (y|x 1 x 2 ) + (1 -ε 1 )ε 2 W BAC (y|x 1 x 2 ) + ε 1 ε 2 W BAC (y|x 1 x 2 ) .
(4.50)

First, let us note that the zero-error non-signaling assisted capacity region of W BAC,ε 1 ,ε 2 is trivial for ε ∈ (0, 1):

Proposition 4.21. If ε 1 , ε 2 ∈ (0, 1), then C NS 0 (W BAC,ε 1 ,ε 2 ) = {(0, 0)}. 89 4. M A C C W N S C Proof. If S NS (W ⊗n , k 1 , k 2 ) = 1, then ∀y n , x n 1 , x n 2 : W ⊗n (y n |x n 1 x n 2 ) > 0 =⇒ r x n 1 ,x n 2 ,y n = p x n 1 ,x n
2 . Indeed, we have for an optimal p, r that:

S NS (W ⊗n , k 1 , k 2 ) = 1 k 1 k 2 x n 1 ,x n 2 ,y n W ⊗n (y n |x n 1 x n 2 )r x n 1 ,x n 2 ,y n ≤ 1 k 1 k 2 x n 1 ,x n 2 ,y n W ⊗n (y n |x n 1 x n 2 )p x n 1 ,x n 2 = 1 , (4.51)
which implies the previous statement. But, for W ⊗n BAC,ε 1 ,ε 2 , one can easily check that for all

y n , x n 1 , x n 2 , W ⊗n (y n |x n 1 x n 2 ) > 0 since ε 1 , ε 2 ∈ (0, 1
). Indeed, one can ip the inputs to a valid preimage of the output. Thus if

S N S (W ⊗n BAC,ε 1 ,ε 2 , k 1 , k 2 ) = 1, we have that ∀y n , x n 1 , x n 2 , r x n 1 ,x n 2 ,y n = p x n 1 ,x n 2 .
In particular, this implies that

x n 1 ,x n 2 r x n 1 ,x n 2 ,y n = x n 1 ,x n 2 p x n 1 ,x n 2 , therefore 1 = k 1 k 2 , so k 1 = 1 and k 2 = 1. Thus S N S (W ⊗n , 2 nR 1 , 2 nR 2 ) = 1 implies that (R 1 , R 2 ) = (0, 0).
We have then applied the numerical method described in Proposition 4.20 to W BAC,ε 1 ,ε 2 for the symmetric case ε 1 = ε 2 = ε := 10 -3 . Since it is hard to go through all nonsignaling assisted codes P and product distributions q 1 , q 2 , we have applied the heuristic of using non-signaling assisted codes obtained while optimizing S NS (W ⊗n , k 1 , k 2 ) in the symmetrized linear program. We have combined them with uniform q 1 , q 2 , as the form of those non-signaling assisted codes coming from our optimization program is symmetric. We have evaluated the achievable corner points for all k 1 , k 2 ≤ 2 n for n ≤ 5 copies which led to Figure 4.5: Compared to the noiseless binary adder channel, we can rst notice that the classical capacity region is slightly smaller, with a classical sum-rate capacity of 1.478 at most. On the other hand, although the zero-error non-signaling assisted capacity of W BAC,ε,ε is completely trivial, we have with our concatenated codes strategy found signi cant rate pairs achievable with non-signaling assistance. In particular, we have reached a non-signaling assisted sum-rate capacity of 1.493 which beats the best classical sum-rate capacity. Thus, it shows that non-signaling assistance can improve the capacity of the noisy binary adder channel as well.

Relaxed Non-Signaling Assisted Capacity Region and Outer Bounds

A natural question that arises when studying the strength of non-signaling assistance is whether a result similar to Theorem 4.1 can be found to describe by a single-letter formula the non-signaling assisted capacity region of MACs. In particular, dropping the constraint that (X 1 , X 2 ) is in product form in Theorem 4.1 seems to be a particularly good candidate to characterize the non-signaling assisted capacity region of MACs, as this looks quite similar to allowing correlations between parties.

We have not been able to show the equivalence between this region and the non-signaling assisted capacity region; however, it turns out to be equivalent to the capacity region de ned by a slight relaxation of non-signaling assistance, which we call S NS (W, k 1 , k 2 ). In particular, this will give us the best known outer bound on the non-signaling capacity.

S NS (W, k 1 , k 2 ) := maximize r,p 1 k 1 k 2 x 1 ,x 2 ,y W (y|x 1 x 2 )r x 1 ,x 2 ,y subject to x 1 ,x 2 r x 1 ,x 2 ,y ≤ 1 x 1 ,x 2 p x 1 ,x 2 = k 1 k 2 x 1 p x 1 ,x 2 ≥ k 1 x 1 r x 1 ,x 2 ,y x 2 p x 1 ,x 2 ≥ k 2 x 2 r x 1 ,x 2 ,y 0 ≤ r x 1 ,x 2 ,y ≤ p x 1 ,x 2 (4.52)
The following proposition shows that this is indeed a relaxation of the non-signaling constraint.

Proposition 4.22.

S NS (W, k 1 , k 2 ) ≤ S NS (W, k 1 , k 2 ). 91 4. M A C C W N S C
Proof. Let us take a solution (p

x 1 ,x 2 , r x 1 ,x 2 ,y , r 1 x 1 ,x 2 ,y , r 2 x 1 ,x 2 ,y ) x 1 ∈X 1 ,x 2 ∈X 2 ,y∈Y of the linear program computing S NS (W, k 1 , k 2 ). Let us show that (p x 1 ,x 2 , r x 1 ,x 2 ,y ) x 1 ∈X 1 ,x 2 ∈X 2 ,
y∈Y is a solution of the linear program computing S NS (W, k 1 , k 2 ) with a same objective value, from which the proposition follows.

They have indeed the same value, since the de nition which is the same for both programs depends only on r x 1 ,x 2 ,y . Let us show that all constraints are satis ed for the solution

(p x 1 ,x 2 , r x 1 ,x 2 ,y ) x 1 ∈X 1 ,x 2 ∈X 2 ,y∈Y .
We have x 1 ,x 2 r x 1 ,x 2 ,y = 1 ≤ 1 so the rst constraint is satis ed. We have then that:

x 1 ,x 2 p x 1 ,x 2 = k 1 x 1 ,x 2 r 2 x 1 ,x 2 ,y = k 1 k 2 x 1 ,x 2 r x 1 ,x 2 ,y = k 1 k 2 ,
so the second constraint is satis ed.

For the third constraint (and symmetrically the fourth constraint), we have:

x 1 p x 1 ,x 2 = k 1 x 1 r 2 x 1 ,x 2 ,y ≥ k 1 x 1 r x 1 ,x 2 ,y .
Finally, we have directly 0 ≤ r x 1 ,x 2 ,y ≤ p x 1 ,x 2 , so the last constraint is satis ed.

We can now de ne the relaxed non-signaling assisted capacity region of a MAC W using De nition 2.12 by

C NS (W ) := C[S NS ](W ).
Proposition 4.23. C NS (W ) is convex.

Proof. We only need to show that for all MACs W, W and integers k 1 , k 2 , k 1 , k 2 :

S NS (W ⊗ W , k 1 k 1 , k 2 k 2 ) ≥ S NS (W, k 1 , k 2 ) • S NS (W , k 1 , k 2 ) ,
thanks to Proposition 2.15. Let us consider optimal solutions (p

x 1 ,x 2 , r x 1 ,x 2 ,y ) x 1 ∈X 1 ,x 2 ∈X 2 ,y∈Y and (p x 1 ,x 2 , r x 1 ,x 2 ,y ) x 1 ∈X 1 ,x 2 ∈X 2 ,y ∈Y of the programs computing S NS (W, k 1 , k 2 ) and S NS (W , k 1 , k 2 ) respectively.
We construct a solution of the program computing the success probability S NS (W ⊗ W , k 1 k 1 , k 2 k 2 ) by de ning:

p * x 1 x 1 ,x 2 x 2 := p x 1 ,x 2 p x 1 ,x 2 and r * x 1 x 1 ,x 2 x 2 ,yy := r x 1 ,x 2 ,y r x 1 ,x 2 ,y , as W ⊗ W : (X 1 × X 1 ) × (X 2 × X 2 ) → (Y × Y ).
It is straightforward to check that it is a valid solution. Indeed, linear constraints are preserved by product, except for coe cients k 1 , k 2 which are replaced by products k 1 k 1 and k 2 k 2 . Therefore, we have that

S NS (W ⊗ W , k 1 k 1 , k 2 k 2 )
is larger than or equal to:

1 k 1 k 1 k 2 k 2 x 1 ,x 1 ,x 2 ,x 2 ,y,y W ⊗ W (yy |x 1 x 1 x 2 x 2 )r * x 1 x 1 ,x 2 x 2 ,yy = 1 k 1 k 1 k 2 k 2 x 1 ,x 1 ,x 2 ,x 2 ,y,y W (y|x 1 x 2 )W (y |x 1 x 2 )r x 1 ,x 2 ,y r x 1 ,x 2 ,y = 1 k 1 k 2 x 1 ,x 2 ,y W (y|x 1 x 2 )r x 1 ,x 2 ,y •   1 k 1 k 2 x 1 ,x 2 ,y W (y |x 1 x 2 )r x 1 ,x 2 ,y   = S NS (W, k 1 , k 2 ) • S NS (W , k 1 , k 2 ) . (4.53) 4. M A C C W N S C Let us call B 1 (P X 1 X 2 ) := H(X 1 |X 2 ), B 2 (P X 1 X 2 ) := H(X 2 |X 1 ) and B 12 (P X 1 X 2 ) := H(X 1 + X 2
) the three bounds. Let us call P X 1 X 2 = (p 11 , p 10 , p 01 , p 00 ). One can notice that:

B 1 (P X 1 X 2 ) = H(X 1 |X 2 ) = H(1 -X 1 |1 -X 2 ) = H(X 1 |X 2 ) = B 1 (P X 1 X 2 ) , B 2 (P X 1 X 2 ) = H(1 -X 2 |1 -X 1 ) = H(X 2 |X 1 ) = B 2 (P X 1 X 2 ) , B 12 (P X 1 X 2 ) = H(X 1 + X 2 ) = H(1 -X 1 + 1 -X 2 ) = H(2 -(X 1 + X 2 )) = H(X 1 + X 2 ) = B 12 (P X 1 X 2 ) . (4.54) Since B 12 (P X 1 X 2 ) = H(X 1 + X 2 ) = H(p 00 , p 11 , p 01 + p 10 ), it is concave in P X 1 X 2 as H is concave and (p 00 , p 11 , p 01 + p 10 ) is linear in P X 1 X 2 . Also, B 1 (P X 1 X 2 ) = H(X 1 |X 2 ) = -D(P X 1 X 2 ||I ⊗ P X 2 ) is concave in P X 1 X 2 as the divergence D is jointly convex and I ⊗ P X 2 is linear in P X 1 X 2 . By symmetry, B 2 (P X 1 X 2 ) is as well concave in P X 1 X 2 .
Let us consider any of those three bounds, which we call B. We have by concavity of B and the fact that B(P

X 1 X 2 ) = B(P X 1 X 2 ): B(P X 1 X 2 ) = B(P X 1 X 2 ) + B(P X 1 X 2 ) 2 ≤ B P X 1 X 2 + P X1 X2 2 = B q 2 , 1 -q 2 , 1 -q 2 , q 2 , (4.55)
with q = p 00 + p 11 . This holds for the three bounds at the same time, so we can restrict ourselves to the distributions of the form q 2 , 1-q 2 , 1-q 2 , q 2 for some q ∈ [0, 1], i.e., P X 1 X 2 (0, 0) = P X 1 X 2 (1, 1) = q 2 and P X 1 X 2 (0, 1) = P X 1 X 2 (1, 0) = 1-q 2 . We have P Y (0) = P Y (2) = q 2 and P Y (1) = 1 -q, so:

B 12 (P X 1 X 2 ) = H(Y ) = -q log q 2 -(1 -q) log(1 -q) = -q (log(q) -1) -(1 -q) log(1 -q) = q + h (q) .
(4.56)

We have

P X 2 (0) = P X 1 X 2 (0, 0) + P X 1 X 2 (1, 0) = q 2 + 1-q 2 = 1 2 so P X 2 (1) = 1 2 . Thus: B 1 (P X 1 X 2 ) = H(X 1 |X 2 ) = 1 2 H(X 1 |X 2 = 0) + 1 2 H(X 1 |X 2 = 1) .
We have

P X 1 |X 2 =0 (0) = P X 1 X 2 (0,0) P X 2 (0) = q so H(X 1 |X 2 = 0) = h (q).
On the other hand, we have

P X 1 |X 2 =1 (1) = P X 1 X 2 (1,1) P X 2 (1)
= q so we get as well H(Y |X 2 = 1) = h (q), and B 1 (P X 1 X 2 ) = H(X 1 |X 2 ) = h (q). Symmetrically, we also get B 2 (P X 1 X 2 ) = h (q). Therefore, we get that C NS (W BAC ) is the closure of the convex hull of:

q∈[0,1] {(R 1 , R 2 ) : R 1 < h (q) , R 2 < h (q) , R 1 + R 2 < q + h (q)} .
However this set is already convex, so we have:

C NS (W BAC ) = q∈[0,1] {(R 1 , R 2 ) : R 1 ≤ h (q) , R 2 ≤ h (q) , R 1 + R 2 ≤ q + h (q)} .
Finally, we can restrict ourselves to q ∈ 1 2 , 2 3 , since h is increasing from 0 to 1 2 (thus q → q + h (q) as well), and the fact that q → q + h (q) achieves its maximum for q = 2 3 with 2 3 + h 2 3 = log 2 (3) and then decreases (whereas h is decreasing from 1 2 to 1), which completes the proof.

As before, one can also de ne a symmetrized version of the relaxed linear program computing the value S NS (W ⊗n , k 1 , k 2 ) in polynomial time in n and compute the zero-error n-shots capacity region by looking at the rates where S NS (W ⊗n , k 1 , k 2 ) = 1. We have computed this up to 7 copies of the binary adder channel, which led to Figure 4.6:

0 0.25 0.5 0.75 1 0 0.25 0.5 0.75 1 R 1 R 2 C(W BAC ) C NS (W BAC ) C NS 0,≤7 (W BAC ) C NS 0,≤7 (W BAC )
Figure 4.6 -Comparison of relaxed and regular non-signaling assisted capacity regions of the binary adder channel. The black dashed curve depicts the classical capacity region C(W BAC ), whereas the grey dotted curve depicts the relaxed non-signaling assisted capacity region C NS (W BAC ) as described in Proposition 4.26. In particular, the curved corners are obtained by taking

R 1 = h(R 2 ) for R 2 ∈ 1 2 , 2
3 and symmetrically by switching the roles played by R 1 and R 2 . The continuous blue (respectively red) curve depicts the zero-error (respectively relaxed) non-signaling assisted achievable rate pairs for 7 copies of the binary adder channel.

The rst noticeable result coming from these curves is that the values S NS and S NS di er. While the highest sum-rate of 2 log 2 (42) 7 1.5406 is achieved on 7 copies of the binary adder channel with zero-error and non-signaling assistance, coming from the fact that S NS (W ⊗7 BAC , 42, 42) = 1, we have that S NS (W ⊗7 BAC , 44, 44) = 1 > S NS (W ⊗7 BAC , 44, 44)

95 4. M A C C W N S C
0.9581 which implies that a sum-rate of 2 log 2 (44) 7

1.5598 is achieved on 7 copies of the binary adder channel with zero-error and relaxed non-signaling assistance. It also largely beats the best found sum-rate of log 2 (72) 4 1.5425 achieved on 8 copies with the regular version. However the fact that the non-signaling assisted capacity region is strictly contained in the relaxed one is still open, as the same rates could potentially be achieved by the cost of using more copies of the channel.

Outer Bound Part of Theorem 4.25

In order to prove Proposition 4.31, we use a connection between hypothesis testing and relaxed non-signaling assisted codes as established in [START_REF] Matthews | A linear program for the nite block length converse of Polyanskiy-Poor-Verdú via nonsignaling codes[END_REF] for point-to-point channels.

De nition 4.5 (Hypothesis Testing). Given distributions P (0) and P (1) on the same space C, we de ne β 1-ε (P (0) , P (1) ) to be the minimum type II error r∈C T r P (1) (r) that can be achieved by statistical tests T which give a type I error no greater than ε, i.e. r∈C T r P (0) (r) ≥ 1 -ε. In other words, we have that:

β 1-ε (P (0) , P (1) ) = minimize Tr r∈C T r P (1) (r) subject to r∈C T r P (0) (r) ≥ 1 -ε 0 ≤ T r ≤ 1 .
(4.57)

Lemma 4.27. For any relaxed non-signaling assisted code (p

x 1 ,x 2 , r x 1 ,x 2 ,y ) x 1 ∈X 1 ,x 2 ∈X 2 ,y∈Y with (k 1 , k 2 ) messages and a probability of success 1 -ε, if P X 1 X 2 (x 1 , x 2 ) = px 1 ,x 2
k 1 k 2 and Y ∈ Y is the outcome of W on inputs X 1 , X 2 , we have:

β 1-ε P X 1 X 2 Y , P X 1 X 2 × P Y |X 2 ≤ 1 k 1 β 1-ε P X 1 X 2 Y , P X 1 X 2 × P Y |X 1 ≤ 1 k 2 β 1-ε (P X 1 X 2 Y , P X 1 X 2 × P Y ) ≤ 1 k 1 k 2 .
(4.58)

Remark. These three bounds are actually achieved with the same statistical test.

Proof. This result is a direct generalization of Theorem 9 in [START_REF] Matthews | A linear program for the nite block length converse of Polyanskiy-Poor-Verdú via nonsignaling codes[END_REF] for point-to-point channels, itself a generalization of Theorem 27 in [START_REF] Yury Polyanskiy | Channel coding rate in the nite blocklength regime[END_REF] without non-signaling assistance.

Let us name W 0 := W and W 1 a MAC yet to be de ned. The coding strategy described by r x 1 ,x 2 ,y and p x 1 ,x 2 leads to a probability of success on channel i ∈ {0, 1} given by:

1 -ε i = 1 k 1 k 2 x 1 ,x 2 ,y r x 1 ,x 2 ,y W i (y|x 1 x 2 ) = x 1 ,x 2 ,y:px 1 ,x 2 >0 r x 1 ,x 2 ,y p x 1 ,x 2 W i (y|x 1 x 2 ) p x 1 ,x 2 k 1 k 2 since 0 ≤ r x 1 ,x 2 ,y ≤ p x 1 ,x 2 = x 1 ,x 2 ,y T x 1 ,x 2 ,y W i (y|x 1 x 2 ) p x 1 ,x 2 k 1 k 2 , (4.59) with T x 1 ,x 2 ,y := rx 1 ,x 2 ,y px 1 ,x 2 if p x 1 ,x 2 > 0, and T x 1 ,x 2 ,y := 0 otherwise.
If now Y is the output of the channel W i , the joint distribution of X 1 , X 2 , Y is given by P

(i) X 1 X 2 Y (x 1 , x 2 , y) = W i (y|x 1 x 2 )P X 1 X 2 (x 1 , x 2 ) = W i (y|x 1 x 2 ) px 1 ,x 2 k 1 k 2 .
On the other hand, we have that for all x 1 , x 2 , y, 0 ≤ T x 1 ,x 2 ,y ≤ 1 since 0 ≤ r x 1 ,x 2 ,y ≤ p x 1 ,x 2 . So we get that:

1 -ε i = x 1 ,x 2 ,y T x 1 ,x 2 ,y P (i) X 1 X 2 Y (x 1 , x 2 , y) . Since x 1 ,x 2 ,y T x 1 ,x 2 ,y P (0) X 1 X 2 Y (x 1 , x 2 , y) ≥ 1 -ε 0 and 0 ≤ T x 1 ,x 2 ,y ≤ 1, we have: β 1-ε 0 (P (0) , P (1) ) ≤ x 1 ,x 2 ,y T x 1 ,x 2 ,y P (1) X 1 X 2 Y (x 1 , x 2 , y) = 1 -ε 1 .
Let us now consider three general cases, depending on the fact that W 1 does not depend on x 1 , x 2 or both:

W 1 (y|x 1 x 2 ) := Q (1) (y|x 2 ); W 1 (y|x 1 x 2 ) := Q (2) (y|x 1 ); W 1 (y|x 1 x 2 ) := Q (0) (
y). These will give respectively the three bounds we want.

First, let us consider the case where W 1 (y|x 1 x 2 ) := Q (1) (y|x 2 ) (the second case where W 1 (y|x 1 x 2 ) := Q (2) (y|x 1 ) being symmetric), we have that:

1 -ε 1 = x 1 ,x 2 ,y T x 1 ,x 2 ,y Q (1) (y|x 2 ) p x 1 ,x 2 k 1 k 2 = 1 k 1 k 2 x 2 ,y Q (1) (y|x 2 ) x 1 T x 1 ,x 2 ,y p x 1 ,x 2 = 1 k 1 k 2 x 2 ,y Q (1) (y|x 2 ) x 1 r x 1 ,x 2 ,y ≤ 1 k 1 k 2 x 2 ,y Q (1) (y|x 2 ) 1 k 1 x 1 p x 1 ,x 2 = 1 k 1 x 1 ,x 2 p x 1 ,x 2 k 1 k 2 y Q (1) (y|x 2 ) = 1 k 1 x 1 ,x 2 p x 1 ,x 2 k 1 k 2 = 1 k 1 .
(4.60)

For the third case, when W 1 (y|x 1 x 2 ) := Q (0) (y), we have:

1 -ε 1 = x 1 ,x 2 ,y T x 1 ,x 2 ,y Q (0) (y) p x 1 ,x 2 k 1 k 2 = 1 k 1 k 2 y Q (0) (y) x 1 ,x 2 T x 1 ,x 2 ,y p x 1 ,x 2 = 1 k 1 k 2 y Q (0) (y) x 1 ,x 2 r x 1 ,x 2 ,y ≤ 1 k 1 k 2 y Q (0) (y) = 1 k 1 k 2 .
(4.61)

In those three cases, we have respectively P

(1)

X 1 X 2 Y = P X 1 X 2 ×Q (1) Y |X 2 ; P X 1 X 2 ×Q (2) Y |X 1 ; and P X 1 X 2 × Q (0) Y . Specializing those cases with Q (1) Y |X 2 := P Y |X 2 ; Q (2) Y |X 1 := P Y |X 1 ; Q (0)
Y := P Y and using the fact that β 1-ε 0 P (0) , P (1) ≤ 1 -ε 1 concludes the proof.

Lemma 4.28. For any relaxed non-signaling assisted code (p

x 1 ,x 2 , r x 1 ,x 2 ,y ) x 1 ∈X 1 ,x 2 ∈X 2 ,y∈Y with (k 1 , k 2 ) messages and a probability of success 1 -ε, if P X 1 X 2 (x 1 , x 2 ) = px 1 ,x 2 k 1 k 2 and Y ∈ Y is the outcome of W on inputs X 1 , X 2 , we have: log(k 1 ) ≤ I(X 1 : Y |X 2 ) + h(ε) 1 -ε , log(k 2 ) ≤ I(X 2 : Y |X 1 ) + h(ε) 1 -ε , log(k 1 ) + log(k 2 ) ≤ I((X 1 , X 2 ) : Y ) + h(ε) 1 -ε . (4.62) 97 4. M A C C W N S C
Proof. Thanks to Lemma 4.27, with the fact that

P X 1 X 2 = P X 1 |X 2 × P X 2 = P X 2 |X 1 × P X 1 ,
we have already:

β 1-ε P X 1 X 2 Y , P X 1 |X 2 × P Y |X 2 × P X 2 ≤ 1 k 1 , β 1-ε P X 1 X 2 Y , P X 2 |X 1 × P Y |X 1 × P X 1 ≤ 1 k 2 , β 1-ε (P X 1 X 2 Y , P X 1 X 2 × P Y ) ≤ 1 k 1 k 2 . (4.63)
Following the steps of section G in [START_REF] Yury Polyanskiy | Channel coding rate in the nite blocklength regime[END_REF], since any hypothesis test is a binary-output transformation, by data-processing inequality for divergence, we have that:

d 1 -ε||β 1-ε P X 1 X 2 Y , P X 1 |X 2 × P Y |X 2 × P X 2 = d β 1-ε (P X 1 X 2 Y , P X 1 X 2 Y ) ||β 1-ε P X 1 X 2 Y , P X 1 |X 2 × P Y |X 2 × P X 2 ≤ D P X 1 X 2 Y || P X 1 |X 2 × P Y |X 2 × P X 2 = I(X 1 : Y |X 2 ) , (4.64)
where the binary divergence is de ned by d(a||b) := a log a b + (1 -a) log 1-a 1-b and satis es d(a||b) ≥ -h(a) -a log(b), and thus:

log 1 b ≤ d(a||b) + h(a) a = d(a||b) + h(1 -a) a .
This leads to:

log(k 1 ) ≤ 1 log β 1-ε P X 1 X 2 Y , P X 1 |X 2 × P Y |X 2 × P X 2 ≤ I(X 1 : Y |X 2 ) + h(ε) 1 -ε .
Similarly for the two other inequalities, since

D P X 1 X 2 Y || P X 2 |X 1 × P Y |X 1 × P X 1 = I(X 2 : Y |X 1 ) and D (P X 1 X 2 Y ||P X 1 X 2 × P Y ) = I((X 1 , X 2 ) : Y ), we get: log(k 1 ) ≤ I(X 1 : Y |X 2 ) + h(ε) 1 -ε , log(k 2 ) ≤ I(X 2 : Y |X 1 ) + h(ε) 1 -ε , log(k 1 ) + log(k 2 ) ≤ I((X 1 , X 2 ) : Y ) + h(ε) 1 -ε . (4.65)
In order to show additivity of the outer bound, we use the following lemma.

Lemma 4.29. For any distribution

P X n 1 X n 2 of (X n 1 , X n 2 ), if Y n ∈ Y n is the outcome of W n on inputs X n
1 , X n 2 , we have:

I(X n 1 : Y n |X n 2 ) ≤ n i=1 I(X 1,i : Y i |X 2,i ) I(X n 2 : Y n |X n 1 ) ≤ n i=1 I(X 2,i : Y i |X 1,i ) I((X n 1 , X n 2 ) : Y n ) ≤ n i=1 I((X 1,i , X 2,i ) : Y i ) .
(4.66)

Proof. Consider n copies of W . Let us write X 1,-i := X 1,1 . . . X 1,i-1 X 1,i+1 . . . X 1,n and Z n := Z 1 . . . Z n . We have:

I(X n 1 : Y n |X n 2 ) = I(X n 1 : Y n |X n 2 ) = n i=1 I(X n 1 : Y i |X n 2 Y i-1 ) by the chain rule = n i=1 I(X 1,i : Y i |X n 2 Y i-1 ) + n i=1 I(X 1,-i : Y i |X n 2 Y i-1 X 1,i ) = n i=1 I(X 1,i : Y i |X n 2 Y i-1 ) , (4.67)
where the last equality comes from Lemma 4.30. As a result,

I(X n 1 : Y n |X n 2 ) = n i=1 H(Y i |X n 2 Y i-1 ) -H(Y i |X n 2 Y i-1 X 1,i ) = n i=1 H(Y i |X n 2 Y i-1 ) -H(Y i |X 2,i X 1,i ) since X 2,-i Y i-1 → (X 1,i , X 2,i ) → Y i Markov chain. ≤ n i=1 H(Y i |X 2,i ) -H(Y i |X 2,i X 1,i ) = n i=1 I(X 1,i : Y i |X 2,i ) . (4.68)
Symmetrically, by switching the roles of X 1 and X 2 , we get the second part of Lemma 4.29.

For the sum-rate case:

I((X n 1 , X n 2 ) : Y n ) = n i=1 I((X n 1 , X n 2 ) : Y i |Y i-1
) by the chain rule

= n i=1 I((X 1,i , X 2,i ) : Y i |Y i-1 ) + n i=1 I((X 1,-i , X 2,-i ) : Y i |Y i-1 X 1,i X 2,i ) = n i=1 I((X 1,i , X 2,i ) : Y i |Y i-1 ) since (X 1,-i , X 2,-i ) → Y i-1 X 1,i X 2,i → Y i Markov chain. = n i=1 H(Y i |Y i-1 ) -H(Y i |Y i-1 X 1,i X 2,i ) = n i=1 H(Y i |Y i-1 ) -H(Y i |X 1,i X 2,i ) since Y i-1 → (X 1,i , X 2,i ) → Y i Markov chain. ≤ n i=1 H(Y i ) -H(Y i |X 2,i X 1,i ) = n i=1 I((X 1,i , X 2,i ) : Y i ) .
We next prove a technical lemma that was used in the previous proof.

Lemma 4.30. For any distribution

P X n 1 X n 2 of (X n 1 , X n 2 ), if Y n ∈ Y n is the outcome of W n on inputs X n 1 , X n 2 , we have I(X 1,-i : Y i |X 1,i X n 2 Y i-1 ) = 0. 99 4. M A C C W N S C
Proof. Let us show that, conditioned on any particular instance of

X 1,i = x i,1 , X n 2 = x n 2 , Y i-1 1 = y i-1 , X 1,-i and Y i are independent.
We have:

P Y i = y i |X 1,i = x i,1 , X n 2 = x n 2 , Y i-1 1 = y i-1 = P (Y i = y i |X 1,i = x i,1 , X 2,i = x 2,i ) = W (y i |x 1,i x 2,i ) , ( 4
.69) by de nition of the law of Y i . On the other hand, we have that:

P (X n 1 = x n 1 , X n 2 = x n 2 , Y n = y n ) = P (X n 1 = x n 1 , X n 2 = x n 2 ) n j=1 W (y j |x 1,j x 2,j ) .
Thus, we have that

P X 1,i = x i,1 , X n 2 = x n 2 , Y i-1 1 = y i-1 is equal to: x 1,-i ,x n 2 ,y i ,...,yn P (X n 1 = x n 1 , X n 2 = x n 2 ) n j=1 W (y j |x 1,j x 2,j ) = x 1,-i ,x n 2 P (X n 1 = x n 1 , X n 2 = x n 2 ) i-1 j=1 W (y j |x 1,j x 2,j ) n j=i   y j W (y j |x 1,j x 2,j )   = x 1,-i ,x n 2 P (X n 1 = x n 1 , X n 2 = x n 2 ) i-1 j=1 W (y j |x 1,j x 2,j ) .
(4.70)

And then,

P X 1,-i = x 1,-i |X 1,i = x i,1 , X n 2 = x n 2 , Y i-1 1 = y i-1 is equal to: x n 2 P (X n 1 = x n 1 , X n 2 = x n 2 ) i-1 j=1 W (y j |x 1,j x 2,j ) x 1,-i ,x n 2 P (X n 1 = x n 1 , X n 2 = x n 2 ) i-1 j=1 W (y j |x 1,j x 2,j )
.

But:

P X 1,-i = x 1,-i , Y i = y i |X 1,i = x i,1 , X n 2 = x n 2 , Y i-1 1 = y i-1 = x n 2 P (X n 1 = x n 1 , X n 2 = x n 2 ) i j=1 W (y j |x 1,j x 2,j ) x 1,-i ,x n 2 P (X n 1 = x n 1 , X n 2 = x n 2 ) i-1 j=1 W (y j |x 1,j x 2,j ) = P X 1,-i = x 1,-i |X 1,i = x i,1 , X n 2 = x n 2 , Y i-1 1 = y i-1 W (y i |x 1,i x 2,i ) = P X 1,-i = x 1,-i |X 1,i = x i,1 , X n 2 = x n 2 , Y i-1 1 = y i-1 × P Y i = y i |X 1,i = x i,1 , X n 2 = x n 2 , Y i-1 1 = y i-1 . (4.71)
Thus, conditioned on any particular instance of

X 1,i = x i,1 , X n 2 = x n 2 , Y i-1 1 = y i-1 , X 1,-i
and Y i are independent, and so I(X 1,-i :

Y i |X 1,i X n 2 Y i-1 ) = 0.
Combining the previous results gives the desired outer bound.

M

A C C W N S C But |T n ε (X 1 , X 2 |y n )| ≤ 2 n(1+ε)H(X 1 ,X 2 |Y ) and for a large enough n, |T n ε (X 1 , X 2 )| ≥ (1 -ε)2 n(1-ε)H(X 1 ,X 2 ) ≥ 2 n((1-ε)H(X 1 ,X 2 )-1 n )
, so in that case:

x n 1 ,x n 2 r x n 1 ,x n 2 ,y n ≤ 2 n(R 1 +R 2 +2εH(X 1 ,X 2 )) 2 n(1+ε)H(X 1 ,X 2 |Y ) 2 n((1-ε)H(X 1 ,X 2 )-1 n ) = 2 n(R 1 +R 2 -I(X 1 ,X 2 :Y )+ 1 n +C 3 ε) ≤ 1 , (4.78) since I(X 1 , X 2 : Y ) = H(X 1 , X 2 ) -H(X 1 , X 2 |Y ) and R 1 + R 2 ≤ I(X 1 , X 2 : Y ) -1 n - C 3 ε, with C 3 := H(X 1 , X 2 |Y ) + 3H(X 1 , X 2 ).
Let us focus on the constraint

x n 1 p x n 1 ,x n 2 ≥ k 1 x n 1 r x n 1 ,x n 2 ,y n (the symmetric constraint x n 2 p x n 1 ,x n 2 ≥ k 2 x n 2 r x n 1 ,x n
2 ,y n will be achieved for symmetric reasons).

Let us x (x n 2 , y n ). If (x n 2 , y n ) ∈ T n ε (X 2 , Y ), then for all x n 1 , (x n 1 , x n 2 , y n ) ∈ T n ε (X 1 , X 2 , Y ), thus r x n 1 ,x n 2 ,y n = 0 and the constraint is ful lled. Let us assume that (x n 2 , y n ) ∈ T n ε (X 2 , Y ). Since r x n 1 ,x n 2 ,y n > 0 implies that (x n 1 , x n 2 , y n ) ∈ T n ε (X 1 , X 2 , Y
), we have that:

x n 1 r x n 1 ,x n 2 ,y n = x n 1 ∈T n ε (X 1 |x n 2 ,y n ) r x n 1 ,x n 2 ,y n = x n 1 ∈T n ε (X 1 |x n 2 ,y n ) p x n 1 ,x n 2 .
Thus:

x n 1 p x n 1 ,x n 2 k 1 x n 1 r x n 1 ,x n 2 ,y n ≥ 1 k 1 x 1 ∈T n ε (X 1 |x n 2 ) P X n 1 X n 2 (x n 1 , x n 2 ) x 1 ∈T n ε (X 1 |x n 2 ,y n ) P X n 1 X n 2 (x n 1 , x n 2 ) ≥ 1 k 1 x 1 ∈T n ε (X 1 |x n 2 ) P X n 1 X n 2 (x n 1 , x n 2 ) x 1 ∈T n ε (X 1 |x n 2 ,y n ) P X n 1 X n 2 (x n 1 , x n 2 ) ≥ 1 k 1 2 -n(1+ε)H(X 1 ,X 2 ) 2 -n(1-ε)H(X 1 ,X 2 ) |T n ε (X 1 |x n 2 )| |T n ε (X 1 |x n 2 , y n )| ≥ 2 n(-R 1 -2εH(X 1 ,X 2 )) |T n ε (X 1 |x n 2 )| |T n ε (X 1 |x n 2 , y n )| . (4.79) But |T n ε (X 1 |x n 2 , y n )| ≤ 2 n(1+ε)H(X 1 |X 2 Y
) and for a large enough n we have

∀x n 2 ∈ T n ε (X 2 ), |T n ε (X 1 |x n 2 )| ≥ (1 -ε)2 n(1-ε)H(X 1 |X 2 ) ≥ 2 n((1-ε)H(X 1 |X 2 )-1 n ) , so we get with C 1 := 2H(X 1 , X 2 ) + H(X 1 |X 2 Y ) + H(X 1 |X 2 ) (symmetrically C 2 := 2H(X 1 , X 2 ) + H(X 2 |X 1 Y ) + H(X 2 |X 1 )): x n 1 p x n 1 ,x n 2 k 1 x n 1 r x n 1 ,x n 2 ,y n ≥ 2 n(H(X 1 |X 2 )-1 n -H(X 1 |X 2 Y )-R 1 -C 1 ε) = 2 n(I(X 1 :Y |X 2 )-1 n -C 1 ε-R 1) ≥ 1 . (4.80)
For a large enough n, all constraints are satis ed, thus (p

x n 4. M A C C W N S C
As 1 -1 2n tends to 1 when n tends to in nity, we get that ∀ε > 0, ∃N ∈ N, ∀n ≥ N :

(1 -ε)S NS SR sum (W ⊗n , 2 nR 1 , 2 nR 1 ) ≤ S sum (W ⊗n , 2 n(R 1 -log(n) n ) , 2 n(R 2 -log(n) n ) ) . Thus, if lim n→+∞ S NS SR sum (W ⊗n , 2 nR 1 , 2 nR 1 ) = 1, we have that for all R 1 < R 1 and R 2 < R 2 : lim n→+∞ S sum (W ⊗n , 2 nR 1 , 2 nR 1 ) ≥ 1 -ε .
Since this is true for all ε > 0, we get in fact that lim

n→+∞ S sum (W ⊗n , 2 nR 1 , 2 nR 1 ) = 1. This implies that C NS SR sum (W ) ⊆ C sum (W )
, and thus that the capacity regions are equal as the other inclusion is always satis ed.

In order to prove Theorem 4.34, we will need the following lemma: Lemma 4.36. If S 1 , S 2 are classical codes (i.e. multisets with elements in X 1 , X 2 ) of size 1 , 2 :

S sum (W, 1 , 2 ) ≥ 1 2   f W 1 S 2 , 2 (S 1 ) 1 + f W 2 S 1 , 1 (S 2 ) 2   ,
where

W 1 S 2 , 2 is the channel de ned by W 1 S 2 , 2 (y|x 1 ) = 1 2 2 i 2 =1 W (y|x 1 S i 2 2 ) and similarly for W 2 S 1 , 1 .
Proof. Let us de ne e 1 (x 1 |i 1 ) := 1 S i 1

1 =x 1 and e 2 (x 2 |i 2 ) := 1 S i 2 2 =x 2
. Then for xed y, let us take

j y 1 ∈ argmax i 1 { 2 i 2 =1 W (y|S i 1 1 S i 2 2 )}, j y 2 ∈ argmax i 2 { 1 i 1 =1 W (y|S i 1 1 S i 2
2 )} and then de ne d(j 1 |y) := 1 j 1 =j y 1 , d(j 2 |y) := 1 j 2 =j y 2 . We have then:

S sum (W, 1 , 2 ) ≥ 1 1 2 i 1 ,i 2 ,x 1 ,x 2 ,y W (y|x 1 x 2 )1 S i 1 1 =x 1 1 S i 2 2 =x 2 1 i 1 =i y 1 + 1 i 2 =i y 2 2 = 1 1 2 i 1 ,i 2 ,y W (y|S i 1 1 S i 2 2 ) 1 i 1 =i y 1 + 1 i 2 =i y 2 2 = 1 1 2 y 1 2   i 2 W (y|S i y 1 1 S i 2 2 ) + i 1 W (y|S i 1 1 S i y 2 2 )   = 1 1 2 y 1 2   max i 1 i 2 W (y|S i 1 1 S i 2 2 ) + max i 2 i 1 W (y|S i 1 1 S i 2 2 )   = 1 2   y max i 1 1 2 i 2 W (y|S i 1 1 S i 2 2 ) 1 + y max i 2 1 1 i 1 W (y|S i 1 1 S i 2 2 ) 2   = 1 2 y max i 1 W 1 S 2 , 2 (y|S i 1 1 ) 1 + y max i 2 W 2 S 1 , 1 (y|S i 2 2 ) 2 = 1 2   f W 1 S 2 , 2 (S 1 ) 1 + f W 2 S 1 , 1 (S 2 ) 2   .
(4.82)

We have now all the tools to prove Theorem 4.34:

Proof of Theorem 4.34. Let us consider an optimal solution r 1 , r 2 , p 1 , p 2 of the program of Proposition 4.8 computing S NS SR sum (W, k 1 , k 2 ). Let us x some multiset S 2 with elements in X 2 of size 2 . Note that r 1 and p 1 are a feasible solution of the program of Proposition 2.17 computing S NS (W 1 S 2 , 2 , k 1 ). As a result, we can apply Theorem 2.18 and get the following statement. For the multiset S 1 obtained by choosing 1 elements of X 1 independently according to the distribution

p 1 x 1 k 1 x 1 ∈X 1 , we
have:

E S 1   f W 1 S 2 , 2 (S 1 ) 1   ≥ k 1 1 1 -1 - 1 k 1 1 • 1 k 1 x 1 ,y W 1 S 2 , 2 (y|x 1 )r 1 x 1 ,y .
Now, let S 2 be the multiset obtained by choosing 2 elements of X 2 independently according to the distribution

p 2 x 2 k 2 x 2 ∈X 2
. We have:

E S 2 1 k 1 x 1 ,y W 1 S 2 , 2 (y|x 1 )r 1 x 1 ,y = E S 2   1 k 1 x 1 ,y 1 2 2 i 2 =1 W (y|x 1 S i 2 2 )r 1 x 1 ,y   = 1 2 2 i 2 =1 E X i 2 2 ∼ p 2 x 2 k 2 1 k 1 x 1 ,y W (y|x 1 X i 2 2 )r 1 x 1 ,y = E X 2 ∼ p 2 x 2 k 2 1 k 1 x 1 ,y W (y|x 1 X 2 )r 1 x 1 ,y = 1 k 1 x 1 ,x 2 ,y p 2 x 2 k 2 W (y|x 1 x 2 )r 1 x 1 ,y = 1 k 1 x 1 ,y W 1 p 2 ,k 2 (y|x 1 )r 1 x 1 ,y . (4.83)
Thus in all, we have:

E S 2   E S 1   f W 1 S 2 , 2 (S 1 ) 1     ≥ E S 2 k 1 1 1 -1 - 1 k 1 1 • 1 k 1 x 1 ,y W 1 S 2 , 2 (y|x 1 )r 1 x 1 ,y = k 1 1 1 -1 - 1 k 1 1 • E S 2 1 k 1 x 1 ,y W 1 S 2 , 2 (y|x 1 )r 1 x 1 ,y ≥ k 1 1 1 -1 - 1 k 1 1 • 1 k 1 x 1 ,y W 1 p 2 ,k 2 (y|x 1 )r 1 x 1 ,y , (4.84)
and symmetrically for

E S 1 E S 2 f W 2 S 1 , 1 (S 2 ) 2
. Since there exists codes S * 1 , S * 2 such that:

1 2   f W 1 S * 2 , 2 (S * 1 ) 1 + f W 2 S * 1 , 1 (S * 2 ) 2   ≥ E S 1 ,S 2   1 2   f W 1 S 2 , 2 (S 1 ) 1 + f W 2 S 1 , 1 (S 2 ) 2     , 107 4. M A C C W N S C
and by applying Lemma 4.36, we get:

S sum (W, 1 , 2 ) ≥ 1 2   f W 1 S * 2 , 2 (S * 1 ) 1 + f W 2 S * 1 , 1 (S * 2 ) 2   ≥ E S 1 ,S 2   1 2   f W 1 S 2 , 2 (S 1 ) 1 + f W 2 S 1 , 1 (S 2 ) 2     = 1 2   E S 2   E S 1   f W 1 S 2 , 2 (S 1 ) 1     + E S 1   E S 2   f W 2 S 1 , 1 (S 2 ) 2       ≥ k 1 2 1 1 -1 - 1 k 1 1 • 1 k 1 x 1 ,y W 1 p 2 (y|x 1 )r 1 x 1 ,y + k 2 2 2 1 -1 - 1 k 2 2 • 1 k 2 x 2 ,y W 2 p 1 (y|x 2 )r 2 x 2 ,y ≥ min k 1 1 1 -1 - 1 k 1 1 , k 2 2 1 -1 - 1 k 2 2 S NS SR sum (W, k 1 , k 2 ) , (4.85)
which concludes the proof.

Remark. In the whole proof of Theorem 4.34, as well as the properties it depends on, we have never used the fact that the output of the channel y was the same for both decoders d 1 and d 2 . This implies that the result also holds for interference channels, i.e. two-sender tworeceiver channels W (y 1 y 2 |x 2 x 2 ). Speci cally, non-signaling assistance shared between the rst sender and the rst receiver and independently shared between the second sender and the second receiver does not change the capacity region of interference channels.

Conclusion

In this chapter, we have addressed the computational complexity of the multiple-access channel coding problem, by showing that no polynomial-time constant approximation exists if a complexity hypothesis on random k-SAT formulas is true. Then, we studied the impact of non-signaling assistance on the capacity of multiple-access channels. We have developed an e cient linear program computing the success probability of the best non-signaling assisted code for a nite number of copies of a multiple-access channel. In particular, this gives lower bounds on the zero-error non-signaling assisted capacity of multiple-access channels. Applied to the binary adder channel, these results were used to prove that a sum-rate of log 2 (72) 4

1.5425 can be reached with zero error, which beats the maximum classical sum-rate capacity of 3 2 . For noisy channels, we have developed a technique giving lower bounds through the use of concatenated codes. Applied to the noisy binary adder channel, this technique was used to show that non-signaling assistance still improves the sum-rate capacity. We have also found an outer bound on the nonsignaling assisted capacity region through a relaxed notion of non-signaling assistance, whose capacity region was characterized by a single-letter formula. Finally, we have shown that independent non-signaling assistance does not change the capacity region.

Our results suggest that quantum entanglement may also increase the capacity of such channels. However, even for the binary adder channel, this question remains open. One could also ask if such e cient methods to compute the best non-signaling assisted codes can be extended to Gaussian multiple-access channels. Finally, establishing a single-letter formula for the non-signaling assisted capacity of multiple-access channels is the main open question left here. It remains open even for the binary adder channel. Proving that non-signaling assistance and relaxed non-signaling assistance coincide asymptotically would directly answer this question and show that the capacity region is described in Theorem 4.25.

Broadcast Channel Coding With Non-Signaling Correlations

Broadcast channels, introduced by Cover in [START_REF] Thomas | Broadcast channels[END_REF], describe the simple network communication setting where one sender aims to transmit individual messages to two receivers. Contrary to point-to-point channels [START_REF] Shannon | A mathematical theory of communication[END_REF] or multiple-access channels [START_REF] Herng | Multiple access channels[END_REF][START_REF] Ahlswede | Multi-way communication channels[END_REF], the capacity region of broadcast channels is known only for particular classes such as the degraded [Ber73, Gal74, AK75], deterministic [START_REF] Marton | The capacity region of deterministic broadcast channels[END_REF][START_REF] Semenovich | Capacity of noiseless broadcast channel[END_REF] and semi-deterministic [START_REF] Izrail | Capacity of a broadcast channel with one deterministic component[END_REF].

Only inner bounds [START_REF] Thomas | An achievable rate region for the broadcast channel[END_REF][START_REF] Edward | Random coding theorems for the general discrete memoryless broadcast channel[END_REF][START_REF] Marton | A coding theorem for the discrete memoryless broadcast channel[END_REF] and outer bounds [Sat78, Mar79, NG07, GN20] on the capacity region are known in the general setting.

On the one hand, it is natural to ask whether quantum-entanglement or even non-signaling correlations change the capacity region of broadcast channels. It is known that gaps exist between the classical, quantum-entanglement assisted and non-signaling assisted capacity regions of two-sender two-receiver interference channels [START_REF] Quek | Quantum and superquantum enhancements to twosender, two-receiver channels[END_REF] and MACs [START_REF] Leditzky | Playing games with multiple access channels[END_REF][START_REF] Seshadri | On the separation of correlation-assisted sum capacities of multiple access channels[END_REF]; see Chapter 4 for a detailed analysis of non-signaling assistance on MACs. However, the in uence of nonlocal resources on broadcast channels has been comparably less studied. We only know that quantum entanglement shared between decoders does not change the capacity region [START_REF] Pereg | Quantum broadcast channels with cooperating decoders: An information-theoretic perspective on quantum repeaters[END_REF].

On the other hand, from an algorithmic point of view, a crucial question in information theory is the complexity of the channel coding problem, which entails maximizing the success probability that can be achieved by sending a xed number of messages over a channel. However, as solving exactly this problem is NP-hard, a natural question that arises is its approximability. For point-to-point channels, Barman and Fawzi found in [START_REF] Barman | Algorithmic aspects of optimal channel coding[END_REF] a (1 -e -1 )-approximation algorithm running in polynomial time. They showed that it is NP-hard to approximate the channel coding problem in polynomial time for any strictly better ratio. For -list-decoding, where the decoder is allowed to output a list of guesses, a polynomial-time approximation algorithm achieving a 1 -e - ! ratio was found in [START_REF] Barman | Tight approximation bounds for maximum multi-coverage[END_REF], and it was shown to be NP-hard to do better in Chapter 3; see Theorem 3.1. For multiple-access channel coding, the complexity of the problem can be linked to the bipartite densest subgraph problem [START_REF] Feige | The dense k-subgraph problem[END_REF], which cannot be approximated within any constant ratio under a complexity hypothesis on random k-SAT formulas [AAM + 11]; see Theorem 4.5 from Chapter 4. However, the approximability of broadcast channel coding has not been addressed in the literature.

111 5. B C C W N S C
In the point-to-point scenario studied in [START_REF] Barman | Algorithmic aspects of optimal channel coding[END_REF], the existence of a constant-ratio approximation algorithm is linked to the equality of the capacity regions with and without non-signaling assistance. Indeed, giving non-signaling assistance to the channel coding problem turns it into a linear program, thus computable in polynomial time. In fact, it is equal to its natural linear relaxation, which is a common strategy towards approximating an integer linear program. Showing that this approximation strategy guarantees a constant ratio is the key ingredient in proving the equality of the capacity regions with and without non-signaling assistance. This raises the following questions on broadcast channels: Does the capacity region of the broadcast channel change when non-signaling resources between parties are allowed? What is the best approximability ratio of the broadcast channel coding problem? How those two questions are related?

Contributions In this chapter, as a rst result, we prove that the sum success probabilities of the broadcast channel coding problem are the same with and without non-signaling assistance between decoders; see Theorem 5.4. This strengthens a result by [START_REF] Pereg | Quantum broadcast channels with cooperating decoders: An information-theoretic perspective on quantum repeaters[END_REF] establishing that entanglement between the decoders does not change the capacity region.

The main focus of this chapter is to study the in uence of sharing a non-signaling resource between the three parties. Our main result shows that for the class of deterministic broadcast channels, non-signaling resources shared between the three parties does not change the capacity region; see Theorem 5.10 and Corollary 5.11. In order to prove this result, we consider the algorithmic problem of optimal channel coding for a deterministic broadcast channel. For this problem, we describe a (1 -e -1 ) 2 -approximation algorithm running in polynomial time. This is achieved through a graph interpretation of the problem, where one aims at partitioning a bipartite graph into k 1 and k 2 parts, such that the resulting quotient graph is the densest possible; see Proposition 5.7 and Theorem 5.8. To prove our result on the limitations of non-signaling assistance for deterministic broadcast channels, we use the same ideas as the ones involved in the analysis of the approximation algorithm.

As far as hardness is concerned, we consider the subproblem of broadcast channel coding where the number of messages one decoder is responsible of is maximum. This subproblem can be interpreted as a social welfare maximization problem. In the theory of fair division [START_REF] Brams | Fair division -from cake-cutting to dispute resolution[END_REF][START_REF] Moulin | Fair division and collective welfare[END_REF], social welfare maximization entails partitioning a set of goods among agents in order to maximize the sum of their utilities. The social welfare problem has been extensively studied through black box approach [START_REF] Blumrosen | On the computational power of iterative auctions[END_REF], which led to a precise analysis of achievable approximation ratio as well as hardness results [START_REF] Dobzinski | An improved approximation algorithm for combinatorial auctions with submodular bidders[END_REF][START_REF] Vahab | Tight information-theoretic lower bounds for welfare maximization in combinatorial auctions[END_REF], depending on the class of utility functions considered and the type of black box access to them. We re ne the hardness result for the class of fractionally sub-additive utility functions to the subclass coming from the broadcast channel coding subproblem interpretation. Speci cally, we show that in the value query model, we cannot achieve a better approximation ratio than Ω 1 √ m in polynomial time, with m the size of one of the outputs of the channel: see Theorem 5.14. Following the previous discussion on the links between constant-ratio approximation algorithms and non-signaling capacity regions, this hardness result is a rst step towards showing that sharing a non-signaling resource between the three parties of a broadcast channel can enlarge its capacity region.

Organization In Section 5.1, we de ne precisely the di erent versions of the broadcast channel coding problem depending on the choice of objective value, and show that they all lead to the same capacity region. In Section 5.2, we de ne the di erent non-signaling assisted versions of the broadcast channel coding problem. In particular, we show that the sum success probabilities with and without non-signaling assistance shared between decoders are the same, and that it implies that the related capacity regions are the same. In Section 5.3, we address both algorithmic aspects and capacity considerations of deterministic broadcast channels. Speci cally, we describe a (1-e -1 ) 2 -approximation algorithm running in polynomial time for that class, and we show that the capacity region for that class is the same with or without non-signaling assistance. Finally, in Section 5.4, we show that in the value query model, we cannot achieve a better approximation ratio than Ω 1 √ m in polynomial time for the general broadcast channel coding problem, with m the size of one of the outputs of the channel.

Broadcast Channel Coding

Broadcast Channels

The coding problem for a broadcast channel W : X → Y 1 × Y 2 can be stated in the following way. We want to encode a pair of messages belonging to [k 1 ] × [k 2 ] into X . The pair is given as input to W , which results in two random outputs in Y 1 and Y 2 . From the output in Y 1 (resp. Y 2 ), we want to decode back the original message in We will call p 1 (W, e, d 1 ) (resp. p 2 (W, e, d 2 )) the probability of successfully decoding the rst (resp. second) message, i.e. that j 1 = i 1 (resp. j 2 = i 2 ), given that the encoder is e and the decoder is d 1 (resp. d 2 ). We will also consider p(W, e, d 1 , d 2 ), the probability of successfully decoding both messages, i.e. that j 1 = i 1 and j 2 = i 2 , given that the encoder is e and the decoders are d 1 , d 2 .

[k 1 ] (resp. [k 2 ]). We will call e : [k 1 ] × [k 2 ] → X the encoder, d 1 : Y 1 → [k 1 ] the rst
We aim to nd the best encoder and decoders according to some gure of merit. However, to do so, we need a one-dimensional real-valued objective to optimize. This leads to two di erent quantities.

The Sum Success Probability

S sum (W, k 1 , k 2 )
We will focus rst on maximizing p 1 (W,e,d 1 )+p 2 (W,e,d 2 ) 2 over all encoders e and decoders d 1 , d 2 . We will call S sum (W, k 1 , k 2 ) the resulting maximum sum probability of successfully encoding and decoding the messages through W , given that the input pair of messages is

113 5. B C C W N S C taken uniformly in [k 1 ] × [k 2 ]. S sum (W, k 1 , k 2 )
is the solution of the following optimization program:

maximize e,d 1 ,d 2 1 k 1 k 2 i 1 ,i 2 ,x,y 1 ,y 2 W (y 1 y 2 |x)e(x|i 1 i 2 ) d 1 (i 1 |y 1 ) + d 2 (i 2 |y 2 ) 2 subject to x∈X e(x|i 1 i 2 ) = 1, ∀i 1 ∈ [k 1 ], i 2 ∈ [k 2 ] j 1 ∈[k 1 ] d 1 (j 1 |y 1 ) = 1, ∀y 1 ∈ Y 1 j 2 ∈[k 2 ] d 2 (j 2 |y 2 ) = 1, ∀y 2 ∈ Y 2 e(x|i 1 i 2 ), d 1 (j 1 |y 1 ), d 2 (j 2 |y 2 ) ≥ 0 (5.1)
Proof. One should note that we allow in fact non-deterministic encoders and decoders for generality reasons, although the value of the program is not changed as it is convex. Besides that remark, let us name I 1 , I 2 , J 1 , J 2 , X, Y 1 , Y 2 the random variables corresponding to i 1 , i 2 , j 1 , j 2 , x, y 1 , y 2 in the coding and decoding process. Then, given e, d 1 , d 2 and W , the objective value of the previous program comes from:

p 1 (W, e, d 1 ) = P (J 1 = I 1 ) = 1 k 1 k 2 i 1 ,i 2 P (J 1 = i 1 |I 1 = i 1 , I 2 = i 2 ) = 1 k 1 k 2 i 1 ,i 2 ,x e(x|i 1 i 2 )P (J 1 = i 1 |i 1 , i 2 , X = x) = 1 k 1 k 2 i 1 ,i 2 ,x,y 1 ,y 2 W (y 1 y 2 |x)e(x|i 1 i 2 )P (J 1 = i 1 |i 1 , i 2 , x, Y 1 = y 1 , Y 2 = y 2 ) = 1 k 1 k 2 i 1 ,i 2 ,x,y 1 ,y 2 W (y 1 y 2 |x)e(x|i 1 i 2 )d 1 (i 1 |y 1 ) ,
(5.2) and symmetrically for p 2 (W, e, d 2 ), which leads to the announced objective value.

One can rewrite this optimization program in a more convenient way, proving that S sum (W, k 1 , k 2 ) depends only on the marginals of W :

Proposition 5.1.

S sum (W, k 1 , k 2 ) = maximize e,d 1 ,d 2 1 2k 1 k 2 i 1 ,x,y 1 W 1 (y 1 |x)d 1 (i 1 |y 1 ) i 2 e(x|i 1 i 2 ) + 1 2k 1 k 2 i 2 ,x,y 2 W 2 (y 2 |x)d 2 (i 2 |y 2 ) i 1 e(x|i 1 i 2 ) subject to x∈X e(x|i 1 i 2 ) = 1, ∀i 1 ∈ [k 1 ], i 2 ∈ [k 2 ] j 1 ∈[k 1 ] d 1 (j 1 |y 1 ) = 1, ∀y 1 ∈ Y 1 j 2 ∈[k 2 ] d 2 (j 2 |y 2 ) = 1, ∀y 2 ∈ Y 2 e(x|i 1 i 2 ), d 1 (j 1 |y 1 ), d 2 (j 2 |y 2 ) ≥ 0 (5.3)
Proof. It follows from the de nitions W 1 (y 1 |x) := y 2 W (y 1 y 2 |x) and W 2 (y 2 |x) := y 1 W (y 1 y 2 |x).

5.1.3

The Joint Success Probability S(W, k 1 , k 2 )

We will now focus on maximizing p(W, e, d 1 , d 2 ) over all encoders e and decoders d 1 , d 2 .

We will call S(W, k 1 , k 2 ) the resulting maximum probability of successfully encoding and decoding the messages through W , given that the input pair of messages is taken uniformly in

[k 1 ] × [k 2 ]. S(W, k 1 , k 2 )
is the solution of the following optimization program:

S(W, k 1 , k 2 ) := maximize e,d 1 ,d 2 1 k 1 k 2 i 1 ,i 2 ,x,y 1 ,y 2 W (y 1 y 2 |x)e(x|i 1 i 2 )d 1 (i 1 |y 1 )d 2 (i 2 |y 2 ) subject to x∈X e(x|i 1 i 2 ) = 1, ∀i 1 ∈ [k 1 ], i 2 ∈ [k 2 ] j 1 ∈[k 1 ] d 1 (j 1 |y 1 ) = 1, ∀y 1 ∈ Y 1 j 2 ∈[k 2 ] d 2 (j 2 |y 2 ) = 1, ∀y 2 ∈ Y 2 e(x|i 1 i 2 ), d 1 (j 1 |y 1 ), d 2 (j 2 |y 2 ) ≥ 0 (5.4)
The proof is the same as in the sum probability scenario. We de ne the (resp. Proof. Let us focus on error probabilities rather than success ones. Call them respectively

E(W, k 1 , k 2 ) := 1 -S(W, k 1 , k 2 ) and E sum (W, k 1 , k 2 ) := 1 -S sum (W, k 1 , k 2 ).
Let us x a solution e, d 1 , d 2 of the optimization program computing S(W, k 1 , k 2 ). Let us remark rst that:

i 1 ,i 2 ,x,y 1 ,y 2 W (y 1 y 2 |x)e(x|i 1 i 2 ) = k 1 k 2 ,
thus, the value of the maximum error for those encoder and decoders is:

E(W, k 1 , k 2 , e, d 1 , d 2 ) := 1 - 1 k 1 k 2   i 1 ,i 2 ,x,y 1 ,y 2 W (y 1 y 2 |x)e(x|i 1 i 2 )d 1 (i 1 |y 1 )d 2 (i 2 |y 2 )   = 1 k 1 k 2 i 1 ,i 2 ,x,y 1 ,y 2 W (y 1 y 2 |x)e(x|i 1 i 2 ) - 1 k 1 k 2 i 1 ,i 2 ,x,y 1 ,y 2 W (y 1 y 2 |x)e(x|i 1 i 2 )d 1 (i 1 |y 1 )d 2 (i 2 |y 2 ) = 1 k 1 k 2   i 1 ,i 2 ,x,y 1 ,y 2 W (y 1 y 2 |x)e(x|i 1 i 2 ) [1 -d 1 (i 1 |y 1 )d 2 (i 2 |y 2 )]   . (5.5) 115 5. B C C W N S C
Similarly, the value of the sum error E sum (W, k 1 , k 2 , e, d 1 , d 2 ) is equal to:

1 - 1 k 1 k 2   i 1 ,i 2 ,x,y 1 ,y 2 W (y 1 y 2 |x)e(x|i 1 i 2 ) d 1 (i 1 |y 1 ) + d 2 (i 2 |y 2 ) 2   = 1 k 1 k 2   i 1 ,i 2 ,x,y 1 ,y 2 W (y 1 y 2 )e(x|i 1 i 2 ) 1 - d 1 (i 1 |y 1 ) + d 2 (i 2 |y 2 ) 2   .
(5.6) However, for x, y ∈ [0, 1], we have that:

1 -xy ≥ max (1 -x, 1 -y) ≥ 1 - x + y 2 ,
and:

1 -xy ≤ (1 -x) + (1 -y) = 2 1 - x + y 2 .
This means that, for all e, d 1 , d 2 , we have:

E sum (W, k 1 , k 2 , e, d 1 , d 2 ) ≤ E(W, k 1 , k 2 , e, d 1 , d 2 ) ≤ 2E sum (W, k 1 , k 2 , e, d 1 , d 2 ) ,
so, maximizing over all e, d 1 , d 2 , we get:

E sum (W, k 1 , k 2 ) ≤ E(W, k 1 , k 2 ) ≤ 2E sum (W, k 1 , k 2 ) .
Thus, up to a multiplicative factor 2, the error is the same. In particular, when one of those errors tends to zero, the other one tends to zero as well. This implies that the capacity regions are the same.

Non-Signaling Assistance

In this section, we will consider the broadcast channel coding problem with additional resources, in order to determine how these resources a ect its success probabilities as well as the capacity regions that can be de ned from them.

Non-Signaling Assistance Between Decoders

Here, we consider the case where the receivers are given non-signaling assistance. This resource, which is a theoretical but easier to manipulate generalization of quantum entanglement, can be characterized as follows. A non-signaling box d(j 1 j 2 |y 1 y 2 ) is any joint conditional probability distribution such that the marginal from one party is independent of the other party's input, i.e. we have:

∀j 1 , y 1 , y 2 , y 2 , j 2 d(j 1 j 2 |y 1 y 2 ) = j 1 d(j 1 j 2 |y 1 y 2 ) , ∀j 2 , y 1 , y 2 , y 1 , j 1 d(j 1 j 2 |y 1 y 2 ) = j 1 d(j 1 j 2 |y 1 y 2 ) .
(5.7) Thus, when receivers are given non-signaling assistance, the product d 1 (j 1 |y 1 )d 2 (j 2 |y 2 ) is replaced by the non-signaling box d(j 1 j 2 |y 1 y 2 ). Thus, we de ne the joint and sum success probabilities S NS dec (W, k 1 , k 2 ) (resp. S NS dec sum (W, k 1 , k 2 )) by: maximize

e,d 1 ,d 2 1 k 1 k 2 i 1 ,i 2 ,x,y 1 ,y 2 W (y 1 y 2 |x)e(x|i 1 i 2 )d(i 1 i 2 |y 1 y 2 ) resp. maximize e,d 1 ,d 2 1 2k 1 k 2 i 1 ,i 2 ,x,y 1 ,y 2 W (y 1 y 2 |x)e(x|i 1 i 2 ) j 2 d(i 1 j 2 |y 1 y 2 ) + 1 2k 1 k 2 i 1 ,i 2 ,x,y 1 ,y 2 W (y 1 y 2 |x)e(x|i 1 i 2 ) j 1 d(j 1 i 2 |y 1 y 2 ) subject to x e(x|i 1 i 2 ) = 1 j 2 d(j 1 j 2 |y 1 y 2 ) = j 1 d(j 1 j 2 |y 1 y 2 ) j 1 d(j 1 j 2 |y 1 y 2 ) = j 1 d(j 1 j 2 |y 1 y 2 ) j 1 ,j 2 d(j 1 j 2 |y 1 y 2 ) = 1 e(x|i 1 i 2 ), d(j 1 j 2 |y 1 y 2 ) ≥ 0
(5.8)

The (resp. sum) capacity region with non-signaling assistance between decoders is de ned using De nition 2.12 by C NS dec (W ) := C[S NS dec ](W ) (resp. C NS dec sum (W ) := C[S NS dec sum ](W )). We will now show that sum and joint capacity regions with non-signaling assistance between decoders are the same.

Proposition 5.3. For any broadcast channel W , C NS dec sum (W ) = C NS dec (W ).

Proof. Given an encoder e and a non-signaling decoding box d, the maximum success probability of encoding and decoding correctly with those is given by:

S NS dec (W, k 1 , k 2 , e, d) := 1 k 1 k 2 i 1 ,i 2 ,x,y 1 ,y 2 W (y 1 y 2 |x)e(x|i 1 i 2 )d(i 1 i 2 |y 1 y 2 ) .
This should be compared to the sum success probability S NS dec sum (W, k 1 , k 2 , e, d) of encoding and decoding correctly with those:

1 k 1 k 2 i 1 ,i 2 ,x,y 1 ,y 2 W (y 1 y 2 |x)e(x|i 1 i 2 ) j 2 d(i 1 j 2 |y 1 y 2 ) + j 1 d(j 1 i 2 |y 1 y 2 ) 2 .
Similarly to what was done in Proposition 5.2, we focus on error probabilities rather than success probabilities. This leads again to:

E NS dec (W, k 1 , k 2 , e, d) = 1 k 1 k 2 i 1 ,i 2 ,x,y 1 ,y 2 W (y 1 y 2 |x)e(x|i 1 i 2 ) [1 -d(i 1 i 2 |y 1 y 2 )] , 117 5. B C C W N S C
and E NS dec sum (W, k 1 , k 2 , e, d) equal to:

1 k 1 k 2 i 1 ,i 2 ,x,y 1 ,y 2 W (y 1 y 2 |x)e(x|i 1 i 2 ) 1 -j 2 d(i 1 j 2 |y 1 y 2 ) 2 + 1 -j 1 d(j 1 i 2 |y 1 y 2 ) 2 .
But we have that:

1 -d(i 1 i 2 |y 1 y 2 ) ≥ max   1 - j 2 d(i 1 j 2 |y 1 y 2 ), 1 - j 1 d(j 1 i 2 |y 1 y 2 )   ≥ 1 -j 2 d(i 1 j 2 |y 1 y 2 ) 2 + 1 -j 1 d(j 1 i 2 |y 1 y 2 ) 2 ,
(5.9) since d(j 1 j 2 |y 1 y 2 ) ∈ [0, 1], and we have that:

1 - j 2 d(i 1 j 2 |y 1 y 2 ) + 1 - j 1 d(j 1 i 2 |y 1 y 2 ) = 1 -d(i 1 i 2 |y 1 y 2 ) + 1 - (j 1 ,j 2 )∈S d(j 1 j 2 |y 1 y 2 ) ≥ 1 -d(i 1 i 2 |y 1 y 2 ) ,
(5.10)

with S := {(i 1 , j 2 ) : j 2 ∈ [k 2 ] -{i 2 }} {(j 1 , i 2 ) : j 1 ∈ [k 1 ] -{i 1 }}.
Thus, this implies that:

E NS dec sum (W, k 1 , k 2 , e, d) ≤ E NS dec (W, k 1 , k 2 , e, d) ≤ 2E NS dec sum (W, k 1 , k 2 , e, d) ,
and by maximizing over all e and d:

E NS dec sum (W, k 1 , k 2 ) ≤ E NS dec (W, k 1 , k 2 ) ≤ 2E NS dec sum (W, k 1 , k 2 ) .
As before, this implies that the capacity regions are the same.

We will now prove that sum success probabilities of the broadcast channel coding problem are the same with and without non-signaling assistance between decoders. In particular, this implies that non-signaling resources shared between decoders does not change the capacity region. Note that, after the publication of [PDB21], Pereg et al. added a remark to the arXiv version of their paper that their result stating that entanglement shared between decoders does not change the capacity of a broadcast channel could be generalized to nonsignaling assistance. The theorem below strengthens this result showing that non-signaling assistance between the decoders cannot increase the sum success probability even in the one-shot setting and for arbitrary broadcast channels.

Theorem 5.4. For any W, k 1 , k 2 , we have S sum (W, k 1 , k 2 ) = S NS dec sum (W, k 1 , k 2 ). As a consequence, C(W ) = C NS dec (W ).

Proof. In the sum scenario, since the objective function does not depend on the product d 1 (j 1 |y 1 )d 2 (j 2 |y 2 ) but only on the marginals d 1 (j 1 |y 1 ) and d 2 (j 2 |y 2 ), the non-signaling box won't give additional decoding power. Indeed, for any encoder e and non-signaling decoding box d, we have that:

S NS dec sum (W, k 1 , k 2 , e, d) := 1 2k 1 k 2 i 1 ,x,y 1 W 1 (y 1 |x)   j 2 d(i 1 j 2 |y 1 y 2 )   i 2 e(x|i 1 i 2 ) + 1 2k 1 k 2 i 2 ,x,y 2 W 2 (y 2 |x)   j 1 d(j 1 i 2 |y 1 y 2 )   i 1 e(x|i 1 i 2 ) .
(5.11)

Full Non-Signaling Assistance

In this section, we will consider the case where the sender and the receivers are given nonsignaling assistance. This means that a three-party non-signaling box P (xj 1 j 2 |(i 1 i 2 )y 1 y 2 ) will replace the product e(x|i 1 i 2 )d 1 (j 1 |y 1 )d 2 (j 2 |y 2 ) in the previous objective values. A joint conditional probability P (xj 1 j 2 |(i 1 i 2 )y 1 y 2 ) is a non-signaling box if the marginal from any two parties is independent of the removed party's input:

∀j 1 , j 2 , i 1 , i 2 , y 1 , y 2 , i 1 , i 2 , x P (xj 1 j 2 |(i 1 i 2 )y 1 y 2 ) = x P (xj 1 j 2 |(i 1 i 2 )y 1 y 2 ) , ∀x, j 2 , i 1 , i 2 , y 1 , y 2 , y 1 , j 1 P (xj 1 j 2 |(i 1 i 2 )y 1 y 2 ) = j 1 P (xj 1 j 2 |(i 1 i 2 )y 1 y 2 ) , ∀x, j 1 , i 1 , i 2 , y 1 , y 2 , y 2 , j 2 P (xj 1 j 2 |(i 1 i 2 )y 1 y 2 ) = j 2 P (xj 1 j 2 |(i 1 i 2 )y 1 y 2 ) .
(5.12)

The scenario is depicted in Figure 5.2. The cyclicity of Figure 5.2 is at rst sight counter-intuitive. Note rst that P being a non-signaling box is completely independent of W : in particular, the variables y 1 , y 2 do not need to follow any laws in the de nition of P being a non-signaling box. Therefore, the remaining ambiguity is the apparent need to encode and decode at the same time. However, since P is a non-signaling box, we won't need to do both at the same time, although the global correlation between the sender and the receivers will be characterized only by P (xj 1 j 2 |(i 1 i 2 )y 1 y 2 ). Indeed, ∀y 1 , y 2 , P (x|(i 1 i 2 )) = P (x|(i 1 i 2 )y 1 y 2 ) by the non-signaling property of P . Thus, one can get the output x on input (i 1 i 2 ) without access to y 1 , y 2 , as that knowledge won't a ect the law of x. Then (y 1 , y 2 ) follows the law given by W given that x. Finally, given access to y 1 , y 2 , the decoding process is described by:

P (j 1 j 2 |(i 1 i 2 )y 1 y 2 x) = P (xj 1 j 2 |(i 1 i 2 )y 1 y 2 ) P (x|(i 1 i 2 )y 1 y 2 ) = P (xj 1 j 2 |(i 1 i 2 )y 1 y 2 ) P (x|(i 1 i 2 )) , 119 5. B C C W N S C e d 1 d 2 j 1 j 2 (i 1 , i 2 ) W y 1 y 2 x P (xj 1 j 2 |(i 1 i 2 )y 1 y 2 ) j 1 j 2 (i 1 , i 2 ) W y 1 y 2 x Figure 5
.2 -A non-signaling box P replacing e, d 1 , d 2 in the coding problem for the broadcast channel W . so we recover P (j

1 j 2 |(i 1 i 2 )y 1 y 2 x)) × P (x|(i 1 i 2 )) = P (xj 1 j 2 |(i 1 i 2 )y 1 y 2 )
, and therefore, the process is not cyclic. Non-signaling boxes de ne exactly the conditional probability distributions where it is possible to consider the conditional probabilities of each party independently. This clari es how one can e ectively encode and then decode messages through a non-signaling box.

We will call the maximum sum success probability S NS sum (W, k 1 , k 2 ), which is given by the following linear program, where the constraints translate precisely the fact that P is a non-signaling box:

S NS sum (W, k 1 , k 2 ) := maximize P 1 2k 1 k 2 i 1 ,x,y 1 W 1 (y 1 |x) i 2 ,j 2 P (xi 1 j 2 |(i 1 i 2 )y 1 y 2 ) + 1 2k 1 k 2 i 2 ,x,y 2 W 2 (y 2 |x) i 1 ,j 1 P (xj 1 i 2 |(i 1 i 2 )y 1 y 2 ) subject to x P (xj 1 j 2 |(i 1 i 2 )y 1 y 2 ) = x P (xj 1 j 2 |(i 1 i 2 )y 1 y 2 ) j 1 P (xj 1 j 2 |(i 1 i 2 )y 1 y 2 ) = j 1 P (xj 1 j 2 |(i 1 i 2 )y 1 y 2 ) j 2 P (xj 1 j 2 |(i 1 i 2 )y 1 y 2 ) = j 2 P (xj 1 j 2 |(i 1 i 2 )y 1 y 2 ) x,j 1 ,j 2 P (xj 1 j 2 |(i 1 i 2 )y 1 y 2 ) = 1 P (xj 1 j 2 |(i 1 i 2 )y 1 y 2 ) ≥ 0 (5.13)
Since it is given as a linear program, the complexity of computing S NS sum (W, k 1 , k 2 ) is polynomial in the number of variables and constraints (see for instance Section 7.1 of [START_REF] Gärtner | Understanding and using linear programming[END_REF]), which is a polynomial in |X |, |Y 1 |, |Y 2 |, k 1 and k 2 .

Similarly, we de ne the maximum joint success probability S NS (W, k 1 , k 2 ) in the following way:

S NS (W, k 1 , k 2 ) := maximize P 1 k 1 k 2 i 1 ,i 2 ,x,y 1 ,y 2 W (y 1 y 2 |x)P (xi 1 i 2 |(i 1 i 2 )y 1 y 2 ) subject to x P (xj 1 j 2 |(i 1 i 2 )y 1 y 2 ) = x P (xj 1 j 2 |(i 1 i 2 )y 1 y 2 ) j 1 P (xj 1 j 2 |(i 1 i 2 )y 1 y 2 ) = j 1 P (xj 1 j 2 |(i 1 i 2 )y 1 y 2 ) j 2 P (xj 1 j 2 |(i 1 i 2 )y 1 y 2 ) = j 2 P (xj 1 j 2 |(i 1 i 2 )y 1 y 2 ) x,j 1 ,j 2 P (xj 1 j 2 |(i 1 i 2 )y 1 y 2 ) = 1 P (xj 1 j 2 |(i 1 i 2 )y 1 y 2 ) ≥ 0 (5.14)
We can rewrite both these programs in more convenient and smaller linear programs:

Proposition 5.5.

S NS sum (W, k 1 , k 2 ) = maximize p,r,r 1 ,r 2 1 2k 1 k 2 x,y 1 W 1 (y 1 |x)r 1 x,y 1 + x,y 2 W 2 (y 2 |x)r 2 x,y 2 subject to x r x,y 1 ,y 2 = 1 x r 1 x,y 1 = k 2 x r 2 x,y 2 = k 1 x p x = k 1 k 2 0 ≤ r x,y 1 ,y 2 ≤ r 1 x,y 1 , r 2 x,y 2 ≤ p x p x -r 1 x,y 1 -r 2 x,y 2 + r x,y 1 ,y 2 ≥ 0 (5.15) S NS (W, k 1 , k 2 ) = maximize p,r,r 1 ,r 2 1 k 1 k 2 x,y 1 ,y 2 W (y 1 y 2 |x)r x,y 1 ,y 2 subject to x r x,y 1 ,y 2 = 1 x r 1 x,y 1 = k 2 x r 2 x,y 2 = k 1 x p x = k 1 k 2 0 ≤ r x,y 1 ,y 2 ≤ r 1 x,y 1 , r 2 x,y 2 ≤ p x p x -r 1 x,y 1 -r 2 x,y 2 + r x,y 1 ,y 2 ≥ 0 (5.16) 121 5. B C C W N S C
Proof. One can check that given a solution of the original program, the following choice of variables is a valid solution of the second program achieving the same objective value:

r x,y 1 ,y 2 := i 1 ,i 2 P (xi 1 i 2 |(i 1 i 2 )y 1 y 2 ) , r 1 x,y 1 := j 2 ,i 1 ,i 2 P (xi 1 j 2 |(i 1 i 2 )y 1 y 2 ) , r 2 x,y 2 := j 1 ,i 1 ,i 2 P (xj 1 i 2 |(i 1 i 2 )y 1 y 2 ) , p x := j 1 ,j 2 ,i 1 ,i 2 P (xj 1 j 2 |(i 1 i 2 )y 1 y 2 ) .
(5.17)

For the other direction, given those variables, a non-signaling probability distribution P (xj 1 j 2 |(i 1 i 2 )y 1 y 2 ) is given by, for j 1 = i 1 and j 2 = i 2 :

P (xi 1 i 2 |(i 1 i 2 )y 1 y 2 ) = r x,y 1 ,y 2 k 1 k 2 , P (xj 1 i 2 |(i 1 i 2 )y 1 y 2 ) = r 2 x,y 2 -r x,y 1 ,y 2 k 1 k 2 (k 1 -1) , P (xi 1 j 2 |(i 1 i 2 )y 1 y 2 ) = r 1 x,y 1 -r x,y 1 ,y 2 k 1 k 2 (k 2 -1) , P (xj 1 j 2 |(i 1 i 2 )y 1 y 2 ) = p x -r 1 x,y 1 -r 2 x,y 2 + r x,y 1 ,y 2 k 1 k 2 (k 1 -1)(k 2 -1)
.

(5.18)

As before, we de ne the (resp. sum) capacity region with non-signaling assistance using De nition 2.12 by

C NS (W ) := C[S NS ](W ) (resp. C NS sum (W ) := C[S NS sum ](W )).
Proposition 5.6. For any broadcast channel W , C NS (W ) = C NS sum (W ).

Proof. Let us show that:

2S NS sum (W, k 1 , k 2 ) -1 ≤ S NS (W, k 1 , k 2 ) ≤ S NS sum (W, k 1 , k 2 ) .
This will imply in particular that:

lim n→+∞ S NS (W ⊗n , 2 R 1 n , 2 R 2 n ) = 1 ⇐⇒ lim n→+∞ S NS sum (W ⊗n , 2 R 1 n , 2 R 2 n ) = 1 ,
thus de ne the same capacity region.

Let us consider an optimal solution p x , r x,y 1 ,y 2 , r 1 x,y 1 , r 2 x,y 2 of the program computing S NS sum (W, k 1 , k 2 ). We have:

S NS sum (W, k 1 , k 2 ) = 1 k 1 k 2 x,y 1 ,y 2 W (y 1 y 2 |x) r 1 x,y 1 + r 2 x,y 2 2 .
However r 1 x,y 1 + r 2 x,y 2 ≤ p x + r x,y 1 ,y 2 so we get that:

S NS sum (W, k 1 , k 2 ) ≤ 1 2k 1 k 2 x,y 1 ,y 2 W (y 1 y 2 |x) (p x + r x,y 1 ,y 2 ) = 1 2 + 1 2 1 k 1 k 2 x,y 1 ,y 2 W (y 1 y 2 |x)r x,y 1 ,y 2 ≤ 1 2 + 1 2 S NS (W, k 1 , k 2 ) , (5.19) since p x , r x,y 1 ,y 2 , r 1 x,y 1 , r 2
x,y 2 is a valid solution of the program computing S NS (W, k 1 , k 2 ). On the other hand, consider now p x , r x,y 1 ,y 2 , r 1

x,y 1 , r 2 x,y 2 an optimal solution of the program computing S NS (W, k 1 , k 2 ). We have that r x,y 1 ,y 2 ≤ r 1

x,y 1 , r 2 x,y 2 so we have that r x,y 1 ,y 2 ≤ r 1

x,y 1 +r 2

x,y 2 2 and thus:

S NS (W, k 1 , k 2 ) = 1 k 1 k 2 x,y 1 ,y 2 W (y 1 y 2 |x)r x,y 1 ,y 2 ≤ 1 k 1 k 2 x,y 1 ,y 2 W (y 1 y 2 |x) r 1 x,y 1 + r 2 x,y 2 2 ≤ S NS sum (W, k 1 , k 2 ) ,
(5.20) since p x , r x,y 1 ,y 2 , r 1 x,y 1 , r 2 x,y 2 is a valid solution of the program computing S NS sum (W, k 1 , k 2 ). This prove the inequalities

2S NS sum (W, k 1 , k 2 ) -1 ≤ S NS (W, k 1 , k 2 ) ≤ S NS sum (W, k 1 , k 2 )
, and thus concludes the proof.

Approximation Algorithm for Deterministic Broadcast Channel Coding

In this section, we will address the question of the approximability of S(W, k 1 , k 2 ), in the restricted scenario of a deterministic broadcast channel W . Speci cally, we study the problem of nding a code e :

[k 1 ] × [k 2 ] → X , d 1 : Y 1 → [k 1 ], d 2 : Y 2 → [k 2 ] that maximizes the program computing S(W, k 1 , k 2 ).
Note that the restriction to deterministic codes does not a ect the value of the objective of the program which is convex, and that the problem is as hard as nding any code maximizing the program computing S(W, k 1 , k 2 ), as a deterministic code with a better or equal value can be retrieved easily from any code.

We say that W is deterministic if ∀x, y 1 , y 2 , W (y 1 y 2 |x) ∈ {0, 1}. We can then de ne (W 1 (x), W 2 (x)) as the only pair (y 1 , y 2 ) such that W (y 1 y 2 |x) = 1, which exists uniquely as W is a conditional probability distribution. Thus, the deterministic broadcast channel coding problem can be de ned in the following way:

De nition 5.1 (D BCC). Given a deterministic channel W and integers k 1 and k 2 , the deterministic broadcast channel coding problem, which we call D BCC, entails maximizing

S(W, k 1 , k 2 , e, d 1 , d 2 ) := 1 k 1 k 2 i 1 ,i 2 1 d 1 (W 1 (e(i 1 i 2 )))=i 1 1 d 2 (W 2 (e(i 1 i 2 )))=i 2 over all functions e : [k 1 ] × [k 2 ] → X , d 1 : Y 1 → [k 1 ], d 2 : Y 2 → [k 2 ]. 123 5. B C C W N S C

Reformulation as a Bipartite Graph Problem

We will reformulate D BCC as a bipartite graph problem. But rst, let us introduce some notations:

De nition 5.2 (Graph notations). Consider a bipartite graph G = (V 1 V 2 , E ⊆ V 1 × V 2 ):
1. G P 1 ,P 2 , the quotient of G by partitions P 1 , P 2 of respectively V 1 , V 2 , is de ned by:

G P 1 ,P 2 := (P 1 P 2 , {(p 1 , p 2 ) ∈ P 1 × P 2 : E ∩ (p 1 × p 2 ) = ∅}) .
2. e G (P 1 , P 2 ) := |E G P 1 ,P 2 | is the number of edges of G P 1 ,P 2 . 5. We will use V 1 , V 2 in previous notations when we do not partition on the left and right part respectively (or identify those to trivial partitions in singletons). For instance,

N P

G V 1 ,V 2 = G.
6. We will use the notations e(P 1 , P 2 ), N P 1 ,P 2 (p) and deg P 1 ,P 2 (p) when the graph G considered is clear from context. Now, let us remark that a deterministic channel W , up to a permutation of elements of X , is characterized by the following bipartite graph:

De nition 5.3 (Bipartite Graph G W associated with the deterministic channel W ).

G W := (Y 1 Y 2 , E = {(y 1 , y 2 ) ∈ Y 1 × Y 2 : ∃x ∈ X , y 1 = W 1 (x) and y 2 = W 2 (x)}) .
Indeed, permuting the elements of X does not change G W nor S(W, k 1 , k 2 ). As a consequence, up to a multiplicative factor k 1 k 2 , we will show that D BCC is equivalent to the following bipartite graph problem:

De nition 5.4 (D G ). Given a bipartite graph G = (V 1 V 2 , E) and integers k 1 , k 2 , the problem D G
entails maximizing e G (P 1 , P 2 ), the number of edges of the quotient graph of G by P 1 , P 2 , over all partitions

P 1 of V 1 in k 1 parts and P 2 of V 2 in k 2 parts. Proposition 5.7. Given a deterministic channel W and integers k 1 , k 2 , it is equivalent to solve D BCC on W, k 1 , k 2 or D Q G on G W , k 1 , k 2 .
That is to say, given an optimal solution of one of those problems, one can e ciently construct an optimal solution of the other. Furthermore, their optimal values satisfy k

1 k 2 D BCC(W, k 1 , k 2 ) = D Q G (G W , k 1 , k 2 ).
k 1 k 2 S(W, k 1 , k 2 , e, d 1 , d 2 ) = i 1 ,i 2 1 i 1 =d 1 (W 1 (e(i 1 ,i 2 ))) 1 i 2 =d 2 (W 2 (e(i 1 ,i 2 ))) = i 1 ,i 2 1 W 1 (e(i 1 ,i 2 ))∈P i 1 1 1 W 2 (e(i 1 ,i 2 ))∈P i 2 2 .
(5.21)

However, since we consider an optimal solution, we have that:

1 W 1 (e(i 1 ,i 2 ))∈P i 1 1 1 W 2 (e(i 1 ,i 2 ))∈P i 2 2 = max x∈X 1 W 1 (x)∈P i 1 1 1 W 2 (x)∈P i 2 2
, as e(i 1 , i 2 ) appears only here in the objective value. Thus:

k 1 k 2 S(W, k 1 , k 2 , e, d 1 , d 2 ) = i 1 ,i 2 max x∈X 1 W 1 (x)∈P i 1 1 1 W 2 (x)∈P i 2 2 = i 1 ,i 2 1 ∃(y 1 ,y 2 )∈E G W :y 1 ∈P i 1 1 and y 2 ∈P i 2 2 = i 1 ,i 2 1 E G W ∩ P i 1 1 ×P i 2 2 =∅ = e G W (P 1 , P 2 ) , (5.22)
which proves that given an optimal solution of D BCC, one can e ciently construct a solution P 1 , P 2 of D G such that:

e G W (P 1 , P 2 ) = k 1 k 2 D BCC(W, k 1 , k 2 ) .
For the other direction, consider an optimal solution P 1 , P 2 of D G . We have as before that:

e G W (P 1 , P 2 ) = i 1 ,i 2 max x∈X 1 W 1 (x)∈P i 1 1 1 W 2 (x)∈P i 2 2 . Now, let us de ne e(i 1 , i 2 ) ∈ argmax x∈X 1 W 1 (x)∈P i 1 1 1 W 2 (x)∈P i 2 2 and d b (y b ) the index i b such that y b ∈ P i b b , for b ∈ {1, 2}.
With those de nitions, we get again that:

max x∈X 1 W 1 (x)∈P i 1 1 1 W 2 (x)∈P i 2 2 = 1 W 1 (e(i 1 ,i 2 ))∈P i 1 1 1 W 2 (e(i 1 ,i 2 ))∈P i 2 2 = 1 i 1 =d 1 (W 1 (e(i 1 ,i 2 ))) 1 i 2 =d 2 (W 2 (e(i 1 ,i 2 ))) ,
(5.23) and thus we have:

e G W (P 1 , P 2 ) = i 1 ,i 2 max x∈X 1 W 1 (x)∈P i 1 1 1 W 2 (x)∈P i 2 2 = i 1 ,i 2 1 i 1 =d 1 (W 1 (e(i 1 ,i 2 ))) 1 i 2 =d 2 (W 2 (e(i 1 ,i 2 ))) = k 1 k 2 S(W, k 1 , k 2 , e, d 1 , d 2 ) , (5.24)
which proves that given an optimal solution of D G , one can eciently construct a solution e, d 1 , d 2 of D BCC such that:

k 1 k 2 S(W, k 1 , k 2 , e, d 1 , d 2 ) = D G (G W , k 1 , k 2 ) . 125 5. B C C W N S C
In particular, this implies that the optimal objective values satisfy:

k 1 k 2 D BCC(W, k 1 , k 2 ) = D G (G W , k 1 , k 2 ) .
Therefore, the solutions of both problems constructed throughout the proof are in fact optimal.

Remark. Note that all bipartite graphs can be written as G W for some deterministic broadcast channel W , with W unique up to a permutation of X .

Approximation Algorithm for D G

In this section, we will sort out how hard is D G , and thanks to Proposition 5.7, how hard is it to solve D BCC.

Theorem 5.8. There exists a polynomial-time (1-e -1 ) 2 -approximation algorithm for D Q G . Furthermore, it is NP-hard to solve exactly D Q G .

Corollary 5.9. There exists a polynomial-time (1-e -1 ) 2 -approximation algorithm for D BCC. Furthermore, it is NP-hard to solve exactly D BCC.

The approximation algorithm is a two-step process. First, we consider the problem of maximizing

k 2 i 2 =1 min k 1 , deg V 1 ,P 2 (P i 2
2 ) over all partitions P 2 of V 2 in k 2 parts. We will show that this is a special case of the submodular welfare problem, which can be approximated within a factor 1 -e -1 in polynomial time [START_REF] Vondrák | Optimal approximation for the submodular welfare problem in the value oracle model[END_REF]. We then choose the partition P 1 on V 1 in k 1 parts uniformly at random. This partition pair will give an objective value e(P 1 , P 2 ) within a (1 -e -1 ) 2 factor from the optimal solution in expectation.

Proof of Theorem 5.8. Consider rst the hardness result. Let us show that the decision version of D G is NP-complete. It is in NP, the certi cate being the two partitions and the selection of edges between those partitions. It is NP-hard as one of its particular cases is the S S problem (see for instance [START_REF] Garey | Computers and Intractability: A Guide to the Theory of NP-Completeness[END_REF]), in the case where k 1 = 2 and k 2 = |V 2 |, by interpreting the neighbors of v 2 ∈ V 2 as a set covering elements of V 1 .

We will show nonetheless that this problem can be approximated within a factor (1 -e -1 ) 2 in polynomial time. First we consider the case where k 2 = |V 2 |. We can then always assume that the right partition is P 2 := {{v 2 } : v 2 ∈ V 2 }, which leads necessarily to a greater or equal number of edges in the quotient graph that with any other right partition. So, in that setting, we only need to nd a partition of V 1 in k 1 parts maximizing the number of edges between vertices in the right part and the quotient of the left vertices.

First, the maximum value we can get is upper bounded by v 2 ∈V 2 min (k 1 , deg(v 2 )). Indeed, each vertex of v 2 van be connected at most to the k 1 parts of V 1 , so its contribution is bounded by k 1 , and there needs to be an edge to each part it is connected, so its contribution is also bounded by deg(v 2 ). Let us show that if we take a partition P 1 of V 1 uniformly at random, we get:

E P 1 [e(P 1 , V 2 )] ≥ 1 -1 - 1 k 1 k 1 v 2 ∈V 2 min (k 1 , deg(v 2 )) ≥ (1 -e -1 ) max P 1 e(P 1 , V 2 ) .
(5.25)

We have e

(P 1 , V 2 ) = v 2 ∈V 2 deg P 1 ,V 2 (v 2 ), so by linearity of expectation E P 1 [e(P 1 , V 2 )] = v 2 ∈V 2 E P 1 [deg P 1 ,V 2 (v 2 )]. However deg P 1 ,V 2 (v 2 ) = |{i 1 ∈ [k 1 ] : N (v 2 ) ∩ P i 1 1 = ∅}|. Recall also that for any v 1 , P v 1 ∈ P i 1 1 = 1
k 1 since the partition is taken uniformly at random. Thus, we get:

E P 1 [deg P 1 ,V 2 (v 2 )] = E P 1 |{i 1 ∈ [k 1 ] : N (v 2 ) ∩ P i 1 1 = ∅}| = E P 1   k 1 i 1 =1 1 N (v 2 )∩P i 1 1 =∅   = k 1 i 1 =1 E P 1 1 N (v 2 )∩P i 1 1 =∅ = k 1 i 1 =1 P N (v 2 ) ∩ P i 1 1 = ∅ = k 1 i 1 =1 1 -P N (v 2 ) ∩ P i 1 1 = ∅ = k 1 i 1 =1   1 - v 1 ∈N (v 2 ) P v 1 ∈ P i 1 1   = k 1 i 1 =1   1 - v 1 ∈N (v 2 ) P v 1 ∈ P i 1 1   = k 1 1 -1 - 1 k 1 deg(v 2 ) , (5.26) since P v 1 ∈ P i 1 1 = 1 -1 k 1 and |N (v 2 )| = deg(v 2 ).
So, in all:

E P 1 [e(P 1 , V 2 )] = v 2 ∈V 2 E P 1 [deg P 1 ,V 2 (v 2 )] = k 1 v 2 ∈V 2 1 -1 - 1 k 1 deg(v 2 )
.

However, the function f :

x → 1 -1 -1 k 1 x is nondecreasing concave with f (0) = 0, so f (x)
x ≥ f (y) y for x ≤ y. In particular, we have that:

f (min(k 1 , deg(v 2 ))) ≥ min(k 1 , deg(v 2 )) k 1 f (k 1 ) ,
and thus:

E P 1 [e(P 1 , V 2 )] ≥ k 1 v 2 ∈V 2 1 -1 - 1 k 1 min(k 1 ,deg(v 2 )) ≥ k 1 v 2 ∈V 2 min(k 1 , deg(v 2 )) k 1 1 -1 - 1 k 1 k 1 ≥ 1 -1 - 1 k 1 k 1 v 2 ∈V 2 min (k 1 , deg(v 2 ))
≥ (1 -e -1 ) max

P 1 e(P 1 , V 2 ) .
(5.27)

Let us now consider the general case with k 2 unconstrained. We apply the previous discussion on the graph G V 1 ,P 2 for some xed partition P 2 of V 2 . Since e G V 1 ,P 2 (P 1 , P 2 ) = e(P 1 , P 2 ), we have the upper bound:

max P 1 e(P 1 , P 2 ) ≤ k 2 i 2 =1 min k 1 , deg V 1 ,P 2 (P i 2 2 ) , 127 5. B C C W N S C
and the previous algorithm gives us a partition P 1 of V 1 such that:

e(P 1 , P 2 ) ≥ (1 -e -1 ) k 2 i 2 =1 min k 1 , deg V 1 ,P 2 (P i 2 2 ) .
Therefore, let us focus on the following optimization problem:

max P 2 k 2 i 2 =1 min k 1 , deg V 1 ,P 2 (P i 2 2 ) ,
We will give a (1 -e -1 )-approximation algorithm running in polynomial time for this problem. In all, this will allow us to get in polynomial time a partition pair (P 1 , P 2 ) such that:

e(P 1 , P 2 ) ≥ (1 -e -1 ) k 2 i 2 =1 min k 1 , deg V 1 ,P 2 (P i 2 2 ) ≥ (1 -e -1 ) 2 max P 2 k 2 i 2 =1 min k 1 , deg V 1 ,P 2 (P i 2 2 )
≥ (1 -e -1 ) 2 max P 1 ,P 2 e(P 1 , P 2 ) .

(5.28)

The problem max P 2

k 2 i 2 =1 min k 1 , deg V 1 ,P 2 (P i 2 2
) is a particular instance of the submodular welfare problem from [START_REF] Vondrák | Optimal approximation for the submodular welfare problem in the value oracle model[END_REF]. Note that deg

V 1 ,P 2 (P i 2 2 ) = deg V 1 ,{P i 2 2 ,V 2 -P i 2 2 } (P i 2
2 ), as the degree of P i 2 2 does not depend on the rest of the partition P 2 . Then, h(S 2 ) :

= min k 1 , deg V 1 ,{S 2 ,V 2 -S 2 } (S 2 ) , for S 2 ⊆ V 2 , is a nondecreasing submodular function, as S 2 → deg V 1 ,{S 2 ,V 2 -S 2 } (S 2
) is a nondecreasing submodular function on V 2 and min(k 1 , •) is nondecreasing concave. Thus, we want to maximize

k 2 i 2 =1 h(S i 2 ) where (S i 2 ) i 2 ∈[k 2 ]
is a partition of items in V 2 among k 2 bidders. It is a particular case of the submodular welfare problem where each nondecreasing submodular utility weight is the same for all bidders and equal to h. Thus, thanks to [START_REF] Vondrák | Optimal approximation for the submodular welfare problem in the value oracle model[END_REF], there exists a polynomial-time (1-e -1 )-approximation

of max P 2 k 2 i 2 =1 min k 1 , deg V 1 ,P 2 (P i 2
2 ) .

Non-Signaling Assisted Capacity Region for Deterministic Channels

Thanks to Theorem 5.8 and Proposition 5.7, there exists a constant-factor approximation algorithm for the broadcast channel coding problem running in polynomial time. We aim to show here that the non-signaling assisted value is linked by a constant factor to the unassisted one. Indeed, the hope is that the non-signaling assisted program is linked to the linear relaxation of the unassisted problem, thus is likely a good approximation since the broadcast channel coding problem can be approximated in polynomial time.

This turns out to be true, and will be proved through the following theorem:

Theorem 5.10. If W is a deterministic broadcast channel, then for all 1 ≤ k 1 and 2 ≤ k 2 :

1 - k k 1 1 e -k 1 k 1 ! 1 -1 - 1 1 k 1 1 -1 - 1 2 k 2 S NS (W, k 1 , k 2 ) ≤ S(W, 1 , 2 ) .
Corollary 5.11. For any deterministic broadcast channel W , C NS (W ) = C(W ).

Proof. We apply Theorem 5.10 on the deterministic broadcast channel W ⊗n .

We

x k 1 = 2 nR 1 , k 2 = 2 nR 2 and 1 = 2 nR 1 n , 2 = 2 nR 2 n . Since 1 -1 -1 k ≥ 1 -e -k , we get: 1 - k k 1 1 e -k 1 k 1 ! 1 -e -n 2 S NS (W ⊗n , 2 nR 1 , 2 nR 1 ) ≤ S W ⊗n , 2 nR 1 n , 2 nR 2 n . As 1 - k k 1 1 e -k 1 k 1 !
(1 -e -n ) 2 tends to 1 when n tends to in nity, we get ∀ε > 0, ∃N ∈ N, ∀n ≥ N :

(1 -ε)S NS (W ⊗n , 2 nR 1 , 2 nR 1 ) ≤ S(W ⊗n , 2 n(R 1 -log(n) n ) , 2 n(R 2 -log(n) n ) ) . Thus, if lim n→+∞ S NS (W ⊗n , 2 nR 1 , 2 nR 1 ) = 1, we have that for all R 1 < R 1 and R 2 < R 2 : lim n→+∞ S(W ⊗n , 2 nR 1 , 2 nR 1 ) ≥ 1 -ε .
Since this is true for all ε > 0, we get in fact that lim n→+∞ S(W ⊗n , 2 nR 1 , 2 nR 1 ) = 1. This implies that C NS (W ) ⊆ C(W ), and thus that the capacity regions are equal as the other inclusion is always satis ed.

Let us now prove the main result:

Proof of Theorem 5.10. The proof will be done in three parts. We will work on the graph G W (see De nition 5.3).

1. First, we prove that for any partition P 2 of Y 2 in 2 parts:

S(W, 1 , 2 ) ≥ 1 -1 - 1 1 k 1 2 i 2 =1 min k 1 , deg Y 1 ,P 2 (P i 2 2 ) k 1 2 .
2. Then, we show that there exists a partition P 2 such that:

2 i 2 =1 min k 1 , deg Y 1 ,P 2 (P i 2 2 ) k 1 2 ≥ 1 - k k 1 1 e -k 1 k 1 ! 1 -1 - 1 2 k 2 min k 1 k 2 , y 1 min(k 2 , deg(y 1 )) k 1 k 2 .
(5.29)

3. Finally, we prove that:

min k 1 k 2 , y 1 min(k 2 , deg(y 1 )) k 1 k 2 ≥ S NS (W, k 1 , k 2 ) . 129 5. B C C W N S C
By combining these three inequalities, we get precisely the claimed result.

1. This part shares a lot of similarities with the proof of Theorem 5.8, which we will adapt to this particular situation. Let us show that if we take a partition P 1 of Y 1 of size 1 uniformly at random, we get, for some xed P 2 of size 2 :

E P 1 [e G W (P 1 , P 2 )] ≥ 1 k 1 1 -1 - 1 1 k 1 2 i 2 =1 min k 1 , deg Y 1 ,P 2 (P i 2 2 ) . Since 1 2 S(W, 1 , 2 ) = maximize P 1 in 1 parts,P 2 in 2 parts
e G W (P 1 , P 2 ) by Proposition 5.7, this will imply that:

S(W, 1 , 2 ) ≥ 1 1 2 E P 1 [e G W (P 1 , P 2 )] ≥ 1 -1 - 1 1 k 1 2 i 2 =1 min k 1 , deg Y 1 ,P 2 (P i 2 2 ) k 1 2 .
(5.30)

We have that e G W (P 1 , P 2 ) = 2 i 2 =1 deg P 1 ,P 2 (P i 2 2 ), so by linearity of expectation, we have that

E P 1 [e G W (P 1 , P 2 )] = 2 i 2 =1 E P 1 [deg P 1 ,P 2 (P i 2
2 )], so we will focus on the contribution of one particular P i 2 2 . Then, we have that deg

P 1 ,P 2 (P i 2 2 ) = |{i 1 ∈ [ 1 ] : N Y 1 ,P 2 (P i 2 2 ) ∩ P i 1 1 = ∅}|. Recall that P v 1 ∈ P i 1 1 = 1
1 for any v 1 since the partition is taken uniformly at random. Thus:

E P 1 [deg P 1 ,P 2 (P i 2 2 )] = E P 1 |{i 1 ∈ [ 1 ] : N Y 1 ,P 2 (P i 2 2 ) ∩ P i 1 1 = ∅}| = E P 1   1 i 1 =1 1 N Y 1 ,P 2 (P i 2 2 )∩P i 1 1 =∅   = 1 i 1 =1 E P 1 1 N Y 1 ,P 2 (P i 2 2 )∩P i 1 1 =∅ = 1 i 1 =1 P N Y 1 ,P 2 (P i 2 2 ) ∩ P i 1 1 = ∅ = 1 i 1 =1 1 -P N Y 1 ,P 2 (P i 2 2 ) ∩ P i 1 1 = ∅ = 1 i 1 =1   1 - v 1 ∈N (P i 2 2 ) P v 1 ∈ P i 1 1 )    = 1   1 -1 - 1 1 deg Y 1 ,P 2 (P i 2 2 )   .
(5.31)

So, in all we have that:

E P 1 [e G W (P 1 , P 2 )] = 2 i 2 =1 E P 1 [deg P 1 ,P 2 (P i 2 2 )] = 1 2 i 2 =1   1 -1 - 1 1 deg Y 1 ,P 2 (P i 2 2 )   .
(5.32)

However the function f :

x → 1 -1 -1 1 x is nondecreasing concave with f (0) = 0, so f (x)
x ≥ f (y) y for x ≤ y. In particular, we have that:

f (min(k 1 , deg Y 1 ,P 2 (P i 2 2 ))) ≥ min(k 1 , deg Y 1 ,P 2 (P i 2 2 ))) k 1 f (k 1 ) ,
and thus:

E P 1 [e G W (P 1 , P 2 )] ≥ 1 2 i 2 =1   1 -1 - 1 1 min(k 1 ,deg Y 1 ,P 2 (P i 2 2 ))   ≥ 1 2 i 2 =1 min(k 1 , deg Y 1 ,P 2 (P i 2 2 )) k 1 1 -1 - 1 1 k 1 = 1 k 1 1 -1 - 1 1 k 1 2 i 2 =1 min k 1 , deg Y 1 ,P 2 (P i 2 2 ) ,
(5.33) which concludes the rst part of the proof.

2. Let us take P 2 a partition of Y 2 of size 2 uniformly at random, and let us prove that:

E   2 i 2 =1 min k 1 , deg Y 1 ,P 2 (P i 2 2 )  
is greater than or equal to:

2 k 2 1 - k k 1 1 e -k 1 k 1 ! 1 -1 - 1 2 k 2 min k 1 k 2 , y 1 min(k 2 , deg(y 1 )) .
First,

2 i 2 =1 min k 1 , deg Y 1 ,P 2 (P i 2 2 ) = 2 i 2 =1 ϕ(deg Y 1 ,P 2 (P i 2 2
)) with ϕ(j) := min(k 1 , j) which is a concave function. Recall the de nition of the Poisson concavity ratio 3.1

α ϕ = inf x∈R + E[ϕ(Poi(x))] ϕ(x) which is equal to 1 - k k 1 1 e -k 1 k 1 ! for that particular function. Let us nd the law of deg Y 1 ,P 2 (P i 2 2 ): deg Y 1 ,P 2 (P i 2 2 ) = y 1 1 N (y 1 )∩P i 2 2 =∅ = y 1 1 -1 N (y 1 )∩P i 2 2 =∅ = y 1 1 -1 ∀y 2 ∈N (y 1 ),y 2 ∈P i 2 2 = y 1 Ber 1 -1 - 1 2 deg ( y 1 )
(5.34)

Thus:

E ϕ(deg Y 1 ,P 2 (P i 2 2 )) = E ϕ y 1 Ber 1 -1 - 1 2 deg(y 1 ) ≥ E ϕ Poi y 1 1 -1 - 1 2 deg(y 1 ) by Proposition 2.2 ≥ α ϕ ϕ y 1 1 -1 - 1 2 deg(y 1 )
by de nition of α ϕ .

( But:

y 1 1 -1 - 1 2 deg(y 1 ) ≥ y 1 1 -1 - 1 2 min(k 2 ,deg(y 1 )) ≥ 1 -1 - 1 2 k 2 1 k 2 y 1 min (k 2 , deg(y 1 )) ,
(5.36) as before. Since ϕ is concave and ϕ(0) = 0, we have in particular that for all 0 ≤ c ≤ 1 and x ∈ R, ϕ(cx) ≥ cϕ(x). We know also that ϕ is nondecreasing. This implies that:

ϕ y 1 1 -1 - 1 2 deg(y 1 ) ≥ ϕ 1 -1 - 1 2 k 2 1 k 2 y 1 min (k 2 , deg(y 1 )) ≥ 1 -1 - 1 2 k 2 ϕ 1 k 2 y 1 min (k 2 , deg(y 1 )) ,
(5.37)

as 0 ≤ 1 -1 -1 2 k 2 ≤ 1. Thus E ϕ(deg Y 1 ,P 2 (P i 2 2 ))
) is larger than or equal to:

α ϕ 1 -1 - 1 2 k 2 min k 1 , 1 k 2 y 1 min (k 2 , deg(y 1 )) = 1 k 2 1 - k k 1 1 e -k 1 k 1 ! 1 -1 - 1 2 k 2 min k 1 k 2 , y 1 min (k 2 , deg(y 1 )) (5.38) since α ϕ = 1 - k k 1 1 e -k 1 k 1 ! . Finally, E 2 i 2 =1 min k 1 , deg Y 1 ,P 2 (P i 2 2 ) = 2 i 2 =1 E ϕ(deg Y 1 ,P 2 (P i 2 2
)) , so we get that:

E   2 i 2 =1 min k 1 , deg Y 1 ,P 2 (P i 2 2 )  
is larger than or equal to:

2 k 2 1 - k k 1 1 e -k 1 k 1 ! 1 -1 - 1 2 k 2 min k 1 k 2 , y 1 min (k 2 , deg(y 1 )) .
Thus, in particular, there exists some partition P 2 that satis es the same inequality, which concludes the second part of the proof.

3. Let us consider an optimal solution r x,y 1 ,y 2 , p x , r 1 x,y 1 , r 2 x,y 2 of the program computing

S NS (W, k 1 , k 2 ), so that S NS (W, k 1 , k 2 ) = 1 k 1 k 2 x r x,W 1 (x),W 2 (x) . a) It comes directly from r x,y 1 ,y 2 ≤ p x that: x r x,W 1 (x),W 2 (x) ≤ x p x = k 1 k 2 . b) x r x,W 1 (x),W 2 (x) = y 1 x:W 1 (x)=y 1 r x,y 1 ,W 2 (x) and we have that: i. x:W 1 (x)=y 1 r x,y 1 ,W 2 (x) ≤ x:W 1 (x)=y 1 1 = deg(y 1 ) , ii. x:W 1 (x)=y 1 r x,y 1 ,W 2 (x) ≤ x:W 1 (x)=y 1 r 1 x,y 1 ≤ x r 1 x,y 1 = k 2 , so x:W 1 (x)=y 1 r x,y 1 ,W 2 (x) ≤ min(k 2 , deg(y 1
)), and thus:

x r x,W 1 (x),W 2 (x) ≤ y 1 min(k 2 , deg(y 1 )) .
In all, we get that:

S NS (W, k 1 , k 2 ) = 1 k 1 k 2 x r x,W 1 (x),W 2 (x) ≤ min k 1 k 2 , y 1 min(k 2 , deg(y 1 )) k 1 k 2 ,
which concludes the third and last part of the proof.

Hardness of Approximation for Broadcast Channel Coding

Since broadcast channels are more general than point-to-point channels (by de ning W 1 (y 1 |x) := Ŵ (y 1 |x) for Ŵ a point-to-point channel and taking W 2 (y 2 |x) = 1 |Y 2 | a completely trivial channel), computing a single value S(W, k 1 , k 2 ) is NP-hard, and it is even NP-hard to approximate within a better factor than 1 -e -1 , as a consequence of the hardness result for point-to-point channels from [START_REF] Barman | Algorithmic aspects of optimal channel coding[END_REF].

The goal of this section is to give some evidence for the hardness of approximation of the general broadcast channel coding problem, speci cally that it cannot be approximated in polynomial time within a Ω(1) factor. It will be a good insight that non-signaling assistance will enlarge the capacity region of the channel as discussed in the introduction.

Formally, one would want to show that it is NP-hard to approximate this problem within a Ω(1) factor in polynomial time. As a rst step towards this goal, we will prove a Ω 1 √ mapproximation hardness in the value query model. Proposition 5.12 (Equivalent formulation of BCC). Given a channel W and integers k 1 and k 2 , the broadcast channel coding problem, which we call BCC, entails maximizing:

1 k 1 k 2 i 1 ,i 2 max x y 1 ∈P i 1 1 ,y 2 ∈P i 2 1
W (y 1 y 2 |x) , over all partitions P 1 of Y 1 in k 1 parts and P 2 of Y 2 in k 2 parts.

Social Welfare Reformulation

The social welfare maximization problem is de ned as follows: given a set M of m items as well as k bidders with their associated utilities v i :

2 M → R + i∈[k]
, the goal is to partition M between the bidders to maximize the sum of their utilities. Formally, we want to compute: maximize

P partition in k parts of M k i=1 v i P i .
Let us show that the subproblem of BCC restricted to k 2 = |Y 2 | can be reformulated as a particular instance of the social welfare maximization problem. In that case, it is easy to see that P 2 = ({y 2 }) y 2 ∈Y 2 is always an optimal solution. Indeed, for any partition P 2 , we have:

1 k 1 |Y 2 | i 1 ,i 2 max x y 1 ∈P i 1 1 ,y 2 ∈P i 2 2 W (y 1 y 2 |x) ≤ 1 k 1 |Y 2 | i 1 ,i 2 y 2 ∈P i 2 2 max x y 1 ∈P i 1 1 W (y 1 y 2 |x) = 1 k 1 |Y 2 | i 1 y 2 ∈Y 2 max x y 1 ∈P i 1 1
W (y 1 y 2 |x) .

(5.39) Therefore, the objective function becomes:

S 1 (W, k 1 , P 1 ) := 1 k 1 k 1 i 1 =1 f 1 W (P i 1 1 ) with f 1 W (S 1 ) := 1 |Y 2 | y 2 max x y 1 ∈S 1 W (y 1 y 2 |x) .
Hence, up to a multiplicative factor k 1 , maximizing S 1 (W, k 1 , P 1 ) over all partitions P 1 of size k 1 is a particular case of the social welfare maximization problem with a common utility f 1 W for all k 1 bidders.

Value Query Hardness

Let us rst introduce the value query model. As described in [START_REF] Dobzinski | An improved approximation algorithm for combinatorial auctions with submodular bidders[END_REF][START_REF] Vahab | Tight information-theoretic lower bounds for welfare maximization in combinatorial auctions[END_REF], a value query to a utility v asks for the value of some input set S ⊆ M , and gets as response v(S) ∈ R + .

In the value query model, we aim at solving the social welfare maximization problem accessing the data only through value queries to

(v i ) i∈[k] .
This is more restricted than using any algorithm, but in such a model, it is possible to show unconditional lower bounds on the number of queries needed to solve a given problem within an approximation rate. In the case of the social welfare maximization problem with XOS utility functions, the approximation rate achievable in polynomial time has been proved in [START_REF] Dobzinski | An improved approximation algorithm for combinatorial auctions with submodular bidders[END_REF][START_REF] Vahab | Tight information-theoretic lower bounds for welfare maximization in combinatorial auctions[END_REF] to be of the order of Θ 1 √ m . Speci cally, in [START_REF] Dobzinski | An improved approximation algorithm for combinatorial auctions with submodular bidders[END_REF], a Ω 1 m 1 2 -approximation in polynomial time was given, and in [START_REF] Vahab | Tight information-theoretic lower bounds for welfare maximization in combinatorial auctions[END_REF], it has been shown that any Ω 1 m 1 2 -ε -approximation for ε > 0 requires an exponential number of value queries. We will adapt their proof in the particular case of one common XOS utility function of the form f 1 W for some broadcast channel W . But rst, let us introduce the de nition of XOS functions and prove that f 1 W is one of those.

De nition 5.6. A linear valuation function (also known as additive) is a set function a : 2 M → R + that assigns a nonnegative value to every singleton {j} for j ∈ M , and for all S ⊆ M it holds that a(S) = j∈S a({j}).

A fractionally sub-additive function (XOS) is a set function f : 2 M → R + , for which there is a nite set of linear valuation functions A = {a 1 , . . . , a } such that f (S) = max i∈[ ] a i (S) for every S ⊆ M .

Remark. Note that the size of A is not bounded in the de nition.

Proposition 5.13. f 1 W is XOS.

Proof. Remark. As our problem is a particular instance of the social welfare maximization problem with XOS functions, the polynomial-time Ω 1 m 1 2

f 1 W (S) = 1 |Y 2 |
-approximation from [START_REF] Dobzinski | An improved approximation algorithm for combinatorial auctions with submodular bidders[END_REF] works also here.

Proof. The proof is inspired by Theorem 3.1 of [START_REF] Vahab | Tight information-theoretic lower bounds for welfare maximization in combinatorial auctions[END_REF]. We will show using probabilistic arguments that any Ω Let us de ne now Y 2 := [m + k 1 + 1]. We take X := Y 2 = [m + 1 + k 1 ] as well. We can now de ne our broadcast channel W , with some positive constant C to be xed later to guarantee that W is a conditional probability distribution. Let us de ne its value for y 2 = 1:

W (y 1 1|x) := C ×        m 2δ 1 y 1 =x when 1 ≤ x ≤ m , 1 m 1 2 -δ when x = m + 1 , 1 y 1 ∈T j when 1 ≤ j := x -(m + 1) ≤ k 1 .
Then, we de ne other y 2 inputs as translations of W (y 1 1|x). Speci cally, we de ne: W (y 1 y 2 |x) := W (y 1 1|t y 2 -1 (x)) with t s (x) := 1 + [(x -1 + s) mod (m + k 1 + 1)] .

All coe cients are nonnegative. So W will be a channel if for all x, y 1 ,y 2 W (y 1 y 2 |x) = 1. However, one has, for some xed x 0 : , which does not depend on x 0 . Thus, we have de ned a correct instance of our problem. Note that on this instance, we have:

f 1 W (S) = 1 |Y 2 | y 2 max x y 1 ∈S W (y 1 y 2 |x) = y 2 max x y 1 ∈S W (y 1 1|t y 2 -1 (x)) = m + k 1 + 1 |Y 2 | max x y 1 ∈S W (y 1 1|x) since t y 2 -1 bijection = C(m + k 1 + 1) |Y 2 | × max        m 2δ |{i} ∩ S| for 1 ≤ i ≤ m 1 m 1 2 -δ |S| |T j ∩ S| for 1 ≤ j ≤ k 1 (5.42)
Let us also consider an alternate broadcast channel W , with the only di erence that 1 y 1 ∈T j is replaced by 1 m 1 2

, for j ∈ [k 1 ]. For that channel, the constant C remains the same (since

j y 1 1 y 1 ∈T j = k 1 × k 1 = k 1 × m × 1 k 1 = j y 1 1 m 1 2
), so we get that: C(m+k 1 +1 f 1 W (S), so distinguishing between v and v is the same as distinguishing between f 1 W and f 1 W . We will prove that it takes an exponential number of value queries to distinguish between v and v . On the one hand, one can easily show that the maximum value of the social welfare problem with v is (k 1 -1)m 2δ + 1 m 1 2 -δ (m -(k 1 -1)) = O(m 1 2 +2δ ), obtained taking (k 1 -1) singletons as the rst components of the partition (the bidders), giving the rest of Y 1 (the items) to the last. On the other hand, the maximum value of the social welfare problem with v is k 1 × k 1 = m, obtained with the partition T 1 , . . . , T k 1 . The fact that it requires an exponential number of value queries to distinguish between the two situations will imply that one cannot get an approximation rate better than Ω 1 m 1 2 -2δ in less than an exponential number of value queries.

f 1 W (S) = C(m + k 1 + 1) |Y 2 | × max          m 2δ |{i} ∩ S| for 1 ≤ i ≤ m 1 m 1 2 -δ |S| 1 m 1 2 |S| for 1 ≤ j ≤ k 1 = C(m + k 1 + 1) |Y 2 | × max    m 2δ |{i} ∩ S| for 1 ≤ i ≤ m
We will now prove that distinguishing between v and v requires an exponential number of value queries. Note rst that v(∅) = v (∅) = 0, so we do not need to consider empty sets.

Let us x some non-empty set S ⊆ [m]. Let us de ne the random boolean variables X i j := 1 i∈T j for j ∈ [k 1 ] and i ∈ [m]. By construction of the random equi-partition T 1 , . . . , T k 1 , (X i j ) i∈[m] is a permutation distribution (see De nition 2.3) of (0, . . . , 0, 1, . . . , 1) with m -k 1 zeros and k 1 ones, each X i j following a Bernouilli law of parameter p := 1 k 1 . Thus it is negatively associated by Proposition 2.8, and the sub-family (X i j ) i∈S is negatively associated as well by Proposition 2.4. Note in particular that |T j ∩ S| = i∈S X i j is a sum of negatively associated Bernouilli variables of the same parameter p, so the version of the Cherno -Hoe ding bound from Proposition 2.10 holds.

Let us rst assume that S is of size 0 < |S| ≤ m

P i∈S X i j > m 2δ = j∈[k 1 ] P   1 |S| i∈S X i j >   1 + m 2δ [S| -1 p   p   ≤ j∈[k 1 ] P   1 |S| i∈S X i j >   1 + m 2δ |S| p   p   ≤ j∈[k 1 ] exp   - p|S| 4 m 2δ p|S| 2 
 by Proposition 2.10

= j∈[k 1 ] exp - 1 4p|S| m 4δ ≤ j∈[k 1 ] exp - m -δ 4 m 4δ since 1 p|S| = k 1 |S| ≥ m -δ = m
1 2 e -m 3δ 4 .

(5.45) Thus, this event occurs with exponentially small probability (on the choice of the partition T 1 , . . . , T k 1 ).

Let us now study the case of S of size |S| > m

P    1 |S| i∈S X i j >   1 + 1 |S|m 1 2 -δ |S| -1 p    p    ≤ j∈[k 1 ] P   1 |S| i∈S X i j >   1 + 1 m 1 2 -δ p   p   = j∈[k 1 ] P 1 |S| i∈S X i j > 1 + m δ p since p = 1 k 1 = 1 m 1 2 ≤ j∈[k 1 ]
exp -p|S| 4 m 2δ by Proposition 2.10

≤ j∈[k 1 ] exp - m δ 4 m 2δ since p|S| = |S| k 1 ≥ m δ = m
1 2 e -m 3δ 4 .

(5.47) Thus, this event occurs with exponentially small probability as well. We have then that for all set S, P (v(S) = v (S)) ≤ p leak := m 1 2 e -m 3δ 4 , which is an exponentially small bound that does not depend on S.

Hence, for every set S, only with exponentially small probability p leak can one distinguish between v and v . For some xed algorithm A, let us consider the sequence L of queries made by A before it is able to distinguish between v and v : L := (S 1 , . . . , S n ), with v(S i ) = v (S i ) for i ∈ [n] and v (S n+1 ) > v(S n+1 ). L is independent of T 1 , . . . T k 1 as no information from this partition is leaked before S n+1 . Thus, for such an algorithm to be correct, it should work for any equi-partition T 1 , . . . T k 1 . We have:

P ∃i ∈ [n] : v(S i ) = v (S i ) ≤ n i=1 P v(S i ) = v (S i ) = np leak by union bound.
In particular, this implies that:

P ∀i ∈ [n] : v(S i ) = v (S i ) ≥ 1 -np leak .
So, if 1-np leak > 0, i.e. n < 1 p leak , then there exists some equi-partition T 1 , . . . T k 1 such that our algorithm outputs a sequence L of queries of length n before being able to distinguish between v and v . In particular, we can take n = 1 2p leak so that L is of exponential size. Hence, for any algorithm A, there exists some equi-partition T 1 , . . . T k 1 such that A needs an exponential number of value queries to distinguish between v and v . This concludes the proof of the theorem for any deterministic algorithm.

Finally, the hardness result holds also for randomized algorithms. Indeed, let us call A s , the running algorithm conditioned on its random bits being s. A s is deterministic so the previous proof holds: with high probability p, the sequence of 1-p p leak queries does not reveal anything to distinguish between v and v , although it is of exponential size in m. Then, averaging over all its random bitstrings, the same result holds, as p leak is independent of the equi-partition T 1 , . . . , T k 1 .

Limitations of the Model

The main weakness of the previous result is that it highly relies on the restriction that one has access to the data only through value queries. Indeed, if one has access to the full data, it is possible to read the partition T 1 , . . . , T k 1 which gives the optimal solution directly. This weakness comes from the fact that our utility function f 1 W can be described by polynomial-size data, as it is characterized by a broadcast channel W , whereas one usually consider any XOS functions, in particular with some having inherently an exponential-size de ning set of linear valuation functions.

On the other hand, one can also remark that when f 1 W is written as a maximum of linear valuation functions, then that de ning set of linear valuation functions is of exponential size. Hence, we think that it is still relevant to study the value query complexity of such a family of functions, as it is not clear how one could recover the partition in polynomial time from this exponential-size set of linear valuations without any additional information.

Conclusion

In this chapter, we have studied several algorithmic aspects and non-signaling assisted capacity regions of broadcast channels. We have shown that sum success probabilities of the broadcast channel coding problem are the same with and without non-signaling assistance between decoders, and that it implied that non-signaling resource shared between decoders does not change the capacity region. For the class of deterministic broadcast channels, we have described a (1 -e -1 ) 2 -approximation algorithm running in polynomial time, and we have shown that the capacity region for that class is the same with or without non-signaling assistance. Finally, we have shown that in the value query model, we cannot achieve a better approximation ratio than Ω 1 √ m in polynomial time for the general broadcast channel coding problem, with m the size of one of the outputs of the channel.

Our results suggest that non-signaling assistance could improve the capacity region of general broadcast channels, which is left as a major open question. An intermediate result would be to show that it is NP-hard to approximate the broadcast channel coding problem within any constant ratio, strengthening our hardness result without relying on the value query model. Finally, one could also try to develop approximations algorithms for other sub-classes of broadcast channels, such as semi-deterministic or degraded ones. This could be a crucial step towards showing that the capacity region for those classes is the same with or without non-signaling assistance.

Conclusion

In this thesis, we have studied several channel coding problems, with and without nonsignaling assistance, from the point of view of algorithmic approximation and of capacity regions.

In Chapter 3, we have introduced a generalization of the maximum -multi-coverage problem which entails maximizing the quantity a∈[n] w a ϕ(|{i ∈ S : a ∈ T i }|) over subsets S ⊆ [m] of cardinality k. We have shown that when ϕ is normalized, nondecreasing and concave, we can obtain an approximation guarantee given by the Poisson concavity ratio α ϕ := min x∈N * E[ϕ(Poi(x))]

ϕ(E[Poi(x)]) and we showed it is tight for sublinear functions ϕ if P = NP. Applied to channel coding, and more speci cally to ϕ-list-decoding, where the length restriction on the list of guesses of the decoder in -list-decoding is replaced by a probability ϕ( ) of correctly decoding a list of guesses of variable size , we have obtained a tight approximation guarantee α ϕ for the class of channels W of the form W (y|x) = 1 t for y ∈ T x with |T x | = t and W (y|x) = 0 elsewhere. A natural open question is whether the NP-hardness guarantee can be extended for ϕ(n) = o(n). Another open problem is to extend the α ϕ -approximation algorithm for ϕ-list-decoding on all channels.

In Chapter 4, we have shown that the multiple-access channel coding problem cannot be approximated within any constant ratio under a complexity hypothesis on random k-SAT formulas. We have shown that optimal non-signaling assisted codes for multiple-access channels can be found in polynomial time in the number of copies of the channel. Applied to the binary adder channel, a non-signaling advantage on its capacity region has been established. We have provided a general single-letter outer bound on the non-signaling capacity region. When non-signaling assistance is not shared between encoders, we have shown that the capacity region is not changed. A remaining open question is whether quantum entanglement may increase the capacity of the binary adder channel. Also, establishing a single-letter formula for the non-signaling assisted capacity of multipleaccess channels is the main open question left here, which could be obtained by achieving the provided single-letter outer bound on the non-signaling capacity region.

In Chapter 5, for the class of deterministic broadcast channels, we have provided a (1-e -1 ) 2approximation algorithm for the unassisted coding problem, and we have shown that their capacity region is not changed with non-signaling assistance. In the value query model, we have shown that we cannot achieve a better approximation ratio than Ω 1 √ m for the general broadcast channel coding problem, with m the output size of the channel. Our 141 6. C results suggest that non-signaling assistance could improve the capacity region of general broadcast channels, which is left as a major open question. An intermediate result would be to show that it is NP-hard to approximate the broadcast channel coding problem within any constant ratio, strengthening our hardness result without relying on the value query model.

Throughout this thesis, we have shown that links between approximation algorithms for channel coding and non-signaling assisted capacity regions are fruitful in both directions. On the one hand, the existence of approximation algorithms with constant ratio for channel coding results often in the absence of non-signaling advantage on the related capacity regions. On the other hand, the hardness of approximation within any constant ratio for the channel coding problem results often in the existence of a non-signaling advantage on the related capacity regions. Although not a formal equivalence, this strong link between those apparently unrelated domains may be the key to address unsolved problems from both topics. In particular, the step achieved in the understanding of the in uence of nonsignaling correlations on broadcast channels, solved now for deterministic channels, goes in that direction, and could be extended to other classes of channels. We believe that this connection deserves more attention. In fact, as the complexity-theoretic point of view on entangled multiplayer games was extremely fruitful [JNV + 20], we expect the algorithmic study of problems arising in information theory to achieve important insights.

Conclusion

Dans cette thèse, nous avons étudié plusieurs problèmes de codage de canal, avec et sans assistance non-signalante, du point de vue de l'approximation algorithmique et des zones de capacité.

Dans le Chapitre 3, nous avons introduit une généralisation du problème de couverture multiple maximale de paramètre où l'on maximise a∈ [n] ϕ(E[Poi(x)]) et nous avons montré qu'elle est optimale pour les fonctions sous-linéaires ϕ si P = NP. Appliqué au codage de canal, et plus spéci quement au décodage de liste de paramètre ϕ, où la restriction de longueur sur la liste de suppositions du décodeur dans le décodage de liste de paramètre est remplacée par une probabilité ϕ( ) de décoder correctement une liste de suppositions de taille variable , nous avons obtenu une stricte garantie d'approximation α ϕ pour la classe de canaux W de la forme W (y|x) = 1 t pour y ∈ T x avec |T x | = t et W (y|x) = 0 ailleurs. Une question ouverte naturelle est de savoir si la garantie de NP-di culté peut être étendue pour ϕ(n) = o(n). Un autre problème ouvert consiste à étendre l'algorithme d'approximation α ϕ pour le décodage de listes de paramètre ϕ sur tous les canaux.

Dans le Chapitre 4, nous avons montré que le problème du codage des canaux à accès multiple ne peut être approximé avec un ratio constant sous une hypothèse de complexité sur les formules k-SAT aléatoires. Nous avons montré que les codes optimaux avec assistance non-signalante pour les canaux à accès multiple peuvent être trouvés en temps polynomial en le nombre de copies du canal. Appliqué au canal additionneur binaire, un avantage non-signalant sur sa zone de capacité a été établi. Nous avons fourni une borne supérieure générale à une seule lettre pour la zone de capacité non-signalante. Lorsque l'assistance non-signalante n'est pas partagée entre les encodeurs, nous avons montré que la zone de capacité n'est pas modi ée. La question de savoir si l'intrication quantique peut augmenter la capacité du canal additionneur binaire reste ouverte. En outre, l'établissement d'une formule à une seule lettre pour la capacité avec assistance non-signalante des canaux à accès multiple reste la principale question ouverte de ce chapitre. Cette dernière pourrait être obtenue en montrant que la borne supérieure à une seule lettre fournie pour la zone de capacité non-signalante peut en fait être atteinte.

Dans le Chapitre 5, pour la classe des canaux de di usion déterministes, nous avons 143 6. C fourni un algorithme d'approximation de ratio (1 -e -1 ) 2 pour le problème du codage sans assistance, et nous avons montré que leur zone de capacité n'est pas modi ée par l'assistance non-signalante. Dans le modèle d'accès par valeur, nous avons montré que nous ne pouvons pas obtenir un meilleur ratio d'approximation que Ω 1 √ m pour le problème de codage du canal de di usion général, avec m la taille de sortie du canal. Nos résultats suggèrent que l'assistance non-signalante pourrait améliorer la zone de capacité des canaux de di usion généraux, ce qui reste une question ouverte majeure. Un résultat intermédiaire consisterait à montrer qu'il est NP-di cile d'approximer le problème de codage des canaux de di usion pour n'importe quel ratio constant, ce qui renforcerait notre résultat de di culté sans s'appuyer sur le modèle d'accès par valeur.

Tout au long de cette thèse, nous avons montré que les di érents liens entre les algorithmes d'approximation pour le codage de canal et les zones de capacité avec assistance non-signalantes sont fructueux dans les deux sens. D'une part, l'existence d'algorithmes d'approximation à rapport constant pour le codage de canal se traduit souvent par l'absence d'avantage non-signalant pour les zones de capacité correspondantes. D'autre part, la di culté d'approximation pour un ratio constant quelconque pour le problème du codage de canal se traduit souvent par l'existence d'un avantage non-signalant dans les zones de capacité correspondantes. Bien qu'il ne s'agisse pas d'une équivalence formelle, ce lien étroit entre ces domaines apparemment sans rapport peut être la clé pour résoudre des problèmes non résolus dans les deux domaines. En particulier, l'étape franchie dans la compréhension de l'in uence des corrélations non-signalantes sur les canaux de di usion, résolue à présent pour les canaux déterministes, va dans cette direction et pourrait être étendue à d'autres classes de canaux. Nous pensons que ce lien mérite plus d'attention. En fait, comme le point de vue de la théorie de la complexité sur les jeux multijoueurs intriqués a été extrêmement fructueux [JNV + 20], nous nous attendons à ce que l'étude algorithmique des problèmes posés par la théorie de l'information aboutisse à des résultats importants.

1 A

 1 the truth value of the statement A k! the factorial of k n k the binomial coe cient of parameters n and k C P the class of problems decidable in polynomial time NP the class of problems decidable in nondeterministic polynomial timeO(f (n))a function at most of the order of f (n) when n tends to in nityΩ(f (n))a function at least of the order of f (n) when n tends to in nityΘ(f (n)) a function of the order of f (n) when n tends to in nity o(f (n))a function dominated by f (n) when n tends to in nityG G = (V,E) an undirected graph with vertices in V and edges in E deg G (v) the degree of v in G, i.e. the number of edges connected to v G G a group g an element of the group G X , Y, Z, . . . sets where G acts x, y, z, . . . elements of X , Y, Z, . . . g • x the action of g on x O G (X ) the set of orbits of X under the action of G xi xii LIST OF SYMBOLS P X , Y, Z, . . . nite sets X, Y, Z, . . . random variables over X , Y, Z, . . . P X , P Y , P Z , . . . the probability distributions of X, Y, Z, . . . P (E) the probability of the event E E [X] the expectation of X σ 2 X the variance of X Ber(p) the Bernouilli distribution of parameter p Bin(n, p) the binomial distribution of parameters n, p Poi(λ) the Poisson distribution of parameter λ I T W a channel, possibly with multiple inputs or multiple outputs W ⊗ W the tensor product of the channels W and W W ⊗n the nth tensor product of W C[S](W ) the capacity region of W for a success probability S C 0 [S](W )

  Dans le Chapitre 3 (basé sur [BFF21]), nous considérons la généralisation suivante du problème de couverture multiple maximale de paramètre dépendant d'une fonction croissante concave ϕ : étant donné des sous-ensembles T 1 , . . . , T m d'un univers [n], des poids strictement positifs w a sur l'univers [n] et d'un entier k, l'objectif est de trouver un sous-ensemble S ⊆ [m] de taille k qui maximise C ϕ (S) := a∈[n] w a ϕ(|S| a ), où |S| a := |{i ∈ S : a ∈ T i }| ; le problème de couverture multiple maximale de paramètre est retrouvé en prenant ϕ(x) := min(x, ). Pour tout ϕ de ce type, nous fournissons un algorithme e cace qui permet d'obtenir un rapport d'approximation égal au rapport de concavité de Poisson de ϕ, dé ni par α ϕ := min x∈N * E[ϕ(Poi(x))]

P

  (xj|iy) ≥ 0

  Lemma 3.7. e∈E |L(e)| = |E|h Recall that our h-uniform hypergraph is regular; call d its regular degree. In particular, we have that d|V | = |E|h. Note also that v∈V |L(v)| = |T | = |V |. Thus: e∈E |L(e)| = e∈E v∈V :v∈e |L(v)| = v∈V e∈E:v∈e

  Figure 3.2 -Comparison between the PoA and α ϕ for the V T A problem. Using the linear program found in[START_REF] Paccagnan | Utility design for distributed resource allocationpart II: Applications to submodular, covering, and supermodular problems[END_REF], we were able to compute the blue curve PoA 20 , the Price of Anarchy of this problem for m = 20 players. Since the PoA only decreases when the number of players grows, this means that PoA < α ϕ in that case. As a comparison, the red curve Curv depicts the general approximation ratio (see[START_REF] Sviridenko | Optimal approximation for submodular and supermodular optimization with bounded curvature[END_REF]) obtained for submodular function with curvature c, with c = 1 -ϕ p (m) + ϕ p (m -1) here.

  Figure 3.3 -Comparison between the PoA and α ϕ for the d-P problem. Using the linear program found in [PM18], we were able to compute the blue curve PoA 20 , the Price of Anarchy of this problem for m = 20 players. Here, the question whether the inequality PoA ≤ α ϕ is tight remains open. As a comparison, the red curve Curv depicts the general approximation ratio (see [SVW17]) obtained for submodular function with curvature c, with c = 1 -ϕ d (m) + ϕ d (m -1) here.

  w a [ϕ(|S ∪ {x}| a ) -ϕ(|S| a ) -(ϕ(|T ∪ {x}| a ) -ϕ(|T | a ))] . (3.14) Let us call g(a) := ϕ(|S ∪ {x}| a ) -ϕ(|S| a ) -(ϕ(|T ∪ {x}| a ) -ϕ(|T | a )): 1. If |T | a = |S| a then thanks to Lemma 3.20, we have that |T ∪ {x}| a = |S ∪ {x}| a , so g(a) = 0 2. Else, we have that |T | a > |S| a : a) If |S ∪ {x}| a = |S| a , then we add elements of T -S using Lemma 3.20 to get that |T ∪ {x}| a = |T | a , so g(a) = 0 in that case. b) Else |S ∪ {x}| a = |S| a . So with |S| a = k, we get that |S ∪ {x}| a = k + 1 and

De nition 3

 3 .5. A Label Cover instance L = (A B, E, [L], [R], {π e } e∈E ) consists of a bi-regular bipartite graph (A B, E) with right degree t, alphabet sets [L], [R] and for every edge e ∈ E, a constraint πe : [L] → [R]. A labeling of L is a function σ : A → [L].We say that σ strongly satis es a right vertex v ∈ B if for every two neighbors u, u of v, we have π (u,v) (σ(u)) = π (u ,v) (σ(u )). Moreover, we say that σ weakly satis es a right vertexv ∈ B if there exists two neighbors u, u of v such that π (u,v) (σ(u)) = π (u ,v) (σ(u )).Theorem 3.34 (δ-Gap-Label-Cover(t, R) from[START_REF] Dudycz | Tight approximation for proportional approval voting[END_REF]). For any xed integer t ≥ 2 and xed δ > 0, there exists R 0 such that for any integer R ≥ R 0 , it is NP-hard for Label Cover instances L = (A B, E, [L], [R], {π e } e∈E ) with right degree t and right alphabet [R] to distinguish between: There exists a labeling σ that strongly satis es all the right vertices.NO: No labeling weakly satis es more than δ fraction of the right vertices.The reduction is the following. From δ-Gap-Label-Cover(t, R), we take h = t and the same parameters δ, R.Given an instance L = (A B, E, [L], [R], {π e } e∈E ), we take G = (A, E , [L], [R], {π e ,v } e ∈E ,v∈e ) with E = {N (b), b ∈ B} with N (b) the set of neighbors of b in L, and π e ,v = π N (b),v := π v,b since v ∈ N (b). Since (A B,E) is bipartite and biregular, we get that our hypergraph has all hyperedges of size h = |N (b)| = t, and that it is regular from the regular left degree of (A B, E). By construction, the notion of weakly and strongly satis ed is the same in both cases, as well as the labelings, and thus we have the NP-hardness of δ, h-A G L C .

Figure 4 . 1 -

 41 Figure 4.1 -Coding for a MAC W .

Figure 4

 4 Figure 4.2 -A non-signaling box P replacing e 1 , e 2 and d in the coding problem for the MAC W .

Figure 4

 4 Figure 4.3 -Non-signaling boxes P 1 , P 2 replacing e 1 , d 1 and e 2 , d 2 in the coding problem for the MAC W .

  .15) It is indeed a MAC as a ,b W ((a , b )|ab) = a ,b 1 a=a 1 b=b = 1 if (a, b) ∈ E and a ,b W ((a , b )|ab) = a ,b 1 |A||B| = 1 otherwise. We claim the following linear relation between the objective functions of the bipartite densest subgraph problem on G and the multiple-access channel coding problem on W G : Lemma 4.6. For G = (A B, E), S ⊆ A and T ⊆ B:

  .16) which implies the claimed equality.

(4. 38 )

 38 Corollary 4.13. For a channel W :X 1 × X 2 → Y, S NS (W ⊗n , k 1 , k 2 ) is the solution of a linear program of size bounded by O n |X 1 |•|X 2 |•|Y|-1, thus it can be computed in polynomial time in n.

  Lemma 4.14. Let τ ∈ O G (A × B), and call ν := τ A and µ := τ B . For x ∈ ν, let us call B x τ := {y : (x, y) ∈ τ }. Then, |B x τ | = |B x τ | =: c ν τ for any x, x ∈ ν, and furthermore, we c ν τ = |τ | |ν| . Symmetrically, the same occurs for A y τ := {x : (x, y) ∈ τ } with y ∈ µ, where one gets that |A y τ | = |A y τ | =: c µ τ = |τ | |µ| for y, y ∈ µ.

Figure 4 . 4 -

 44 Figure 4.4 -Capacity regions of the binary adder channel W BAC . The black dashed curve depicts the classical capacity region C(W BAC ), whereas the grey dashed curve shows the best known inner bound border on the zero-error classical capacity region C 0 (W BAC ), made from results by[START_REF] Mattas | A new bound for the zero-error capacity region of the two-user binary adder channel[END_REF][START_REF] Coebergh Van Den Braak | A family of good uniquely decodable code pairs for the two-access binary adder channel[END_REF][START_REF] Kasami | Graph theoretic approaches to the code construction for the two-user multiple-access binary adder channel[END_REF]; see[START_REF] Mattas | A new bound for the zero-error capacity region of the two-user binary adder channel[END_REF] for a description of this border. On the other hand, the continuous curves depict the best zero-error non-signaling assisted achievable rate pairs for respectively 2, 3 and 7 copies of the binary adder channel.

  BAC,ε,ε ) for ε = 10 -3 Zero-Error Non-Signaling Capacity Region Non-Signaling Achievable Points for 3 Copies Non-Signaling Achievable Points for 5 Copies = 10 -3 Signaling Capacity Region

Figure 4 . 5 -

 45 Figure 4.5 -Capacity regions of the noisy binary adder channel W BAC,ε,ε for ε = 10 -3 . The black dashed curve depicts the classical capacity region C(W BAC,ε,ε ) which was found numerically using Theorem 4.1. The red point depicts the zero-error non-signaling assisted capacity region (Proposition 4.21). The blue curve depicts achievable non-signaling assisted rates pairs obtained from C(W ⊗5 BAC,ε,ε [P ]) through the numerical method described in Proposition 4.20.

Figure 5

 5 Figure 5.1 -Coding for a broadcast channel W .

  sum) capacity region using De nition 2.12 by C(W ) := C[S](W ) (resp. C sum (W ) := C[S sum ](W )). The objective values of those two optimization programs are not the same, but S(W, k 1 , k 2 ) and S sum (W, k 1 , k 2 ) still characterize the same capacity region [Wil90]: Proposition 5.2. For any broadcast channel W , C(W ) = C sum (W ).

  e. C sum (W ) = C NS dec sum (W ) Finally, since C(W ) = C sum (W ) by Proposition 5.2 and C NS dec sum (W ) = C NS dec (W ) by Proposition 5.3, we get that C(W ) = C NS dec (W ).

Proof.

  Consider an optimal solution e, d 1 , d 2 of D BCC. Note that d 1 de nes a partition P 1 of Y 1 in k 1 parts and d 2 de nes a partition P 2 of Y 2 in k 2 parts, with P i b b := {y b ∈ Y b : d b (y) = i b } for b ∈ {1, 2}. Then we have:

First, let us

  introduce formally the problem: De nition 5.5 (BCC). Given a channel W and integers k 1 , k 2 , the broadcast channel coding problem, which we call BCC, entails maximizing:S(W, k 1 , k 2 , e, d 1 , d 2 ) := 1 k 1 k 2 i 1 ,i 2 ,y 1 ,y 2 W (y 1 y 2 |e(i 1 , i 2 ))1 d 1 (y 1 )=i 1 ,d 2 (y 2 )=i 2 , over all functions e : [k 1 ] × [k 2 ] → X , d 1 : Y 1 → [k 1 ] and d 2 : Y 2 → [k 2 ].As in the deterministic case, we restrict ourselves to deterministic encoders and decoders, which does not change the value nor the hardness of the problem. Also, it can be equivalently stated in terms of partitions corresponding to d 1 , d 2 as:

y 2 max x y 1 ∈S 1 W

 1 (y 1 y 2 |x) = max λ:Y 2 →X a λ (S) , wherea λ (S) = 1 |Y 2 | y 2 y 1 ∈S W (y 1 y 2 |λ(y 2 )) = y 1 ∈S 1 |Y 2 | y 2 W (y 1 y 2 |m(y 2 )) = y 1 ∈S a λ ({y 1 })) with a λ ({y 1 })) = 1 |Y 2 | y 2 W (y 1 y 2 |λ(y 2 )) ∈ R + (5.40)So f 1 W is the maximum of the set of a λ for λ ∈ X Y 1 , which are linear valuation functions, thus f 1 W is XOS.Let us now state the value query hardness of approximation of the broadcast channel problem:Theorem 5.14. In the value query model, for any xed ε > 0, a Ω broadcast channel coding problem on W, k 1 , k 2 , restricted to the case of|Y 2 | = k 2 and m = |Y 1 | = k 2 1, requires exponentially many value queries to f 1 W .

-

  approximation algorithm requires an exponential number of value queries. Let us x a small constant δ > 0. We choose k 1 ∈ N as the number of 135 bidders) and the output space Y 1 := [m] with m := k 2 1 (the items). Then, we choose uniformly at random an equi-partition of Y 1 in k 1 parts of size k 1 , which we name T 1 , . . . , T k 1 .

y 1 ,y 2 W

 2 (y 1 y 2 |x 0 ) = y 1 y 2 W (y 1 y 2 |x 0 ) = y 1 y 2 W (y 1 1|t y 2 -1 (x 0 )) =

  |S|. Let us consider normalized versions v(S):= |Y 2 | C(m+k 1 +1 f 1 W (S) and v (S) := |Y 2 |

12 1 2

 1 +δ . Then, we have that 1 m -δ |S| ≤ m 2δ , so we get that v (S) = m 2δ . On the other hand, we have that:v(S) = max m 2δ |T j ∩ S| for 1 ≤ j ≤ k 1 (5v(S) is di erent from v (S) if and only if ∃j ∈ [k 1 ], |T j ∩ S| > m 2δ. But, we have:P ∃j ∈ [k 1 ], |T j ∩ S| > m 2δ ≤ j∈[k 1 ] P |T j ∩ S| > m 2δ by union bound = j∈[k 1 ]

12 1 2

 1 +δ . Then, we have that 1 m -δ |S| > m 2δ , so we get that v (S) = 1 m 1 2 -δ |S|. On the other hand, we have that:|S| |T j ∩ S| for 1 ≤ j ≤ k 1 (5.46) Thus, v(S) is di erent from v (S) if and only if ∃j ∈ [k 1 ], |T j ∩ S| > 1 m 1 2 -δ |S|. But,we 138 5.4. Hardness of Approximation for Broadcast Channel Coding have: P ∃j ∈ [k 1 ], |T j ∩ S| >

  w a ϕ(|{i ∈ S : a ∈ T i }|) sur les sous-ensembles ⊆ [m] de cardinal k. Nous avons montré que lorsque ϕ est normalisée, croissante et concave, nous pouvons obtenir une garantie d'approximation donnée par le rapport de concavité de Poisson α ϕ := min x∈N * E[ϕ(Poi(x))]

  Information Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Channel Coding for Point-to-Point Channels . . . . . . . . . . . . . . . . . 3 Tight Approximation Guarantees for Concave Coverage Problems 3.1 Approximation Algorithm for ϕ-M C . . . . . . . . . . . . . . . 3.1.1 Generalization to Matroid Constraints . . . . . . . . . . . . . . . . 3.2 Hardness of Approximation for ϕ-M C . . . . . . . . . . . . . . 3.2.1 Partitioning System . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 The Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Further Hardness Under Gap-ETH . . . . . . . . . . . . . . . . . . 3.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Generalized List-Decoding . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 General Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.2 Calculations of α ϕ . . . . . . . . . . . . . . . . . . . . . . . . . . . Probabilités . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2 Théorie de l'information . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.3 Problème du codage pour les canaux un-à-un . . . . . . . . . . . . . . . . 19 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 Zones de capacité pour les canaux à accès multiple . . . . . . . . . . . . . 61 Propriétés des codes avec assistance non-signalante . . . . . . . . . . . . . 73 4.4 Bornes sur les taux atteignables avec assistance non-signalante . . . . . . 85 4.5 Zone de capacité avec assistance non-signalante relaxée et bornes extérieures 90 4.6 Zone de capacité avec assistance non-signalante indépendante . . . . . . . 105 4.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 5 Problème du codage pour les canaux de di usion et corrélations nonsignalantes 111 5.1 Problème du codage pour les canaux de di usion . . . . . . . . . . . . . . 113 5.2 Assistance non-signalante . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 5.3 Algorithme d'approximation pour le problème du codage pour les canaux de di usion déterministes . . .

	viii	CONTENTS
	3.5 3.5.3 NP-hardness of δ, h-A G L C Table des matières List of Symbols
	Résumé G	iv
	N	the set of nonnegative integers
	R	the set of real numbers
	Remerciements [k]	v
	1 Introduction 2 Background 2.1 Probability Theory . vii Table des matières Liste des symboles et abréviations 1 Introduction 2 Préliminaires 2.1 3 Strictes garanties d'approximation pour des problèmes de couverture concaves 3.1 Algorithme d'approximation pour ϕ-M C . . . . . . . . . . . . 25 x xi 5 11 21 3.2 Di culté d'approximation pour ϕ-M C . . . . . . . . . . . . . 28 3.3 4 Problème du codage pour les canaux à accès multiple et corrélations non-signalantes 59 4.1 6 Conclusion Bibliographie	143 153

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Usual Distributions and Properties . . . . . . . . . . . . . . . . . . 2.1.2 Negatively Associated Random Variables . . . . . . . . . . . . . . . 2.1.3 Non-Signaling Probability Distributions . . . . . . . . . . . . . . . 2.2 2.2.1 Information Quantities . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Typical Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.4 Capacity Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 3.3.2 Multiwinner Elections . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Resource Allocation in Multi-Agent Systems . . . . . . . . . . . . . 3.3.4 Vehicle-Target Assignment . . . . . . . . . . . . . . . . . . . . . . 3.3.5 Welfare Maximization for ϕ-Coverage . . . . . . . . . . . . . . . . 3.4 4.2 Di culté d'approximation pour le problème du codage pour les canaux à accès multiple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 ix x TABLE DES MATIÈRES 4.3 . . . . . . . . . . . . . . . . . . . . . . . . . 123 5.4 Di culté d'approximation pour le problème du codage pour les canaux de di usion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 5.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

  Bennett et Wiesner ont montré dans[START_REF] Bennett | Communication via one-and two-particle operators on Einstein-Podolsky-Rosen states[END_REF] que si l'intrication quantique est partagée entre les parties, sous la forme de ce que l'on appelle une paire EPR, il est possible de transmettre deux bits d'information en utilisant un seul qubit. C'est ce que l'on appelle aujourd'hui le codage superdense. Pour les canaux point à point (classiques), l'intrication quantique partagée entre l'émetteur et le récepteur peut augmenter la probabilité de succès optimale du codage de canal [CLMW10, PLM + 11], bien qu'elle n'augmente pas la capacité du canal [BBC + 93, BSST99].On peut également abstraire l'intrication quantique en corrélations non-signalantes[START_REF] Cirel'son | Quantum generalizations of Bell's inequality[END_REF][START_REF] Popescu | Quantum nonlocality as an axiom[END_REF]. A n de comprendre ces corrélations, il est utile d'examiner l'interprétation de l'inégalité CHSH mentionnée précédemment sous forme de jeu. Nous considérons un jeu à deux joueurs avec Alice et Bob. Un arbitre donne un bit uniformément aléatoire x (resp. y) à Alice (resp. Bob), et leur but commun est de choisir respectivement les bits a et b tels que a ⊕ b = x ∧ y sans communiquer, où ⊕ dénote la disjonction exclusive. Il est facile de voir qu'aucune stratégie ne peut conduire à une meilleure probabilité de succès que 3 4 , même en supposant que des variables cachées sont partagées entre les joueurs. Cependant, comme le montre[START_REF] Clauser | Proposed experiment to test local hidden-variable theories[END_REF], si Alice et Bob partagent une paire EPR intriquée, ils peuvent appliquer des mesures bien choisies et atteindre une probabilité de succès de cos 2 π

	8	0.85, ce qui
	en fait ne peut pas être surpassé

Table 3

 3

	.1.

  ∈ [R], {X a i } a∈[n] are negatively associated because it corresponds to a permutation distribution of (0, . . . , 0, 1, . . . , 1), with n -

	3.5. Appendix
	For xed i xϕn h zeros and a random subset of size xϕn h ; see De nition 2.3 and Proposition 2.8. Then, using the fact xϕn h ones, since it describes that the families {X a i } a∈[n] are mutually independent, we obtain that {X a i } i∈[R],a∈[n] are
	negatively associated by Proposition 2.7.
	by
	Proposition 2.6.

Thus in order to show that {C ϕ a (Q)} a∈

[n]

are negatively associated, it su ces to show that {X a i } i∈[R],a∈

[n]

are negatively associated

  ∈ argmax i 1 ,i 2 {W (y|e 1 (i 1 )e 2 (i 2 ))}. Finally, optimizing over functions e 1 , e 2

	4.2. Hardness of Approximation for Multiple-Access Channel Coding
	by taking d(y)
	70

e 2 (i 2 )) ,

  ). Note also that all the numbers occurring in this linear program are integers or fractions of integers, with those integers ranging in [(|X 1 ||X 2 ||Y|) n ], thus of size O(n log(|X 1 ||X 2 ||Y|)). So the size of this linear program is bounded by O

  so y = y . Thus we get that|B x τ | ≤ |B x τ |. By a symmetric argument with x replacing x and g -1 replacing g,

	we get that |B x τ | ≤ |B x τ |, and so |B x τ | = |B x τ | =: c ν τ .
	Furthermore, {B x

τ } x∈ν is a partition of τ , so x∈ν |B x τ | = |ν|c ν τ = |τ |, and c ν τ = |τ | |ν| . Lemma 4.15. For any (x, y) ∈ A × B and v (x,y) G variable indexed by orbits of A × B, let us de ne the variable v x,y

  as a consequence of Theorem 4.1. Its zero-error classical capacity C 0 (W BAC ) is not yet characterized. A lot of work has been done in nding outer and inner bounds on this region [Lin69, vT78, KL78, Jr.78, KLWY83, vdBvT85, BB98, UL98, AB99, MÖ05, OS15]. To date, the best lower bound on the sum-rate capacity is log 2 (240/6) 1.3178 [MÖ05]. Thanks to Proposition 4.17, we were able to compute the regions C NS 0,≤n (W ) for n going up to 7, which led to Figure 4.4. The code can be found on GitHub. It uses Mosek linear programming solver [AA00]. Note that the linear program from Theorem 4.12 has still a large number of variables and constraints although polynomial in n. Speci cally, for n = 2, it has 244 variables and 480 constraints; for n = 3, it has 1112 variables and 2054 constraints; for n = 7, it has 95592 variables and 162324 constraints; nally, for n = 8, it has 226911 variables and 383103 constraints.

  BAC ), and as a consequence, we have that C(W BAC ) C NS (W BAC ).

					log 2 (72) 8	, log 2 (72) 8	∈ C NS 0 (W BAC ) but log 2 (72) 8	, log 2 (72) 8	∈
	C(W Proof. Since 2 8 log 2 (72) 8	= 72 and numerically S NS (W ⊗8 BAC , 72, 72) = 1 thanks to Corol-
	lary 4.13, we get that log 2 (72) 8	, log 2 (72) 8	∈ C NS 0 (W BAC ) by Proposition 4.11. However,
	log 2 (72) 8	+ log 2 (72) 8	> 3 2 so log 2 (72) 8	, log 2 (72)

8

∈ C(W BAC ) by Theorem 4.1 applied to

W BAC . Since C(W BAC ) ⊆ C NS (W BAC ) and C NS 0 (W BAC ) ⊆ C NS (W BAC ), we thus get that C(W BAC ) C NS (W BAC ).

  Thus, by choosing d 1 (j 1 |y 1 ) := j 2 d(j 1 j 2 |y 1 y 2 ) and d 2 (j 2 |y 2 ) := j 1 d(j 1 j 2 |y 1 y 2 ), which are well-de ned since d is a non-signaling box, we haveS sum (W, k 1 , k 2 , e, d 1 , d 2 ) = S NS dec sum (W, k 1 , k 2 , e, d). By optimizing over all e and d, S NS dec sum (W, k 1 , k 2 ) ≤ S sum (W, k 1 , k 2 ). Since the inequality is obvious in the other direction, as d(j 1 j 2 |y 1 y 2 ) := d 1 (j 1 |y 1 )d 2 (j 2 |y 2 ) is always a non-signaling box, we have that S sum (W, k 1 , k 2 ) = S NS dec sum (W, k 1 , k 2 ). This implies in particular that the capacity regions are the same, i.

  is the degree, i.e. the number of neighbors, of p in the graph G P 1 ,P 2 .

1 ,P 2 G (p) := N G P 1 ,P 2 (p) is the set of neighbors of p in the graph G P 1 ,P 2 . 4. Similarly, deg P 1 ,P 2 G (p) := deg G P 1 ,P 2 (p)

One can always replace a generic ϕ to a normalized one without changing the optimal solutions through a simple a ne transformation.

We require ϕ to be de ned for nonnegative integers and will extend it over R+ by considering its piecewise linear extension.

Indeed, for a subset of candidates S ⊆ [m], the utility of a voter a ∈ [n] is equal to ϕ(|S| a ), with |S| a = |{i ∈ S : a ∈ Ti}|.

Formally, to capture speci c welfare-maximization problems in their entirety we have to a consider ϕ-M C with a matroid constraint, and not just bound the number of selected subsets by k. Details pertaining to matroid constraints and the reduction appear in Section 3.1.1 and 3.3.3, respectively.

rather than P (J 1 = I 1 , J 2 = I 2 ) as an objective value, which leads

,x n

, r x n 1 ,x n 2 ,y n ) is a valid solution.

Remerciements

S NS (W , 1, 1) = S NS (W, k 1 , k 2 ), since S NS (W , 1, 1) = 1 by the rst property of Proposition 4.9.

Corollary 4.10.

1. C NS (W ) is convex.

2. If (R 1 , R 2 ) is achievable with non-signaling assistance, then we have

3. If (R 1 , R 2 ) is achievable with non-signaling assistance, then for all

is achievable with non-signaling assistance.

Proof.

1. It comes directly from Proposition 2.15 and the fourth property of Proposition 4.9.

2. By the second property of Proposition 4.9, we have

is not achievable with non-signaling assistance.

Symmetrically, R 2 > log 2 |X 2 | is not achievable with non-signaling assistance.

Furthermore, if one takes R 1 +R 2 > log 2 |Y|, then in particular k 1 k 2 > |Y|, so by the second property of Proposition 4.9,

Thus, R 1 + R 2 > log 2 |Y| is not achievable with non-signaling assistance.

3. Since (R 1 , R 2 ) is achievable with non-signaling assistance, we have by de nition that S NS (W ⊗n , 2 nR 1 , 2 nR 2 ) → n→+∞ 1. But, for all positive integers n, we have that 2 nR 1 ≤ 2 nR 1 and 2 nR 2 ≤ 2 nR 2 , so by the third property of Proposition 4.9, we have that S NS (W ⊗n , 2 nR 1 , 2 nR 2 ) ≥ S NS (W ⊗n , 2 nR 1 , 2 nR 2 ). Thus S NS (W ⊗n , 2 nR 1 , 2 nR 2 → n→+∞ 1 since it is upper bounded by 1, and so (R 1 , R 2 ) is achievable with non-signaling assistance.

Proposition 4.11. C NS 0 (W ) is the closure of the set of rate pairs (R 1 , R 2 ) such that:

, which is the closure of the former rate pairs, is in particular included in the closure of the latter rate pairs. For the other inclusion, consider a rate pair (R 1 , R 2 ) and let us assume that there exists some positive integer n such that

which is enough to conclude, since we consider only closure of such sets.

First, for all positive integers m, we have that S NS (W ⊗nm , 2 R 1 nm , 2 R 2 nm ) = 1. By the fourth property of Proposition 4.9, we have that

A direct property that follows from this de nition and Proposition 4.22 is the fact that the non-signaling assisted capacity region is included in the relaxed non-signaling assisted capacity region.

Corollary 4.24. C NS (W ) ⊆ C NS (W ).

We present now the main result of this section, the characterization of C NS (W ) by a single-letter formula.

Theorem 4.25 (Characterization of C NS (W )). C NS (W ) is the closure of the convex hull of all rate pairs (R 1 , R 2 ) satisfying:

Remark. Note that the only di erence with the classical capacity region of MACs in Theorem 4.1 is that the joint distribution of X 1 and X 2 does not have a product form constraint here.

The proof of Theorem 4.25 will be divided in Proposition 4.31 (outer bound part) and Proposition 4.33 (achievability part). But rst, let us apply these results to the binary adder channel.

Application to the Binary Adder Channel Let us determine the relaxed non-signaling assisted capacity of the binary adder channel which will be plotted in Figure 4.6.

Proposition 4.26. C NS (W BAC ) has the following description:

Remark. Note that for q = 1 2 , the bound becomes

Proof. We use the characterization of C NS provided by Theorem 4.25. Let us consider an arbitrary P X 1 X 2 = (p 00 , p 01 , p 10 , p 11 ). First, we have that

Then, we have that

Symmetrically we have as well

. In all, C NS (W BAC ) is equal to:

Proposition 4.31 (Outer bound part of Theorem 4.25). If a rate pair is achievable with relaxed non-signaling assistance then it is in the closure of the convex hull of all (R 1 , R 2 ) satisfying:

Proof. Consider (R 1 , R 2 ) achievable with relaxed non-signaling assistance: we have a sequence of relaxed non-signaling assisted codes for n copies of the MAC W with k 1 = 2 nR 1 , k 2 = 2 nR 2 messages and an error probability ε n → n→+∞ 0, along with associated distributions of X n 1 X n 2 Y n . Thus combining Lemma 4.28 and Lemma 4.29, we have that:

Then let us consider some random variable Q uniform on [n] and independent of (X n 1 , X n 2 , Y n ). Then we can write:

Since Y Q conditioned on X 1,Q and X 2,Q still follows the law of the MAC W (y|x 1 x 2 ), we can take X 1 = X 1,Q , X 2 = X 2,Q , and then the output of the channel Y satis es Y = Y Q , and thus we obtain:

Doing this similarly on the other conditional mutual informations, we get:

(4.73)

By taking the limit as n goes to in nity, since the limit of ε n is 0, then the limit of h(εn) n is 0 as well and we get that (R 1 , R 2 ) must be in the set of rate pairs such that:

for some uniform Q in a nite set, (X 1 , X 2 ) any joint law depending on Q, and Y the output of W on inputs (X 1 , X 2 ).

Finally, in order to show that this is the right region, one has only to see that the corner points of this region, such as for instance (I(X 1 : Y |Q), I(X 2 : Y |X 1 , Q)), are nite convex combination of the points (I(X 1 : Y |Q = q), I(X 2 : Y |X 1 , Q = q)) which are all in the capacity region of the theorem by taking (X 1 X 2 ) ∼ P X 1 X 2 |Q=q . This implies that (R 1 , R 2 ) is in the convex hull of that region, so we can drop the random variable Q and the proof is completed.

The main consequence of that outer bound on the relaxed non-signaling assisted capacity region is that it holds also for the non-signaling assisted capacity region thanks to Corollary 4.24:

Corollary 4.32 (Outer Bound on the Non-Signaling Assisted Capacity Region). If a rate pair is achievable with non-signaling assistance, then it is in the closure of the convex hull of all (R 1 , R 2 ) satisfying:

Achievability Part of Theorem 4.25

In order to construct the relaxed non-signaling assisted code for achievability, we will use the notions of jointly and conditional typical sets described in Section 2.2.2:

Proposition 4.33 (Achievability part of Theorem 4.25). If a rate pair is in the closure of the convex hull of (R 1 , R 2 ) satisfying:

Let n ∈ N which will be chosen large enough during the proof. We consider n independent random variables (X

the law of their product. We have then

(4.75) Thus, Ŷ follows the product law of Y i , i.e. Ŷ = Y n .

Let us consider C 1 , C 2 , C 3 some positive numbers independent of n and ε which we will de ne later, k 1 = 2 nR 1 , k 2 = 2 nR 2 integers with (R 1 , R 2 ) positive rates such that:

We de ne a solution of S NS (W ⊗n , 2 nR 1 , 2 nR 2 ) in the following way:

and:

2 is satis ed. We have also that:

, so in that case:

Otherwise, if y n ∈ T n ε (Y ), then:

.77)

Then:

since typical sets cover asymptotically the whole probability mass. Thus, we get that

Therefore, for suciently large n we can achieve a rate pair arbitrarily close to the outer bound. Finally, since C NS (W ) is closed and convex, a rate pair that is in the closure of the convex hull of the initial region is also in C NS (W ), and thus the proof is completed.

Independent Non-Signaling Assisted Capacity Region

The goal of this section is to show that independent non-signaling assistance does not change the capacity region of a MAC W , i.e. that C NS SR (W ) = C(W ). In order to prove this result, we will need some properties in the one-sender one-receiver case from [START_REF] Barman | Algorithmic aspects of optimal channel coding[END_REF] recalled in Section 2.3.

Let us state our result on independent non-signaling assistance, which says that even in one-shot scenarios, the success probability with and without that assistance are close: Theorem 4.34. For any 1 , k 1 , 2 , k 2 , S sum (W, 1 , 2 ) is larger than or equal to:

In particular, this will imply that the capacity regions are the same: Proof. We will show that C NS SR sum (W ) = C sum (W ), which is enough to conclude thanks to Proposition 4.2 and Proposition 4.3. We apply Theorem 4.34 on the MAC W ⊗n .

Let us x k 1 = 2 nR 1 , k 2 = 2 nR 2 and 1 = 2 nR 1 n , 2 = 2 nR 2 n . Since:

and 1 -1 2k 1 = 1 -2 2k 2 = 1 -1 2n , we get:

1 -1 2n S NS SR sum (W ⊗n , 2 nR 1 , 2 nR 1 ) ≤ S sum W ⊗n ,