Dr Martina Cardone
email: cardone@eurecom.fr

Dr Daniela Tuninetti
email: danielat@uic.edu

Raymond Knopp
email: knopp@eurecom.fr

Umer Salim
email: umer.salim@intel.com

On the Capacity of the Two-user Gaussian Causal Cognitive Interference Channel

Keywords: Binning, causal cooperation, cognitive radio, constant gap, cooperative communication, dirty paper coding, generalized degrees of freedom, interference channel, superposition coding, Z-channel

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I. INTRODUCTION

This work considers the cognitive radio overlay paradigm [START_REF] Goldsmith | Breaking spectrum gridlock with cognitive radios: An information theoretic perspective[END_REF] that consists of two source-destination pairs sharing the same channel in which the pair with cognitive abilities attains its communication goals while helping the other (non cognitive) pair. The sources are indicated as PTx and CTx, and the destinations as PRx and CRx. PTx and PRx are referred to as the primary pair, while CTx and CRx as the cognitive pair. The prime features of overlay cognitive radio are to firstly allow the cognitive nodes to communicate without hindering the communication of the primary nodes, and secondly to enhance the communication reliability of the primary nodes. To this end, the CTx is assumed to operate in a full-duplex mode on the same channel as the PTx. Due to the broadcast property of the wireless media, the CTx overhears the PTx through a lossy communication link. Contrary to the commonly studied cognitive radio model that assumes perfect non-causal primary message knowledge available at the CTx [START_REF] Devroye | Achievable rates in cognitive radio channels[END_REF], in this work we treat the causal case, that is, the CTx has access only to primary information it receives over the air. We refer to this system as the Causal Cognitive Interference Channel (CCIC).

From an application standpoint, the CCIC fits future 4G networks with heterogeneous deployments [START_REF]LTE-A, 3rd Generation Partnership Project; Technical Specification Group Radio Access Network; Evolved Universal Terrestrial Radio Access (EUTRA)[END_REF] where the CTx corresponds to the so-called small-cell base-station, or eNB. In this scenario, the CTx would listen to the PTx transmission but not make use of a dedicated point-to-point backhaul link (i.e., on either another channel or through a wired link). We consider deployment scenarios where the CTx→CRx link is on the same carrier frequency as PTx→PRx link and the CTx operates in a full-duplex mode. This implies that the CTx can listen to the PTx's transmission while transmitting. Fullduplex communication is possible thanks to sophisticated selfinterference cancellation techniques at the CTx [START_REF] Duarte | Experiment-driven characterization of full-duplex wireless systems[END_REF]. Moreover, we assume that the PRx and CRx can implement sophisticated interference-mitigation techniques which exploit knowledge of the codebooks used at both PTx and CTx. These codebooks are conceived for the interference scenario (e.g. superpositioncoding [START_REF] Han | A new achievable rate region for the interference channel[END_REF] or Dirty Paper Coding (DPC) [START_REF] Costa | Writing on dirty paper (corresp.)[END_REF]). It should be noted that, since 4G air-interfaces already specify up to 8level superposition coding for point-to-point MIMO or pointto-multipoint MIMO transmission [START_REF]LTE-A, 3rd Generation Partnership Project; Technical Specification Group Radio Access Network; Evolved Universal Terrestrial Radio Access (EUTRA)[END_REF], it is feasible to assume that extensions for distributed superposition coding could also be envisaged.

Different interference scenarios are considered and can correspond to the choice of appropriate deployment configurations in cognitive radio networks. The first class is the fully connected CCIC where both destinations suffer from interference, i.e., in this case both destinations are in the coverage area of both sources. The second class is the interferenceasymmetric Gaussian CCIC where either the link PTx→CRx is non-existent (referred to as the Z-channel) or the link CTx→PRx is non-existent (referred to as the S-channel). In the noncooperative IC these two asymmetric scenarios are the same, up to a relabeling of the nodes. In the CCIC case, due to the asymmetry in the cooperation, the two scenarios are different and must be treated separately. The Z-channel models a situation such as an indoor CTx→CRx with another receiver (PRx) connected to an outdoor base station (PTx) in the vicinity of CTx. The S-channel models the case where PRx is out-of-range of CTx and the base station (PTx) schedules traffic to both PRx and CTx/CRx concurrently. Both scenarios are relevant for practical cognitive radio deployments and their ultimate performance is investigated in this work.

A. Related past work

The presence of a lossy communication link between PTx and CTx enables CTx to cooperate with PTx. CTx, in fact, through this noisy channel overhears the signal sent by the PTx and gathers information about PTx's message, which serves as the basis for unilateral cooperation between the two sources. Unilateral source cooperation is a special case of the IC with generalized feedback, or bilateral source cooperation [START_REF] Host-Madsen | Capacity bounds for cooperative diversity[END_REF], [START_REF] Yang | Interference channel with generalized feedback (a.k.a. with source cooperation): Part i: Achievable region[END_REF], [START_REF] Prabhakaran | Interference channels with source cooperation[END_REF], [START_REF] Tuninetti | An outer bound region for interference channels with generalized feedback[END_REF], [START_REF] Tandon | Dependence balance based outer bounds for Gaussian networks with cooperation and feedback[END_REF].

1) IC with bilateral source cooperation: Bilateral source cooperation has been actively investigated recently. Host-Madsen [START_REF] Host-Madsen | Capacity bounds for cooperative diversity[END_REF] first studied outer and inner bounds for the Gaussian IC with either source or destination bilateral cooperation. For outer bounds, the author in [START_REF] Host-Madsen | Capacity bounds for cooperative diversity[END_REF] evaluated the different cutset upper bounds and then tightened the sum-rate upper bound by extending the sum-rate outer bounds originally developed by Kramer [START_REF] Kramer | Outer bounds on the capacity of Gaussian interference channels[END_REF] for the Gaussian noncooperative IC in weak and strong interference to the cooperative case. Tuninetti [START_REF] Tuninetti | An outer bound region for interference channels with generalized feedback[END_REF] derived a general outer bound for the IC with bilateral source cooperation by extending Kramer's Gaussian noise sum-rate upper bounds in [14, Theorem 1] to any memoryless IC with source cooperation, and more recently to any form of source and destination cooperation [START_REF] Tuninetti | An outer bound for the memoryless two-user interference channel with general cooperation[END_REF]. Prabhakaran and Viswanath [START_REF] Prabhakaran | Interference channels with source cooperation[END_REF] extended the idea of [START_REF] Etkin | Gaussian interference channel capacity to within one bit[END_REF]Theorem 1] to derive a sum-rate outer bound for a class of injective semideterministic IC with bilateral source cooperation in the spirit of the work by Telatar and Tse [START_REF] Telatar | Bounds on the capacity region of a class of interference channels[END_REF], and evaluated it for the Gaussian channel with independent noises (this assumption is not without loss of generality when cooperation and feedback are involved). Tandon and Ulukus [START_REF] Tandon | Dependence balance based outer bounds for Gaussian networks with cooperation and feedback[END_REF] developed an outer bound for the IC with bilateral source cooperation based on the dependence-balance idea of Hekstra and Willems [START_REF] Hekstra | Dependence balance bounds for singleoutput two-way channels[END_REF] and proposed a novel method to evaluate it for the Gaussian channel with independent noises.

The largest known achievable region for general bilateral source cooperation, to the best of our knowledge, is the one presented in [START_REF] Yang | Interference channel with generalized feedback (a.k.a. with source cooperation): Part i: Achievable region[END_REF]Section V]. In [10, Section V] each source splits its message into two parts, i.e., a common and a private message, as in the Han-Kobayashi's scheme for the noncooperative IC [START_REF] Han | A new achievable rate region for the interference channel[END_REF]; these two messages are further subdivided into a noncooperative and a cooperative part. The noncooperative messages are transmitted as in the noncooperative IC [START_REF] Han | A new achievable rate region for the interference channel[END_REF], while the cooperative messages are delivered to the destinations by exploiting the cooperation among the two sources. In [10, Section V] each source, e.g. source 1, after learning the cooperative messages of source 2, sends the common cooperative message of source 2 and uses Gelfand-Pinsker's binning [START_REF] Gelfand | Coding for channel with random parameters[END_REF], or Dirty Paper Coding (DPC) [START_REF] Costa | Writing on dirty paper (corresp.)[END_REF] in the Gaussian noise case, against the private cooperative message of source 2 in an attempt to rid its own receiver of this interference. The achievable scheme in [10, Section V] only uses partial-decode-and-forward for cooperation. A possibly larger achievable region could be obtained by also including compress-and-forward as cooperation mechanism in the spirit of [START_REF] Cover | Capacity theorems for the relay channel[END_REF] for the relay channel.

For the two-user Gaussian noise IC with bilateral source cooperation, under the assumption that the cooperation links have same strength, the scheme of [10, Section V] was sufficient to match the sum-capacity upper bounds of [START_REF] Tuninetti | An outer bound region for interference channels with generalized feedback[END_REF], [START_REF] Prabhakaran | Interference channels with source cooperation[END_REF] to within a constant gap [START_REF] Prabhakaran | Interference channels with source cooperation[END_REF], [START_REF] Yang | Interference channels with source cooperation in the strong cooperation regime: Symmetric capacity to within 2 bits/s/hz with dirty paper coding[END_REF]. [START_REF] Prabhakaran | Interference channels with source cooperation[END_REF] characterized the sumcapacity to within 20/2 bits (in this work we consider the gap per user) of the IC with bilateral source cooperation under the condition that the cooperation links have the same strength, but otherwise arbitrary direct and interfering links. The gap was reduced to 2 bits in the 'strong cooperation regime' in [START_REF] Yang | Interference channels with source cooperation in the strong cooperation regime: Symmetric capacity to within 2 bits/s/hz with dirty paper coding[END_REF] with symmetric direct links, symmetric interfering links and symmetric cooperation links. In this work we seek extensions of these results to the case where the cooperation links have different strengths. In particular, motivated by the cognitive radio technology, we focus on the case of unilateral source cooperation where one of the cooperation links is absent. Moreover, we seek to determine the whole capacity region to within a constant gap, not simply the sum-capacity. To the best of our knowledge, the whole capacity region with source cooperation has never been characterized to within a constant gap in the literature. Moreover, the case of asymmetric cooperation links, of which unilateral cooperation is a special case, to the best of our knowledge, has not been considered in the literature. The major contribution of this work is to approximately determine the capacity for the Gaussian Zand S-channels and for the fully connected GCCIC in some parameters regimes which, roughly speaking, exclude the case of weak interference at both receivers.

2) IC with unilateral source cooperation: Unilateral source cooperation is clearly a special case of the general bilateral cooperation case where the cooperation capabilities of the two sources are not restricted to be the same. This case has been specifically considered in [START_REF] Wu | Partial decode-forward binning schemes for the causal cognitive relay channels[END_REF] where the cooperating transmitter works either in full-duplex or in half-duplex mode. For full-duplex unilateral cooperation, the authors of [START_REF] Wu | Partial decode-forward binning schemes for the causal cognitive relay channels[END_REF] evaluated the performance of two achievable schemes: one that exploits partial-decode-and-forward and binning and a second one that extends the first by adding rate splitting. It was observed, through numerical evaluations, that the proposed inner bounds are not too far from the outer bound of [START_REF] Tandon | Dependence balance based outer bounds for Gaussian networks with cooperation and feedback[END_REF] for certain Gaussian noise channels. In this work we formally prove that the outer bound region obtained from [START_REF] Host-Madsen | Capacity bounds for cooperative diversity[END_REF], [START_REF] Prabhakaran | Interference channels with source cooperation[END_REF], [START_REF] Tuninetti | An outer bound region for interference channels with generalized feedback[END_REF] is achievable to within a constant gap, for the two interferenceasymmetric network scenarios considered, i.e., the Gaussian Zand S-channels, and for the fully-connected GCCIC in some parameters regimes. Moreover, we use as unifying framework the achievable scheme of [10, Section V], of which the schemes of [START_REF] Wu | Partial decode-forward binning schemes for the causal cognitive relay channels[END_REF] are special cases.

An extension of the IC with unilateral source cooperation was studied in [START_REF] Mirmohseni | On the capacity of interference channel with causal and noncausal generalized feedback at the cognitive transmitter[END_REF], where it was assumed that at any given time instant the cognitive source has a non-causal access to L ≥ 0 future channel outputs. The case L = 0 corresponds to the strictly causal case considered in this paper, while the case L → ∞ to the limiting non-causal cognitive IC [START_REF] Devroye | Achievable rates in cognitive radio channels[END_REF]. The authors of [START_REF] Mirmohseni | On the capacity of interference channel with causal and noncausal generalized feedback at the cognitive transmitter[END_REF] derived potentially tighter outer bounds for the CCIC channel (i.e., case L = 0) than those of [START_REF] Prabhakaran | Interference channels with source cooperation[END_REF], [START_REF] Tuninetti | An outer bound region for interference channels with generalized feedback[END_REF] specialized to unilateral source cooperation; unfortunately it is not clear how to evaluate these bounds in Gaussian noise because they are expressed as a function of auxiliary random variables jointly distributed with the inputs and for which no cardinality bounds on the corresponding alphabets are known. The achievable region in [START_REF] Mirmohseni | On the capacity of interference channel with causal and noncausal generalized feedback at the cognitive transmitter[END_REF]Corollary 1] is also no smaller than the region in [10, Section V] specialized to the case of unilateral source cooperation (see [START_REF] Mirmohseni | On the capacity of interference channel with causal and noncausal generalized feedback at the cognitive transmitter[END_REF]Remark 2,point 6]). Although [START_REF] Mirmohseni | On the capacity of interference channel with causal and noncausal generalized feedback at the cognitive transmitter[END_REF]Corollary 1] is, to the best of our knowledge, the largest known achievable region for the general memoryless CCIC with unilateral cooperation, its evaluation in general is quite involved as the rate region is specified by 9 jointly distributed auxiliary random variables and by 30 rate constraints. In [START_REF] Mirmohseni | On the capacity of interference channel with causal and noncausal generalized feedback at the cognitive transmitter[END_REF] inner bounds were compared numerically to the 2 × 2 MIMO outer bound for the GCCIC; the 2 × 2 MIMO outer bound is loose in general compared to the bounds in [START_REF] Host-Madsen | Capacity bounds for cooperative diversity[END_REF], [START_REF] Prabhakaran | Interference channels with source cooperation[END_REF], [START_REF] Tuninetti | An outer bound region for interference channels with generalized feedback[END_REF]. Although it was noted in [START_REF] Mirmohseni | On the capacity of interference channel with causal and noncausal generalized feedback at the cognitive transmitter[END_REF] that, for the simulated set of channel gains, the proposed bounds are not far away from one another, a performance guarantee in terms of (sum-)capacity to within a constant gap was not given. In this work we characterize the capacity to within a constant gap for several channel configurations, i.e., for the Gaussian Z-and S-channels and for the fully-connected GCCIC in some parameters regimes, which, roughly speaking, exclude the case of weak interference at both receivers.

3) Non-causal cognitive radio channel: The cognitive radio channel is commonly modeled following the pioneering work of Devroye et al [START_REF] Devroye | Achievable rates in cognitive radio channels[END_REF] in which the superior capabilities of the cognitive source are modeled as perfect non-causal knowledge of PTx's message at CTx. For this non-causal model the capacity region in Gaussian noise is known exactly for some parameter regimes and to within 1 bit otherwise [START_REF] Rini | Inner and outer bounds for the Gaussian cognitive interference channel and new capacity results[END_REF]. In this work we remove the ideal non-causal message knowledge assumption by considering a more realistic scenario where CTx causally learns the PTx's message through a noisy link. The study of the causal model stems from the question of whether cognitive radio can offer a substantial rate gain over the noncooperative IC. Since the answer was in the positive for the non-causal model [START_REF] Rini | Inner and outer bounds for the Gaussian cognitive interference channel and new capacity results[END_REF], the next question is whether such gains can be attained in practical channels where message knowledge must be obtained through a noisy channel. This work answers this question in the positive. In particular, we identify the set of the channel parameters sufficient to attain, to within a constant gap, the ultimate performance limits of cognitive radio as predicted by the non-causal model [START_REF] Rini | Inner and outer bounds for the Gaussian cognitive interference channel and new capacity results[END_REF].

B. Contributions and paper organization

The rest of the paper is organized as follows. Section II describes the channel model, defines the concept of capacity to within a constant gap and of generalized degrees of freedom (gDoF), and summarizes known inner and outer bounds. Section III characterizes the capacity region of the symmetric GCCIC to within 1 bit for every set of the channel parameters, with the exclusion of a subset of the weak interference regime, for which the sum-capacity to within 3.16 bits is provided (see Theorem 1). Section IV considers the general GCCIC and characterizes its capacity region to within 2 bits when, roughly speaking, the interference is not weak at both receivers (see Theorem 2). In order to better understand the weak interference regime, we analyze the 'interference asymmetric' GCCIC in which one of the interfering links is absent which models different network topologies; we determine the capacity region to within 2 bits for the Z-channel in Section V (see Theorem 3), and to within 2 bits for the S-channel in Section VI (see Theorem 4). Section VII concludes the paper. Most of the proofs are reported in the Appendix. In particular, the Appendix contains the details of the relatively simple proposed achievable schemes, which can be used to provide design insights into practical schemes for future cognitive networks. For all system models considered, we compare the gDoF attained with causal unilateral cooperation with that of other known forms of cooperation to quantify when causal cognitive radio might be worth implementing in practice.

II. SYSTEM MODEL AND BACKGROUND

Throughout the paper we adopt the notation convention of [START_REF] Gamal | Network Information Theory[END_REF]. In particular, [n 1 : n 2 ] denotes the set of integers from n 1 to n 2 ≥ n 1 ; [x] + := max{0, x} for x ∈ R; log + (x) := max{0, log(x)} for x ∈ R + ; Y j is a vector of length j with components (Y 1 , . . . , Y j ). The subscript c (in sans serif font) is used for quantities related to the cognitive pair, while the subscript p (in sans serif font) for those related to the primary pair. The subscript f (in sans serif font) is used to refer to generalized feedback information received at CTx. The subscript c (in roman font) is used to denote common messages, while the subscript p (in roman font) to denote private messages. The notation eq(n) is used to denote the rightmost side of the equation number n.

A. The Gaussian noise channel

A single-antenna full-duplex GCCIC, shown in Fig. 1, is described by the input/output relationship

  Y f Y p Y c   =   √ C ⋆ S p √ I c e jθc I p e jθp √ S c   X p X c +   Z f Z p Z c   (1)
where ⋆ indicates the channel gain that does not affect the capacity region (because CTx can remove its transmit signal X c from its channel output Y f ). The channel gains are constant, and therefore known to all nodes. Without loss of generality

PTx

CTx

PRx

CRx

W p W c + + Z c Z p p S p p I p e jθ p p I c e jθ c X p X c Y p Y c + Z f p S c p C Ŵp Ŵc Y f
Fig. Rc ] for CRx, where N ∈ N denotes the codeword length and R p ∈ R + and R c ∈ R + the transmission rates for PTx and CTx, respectively. The messages W p and W c are independent and uniformly distributed on their respective domains. At time i, i ∈ [1 : N ], PTx maps its message W p into a channel input symbol X p,i (W p ) and CTx maps its message W c and its past channel observations into a channel input symbol X c,i (W c , Y i-1 f ). At time N , PRx makes an estimate of its intended message based on all its channel observations as W p (Y N p ), and similarly CRx outputs W c (Y N c ). The capacity region is the convex closure of all non-negative rate pairs (R p , R c ) such that max u∈{c,p} P[

W u = W u ] → 0 as N → +∞.
The noncooperative Gaussian IC is obtained as a special case of the GCCIC by setting C = 0 and the Gaussian noncausal cognitive IC in the limit for C → +∞. A GCCIC is said to be a Z-channel if I p = 0, i.e., the CRx does not experience interference from PTx, and an S-channel if I c = 0, i.e., the PRx does not experience interference from CTx.

Capacity region to within a constant gap. The capacity region of the GCCIC is said to be known to within GAP bits if we can show an inner bound region I and an outer bound region O such that

(R p , R c ) ∈ O =⇒ ([R p -GAP] + , [R c -GAP] + ) ∈ I.
Generalized Degrees of Freedom (gDoF). The knowledge of the capacity region to within a constant gap implies an exact capacity characterization at high SNR. The gDoF is a performance measure introduced in [START_REF] Etkin | Gaussian interference channel capacity to within one bit[END_REF] for the noncooperative IC to capture the high SNR behavior of the sum-capacity as a function of the relative strengths of direct and interference links. The gDoF represents a more refined characterization of the sum-capacity at high SNR compared to the classical DoF. In order to quantify the gain of causal unilateral source cooperation compared to the noncooperative IC, we shall use the gDoF as a performance measure. Let S > 1 and parameterize S p := S 1 , primary direct link, (2a) S c := S 1 , cognitive direct link, (2b) I p := S αp , α p ≥ 0, interference at CRx from PTx, (2c) I c := S αc , α c ≥ 0, interference at PRx from CTx, (2d)

C := S β , β ≥ 0, cooperation link, ( 2e
)
where α p and α c measure the strength of the interference links compared to the direct link, while β the strength of the cooperation link compared to the direct link. We remark that the parameterization in (2), with direct links of the same strength, is used only for evaluation of the gDoF. Moreover, in order to capture different network topologies, we focus on 1) interference-symmetric channel:

α p = α c = α; 2) Z-channel: α p = 0, α c = α;
3) S-channel: α p = α, α c = 0. The case α p = α c = 0 is not interesting since in this case the GCCIC reduces to two parallel point-to-point links for which cooperation is useless. For the above three cases, the system is parameterized by the triplet (S, α, β), where S is referred to as the (direct link) SNR, α as the interference exponent and β as the cooperation exponent. 1 The gDoF is defined as

d(α, β) := lim S→+∞ max{R p + R c } 2 log(1 + S) (3)
where the maximization is intended over all possible achievable rate pairs (R c , R p ). Without cooperation, the gDoF d(α, 0) reduces to the gDoF characterized in [START_REF] Etkin | Gaussian interference channel capacity to within one bit[END_REF] while for β → +∞ to the gDoF that can be evaluated from the capacity characterization to within 1 bit of [START_REF] Rini | Inner and outer bounds for the Gaussian cognitive interference channel and new capacity results[END_REF]. Here we are interested in determining under which condition on the cooperation exponent β we have d(α, β) > d(α, 0) since a strict improvement in gDoF implies an unbounded gain in terms of sum-capacity as the SNR grows to infinity.

B. Known outer bounds for the GCCIC

In the literature several outer bounds are known for bilateral source cooperation [START_REF] Host-Madsen | Capacity bounds for cooperative diversity[END_REF], [START_REF] Prabhakaran | Interference channels with source cooperation[END_REF], [START_REF] Tuninetti | An outer bound region for interference channels with generalized feedback[END_REF], [START_REF] Tandon | Dependence balance based outer bounds for Gaussian networks with cooperation and feedback[END_REF]. Here we specialize some of them for the GCCIC in [START_REF] Cardone | On the interference channel with causal cognition[END_REF]. We let E [X p X * c ] = ρ, for some ρ ∈ C such that |ρ| ≤ 1. An outer bound region

R c ≤ log (1 + S c ) (4a) R p ≤ min log 1 + ( S p + I c ) 2 , log (1 + C + S p ) (4b) R p + R c ≤ min r (CS) , r (DT) , r (PV) (4c) r (CS) ≤ log (1 + S c ) + min log 1 + ( S p + I c ) 2 , log (1 + C + S p ) (4d) r (DT) ≤ min log 1 + max{I c , S c } 1 + I c + log 1 + ( S p + I c ) 2 , log 1 + C + max{S p , I p } 1 + I p + log 1 + ( S c + I p ) 2 (4e) r (PV) ≤ log     1 + S p max{1, I p } + I c 2     1 + √ S c max{1, I c } + I p 2     + ∆ (4f) ∆ := log      (1 + C) 1 + √ Sc √ max{1,Ic} + √ Ip √ max{1,C} 2 1 + √ Sc √ max{1,Ic} + I p 2      (4g)
for the GCCIC is reported in (4) at the top of next page and is obtained by upper bounding over (ρ, θ c , θ p ) each mutual information term in the bounds in [START_REF] Host-Madsen | Capacity bounds for cooperative diversity[END_REF], [START_REF] Prabhakaran | Interference channels with source cooperation[END_REF], [START_REF] Tuninetti | An outer bound region for interference channels with generalized feedback[END_REF] (the details can be found in Appendix A). In particular, the bounds on the individual rates in (4a) and (4b) are cut-set bounds, and the sum-rate upper bound in (4c) is the minimum of three quantities obtained as follows: from the cut-set bounds on the individual rates we obtain (4d), from [START_REF] Tuninetti | An outer bound region for interference channels with generalized feedback[END_REF] we obtain (4e), and from [START_REF] Prabhakaran | Interference channels with source cooperation[END_REF] we obtain (4f).

The upper bound in (4) for C → +∞ reduces to the upper bound for the Gaussian non-causal cognitive IC in [24, Theorem III.1], which unifies previously known outer bounds for the weak (S c > I c ) and strong (S c ≤ I c ) interference regimes. The region in [24, Theorem III.1] is known to be achievable to within 1 bit in all parameter regimes. However, in weak interference (S c > I c ), the capacity region of the Gaussian non-causal cognitive IC is known exactly and is given by

R p ≤ log 1 + S p + |γ c | 2 I c + 2|γ c | S p I c 1 + (1 -|γ c | 2 )I c (5a) R c ≤ log 1 + (1 -|γ c | 2 )S c (5b)
union over all |γ c | ≤ 1. Therefore, the region in ( 5) is an outer bound for the GCCIC for S c > I c .

From the sum-rate upper bound in (4c), with the parameterization in (2), we can immediately obtain the following gDoF upper bound

d ≤ 1 2 min d (CS) (α c , β) + d (CS) (α p , 0), (6a)
min{d (DT) (α c , 0), d (DT) (α p , β)}, (6b)
d (PV) (α p , α c , β) (6c)
where

d (CS) (α, β) := max{1, min{α, β}} (6d)
d (DT) (α, β) := max{β, α, 1} -α + max{α, 1} (6e)
d (PV) (α p , α c , β) := max {1 -α p , α c } + max {1 -α c + β, α p } . (6f)
The proof follows by using the upper bound in (4c) in the gDoF definition in (3) (the details can be found in Appendix A). The achievability for the interference-symmetric (α p = α c = α) and the interference-asymmetric cases (either α p = 0, α c = α or α p = α, α c = 0) will follow from the constant gap results in the next sections.

C. Known inner bounds for the general memoryless CCIC

To the best of our knowledge, the largest known achievable region for the general memoryless IC with generalized feedback, or bilateral source cooperation, is the superposi-tion+binning region from [10, Section V]. In this scheme, adapted to the case of unilateral source cooperation, the PTx's message is split into four parts: the noncooperative common message and the noncooperative private message are sent as in the Han-Kobayashi's scheme for the noncooperative IC [START_REF] Han | A new achievable rate region for the interference channel[END_REF]; the cooperative common message and the cooperative private message are decoded at CTx in a given slot and retransmitted in the next slot by using a decode-and-forward based block-Markov scheme. The CTx's message is split into two parts: the noncooperative common message and the noncooperative private message that are sent as in the Han-Kobayashi's scheme for the noncooperative IC [START_REF] Han | A new achievable rate region for the interference channel[END_REF]. The common messages are decoded at both destinations while non-intended private messages are treated as noise. For cooperation, the two sources 'beam form' the PTx's cooperative common message to the destinations as in a distributed MIMO system, and the CTx precodes its private messages against the interference created by the PTx's cooperative private message as in a MIMO broadcast channel. The achievable region in [10, Section V] is quite complex to evaluate because it is a function of 11 auxiliary random variables and is described by about 30 rate constraints per source-destination pair. In this work we will use a small subset of these 11 auxiliary random variables in each parameter regime (see Appendices B and C) and show that the corresponding schemes are to within a constant gap from the outer bound in [START_REF] Devroye | Achievable rates in cognitive radio channels[END_REF].

As noted in the Introduction, the largest known achievable region for the IC with unilateral source cooperation is, to the best of our knowledge, the region in [23, Next, in Section III we consider the symmetric GCCIC, where the direct links have the same strength and the interfering links have the same strength. We characterize its capacity to within a constant gap in the strong interference regime and in the weak interference regime when the strength of the cooperation link is greater than a threshold. When the system experiences weak interference and the strength of the cooperation link is below a threshold the sum-capacity to within a constant gap is characterized. This preliminary analysis allows us to identify the key features of the proposed achievable schemes in the strong and weak interference regimes. Moreover, it sets the stage for the constant gap derivation for the general GCCIC in some parameter regimes in Section IV and for the whole capacity characterization to within a constant gap for the general Z-channel in Section V, and for the general S-channel in Section VI.

III. THE CAPACITY REGION TO WITHIN A CONSTANT GAP

FOR THE SYMMETRIC GCCIC The symmetric GCCIC is defined by S p = S c = S and I p = I c = I = S α . Following the naming convention of the noncooperative IC, we say that the symmetric GCCIC has strong interference if S ≤ I, that is 1 ≤ α, and weak interference otherwise. Our main result for the symmetric GCCIC is as follows:

Theorem 1 When S ≤ I, the capacity region of the symmetric GCCIC is achievable to within 1 bit using schemes based on superposition coding. Depending on the strength of the cooperation link, the approximately optimal strategies might or might not require to engage in cooperation. The achievable rate regions for this regime are given in [START_REF] Han | A new achievable rate region for the interference channel[END_REF], in [START_REF] Host-Madsen | Capacity bounds for cooperative diversity[END_REF] and in [START_REF] Prabhakaran | Interference channels with source cooperation[END_REF] below. When S > I and C ≥ ∆ th , with ∆ th := S + I + 2 IS I 1+I (1+I), the capacity region of the symmetric GCCIC is achievable to within 1 bit using a scheme based on superposition coding and DPC whose achievable rate region is given in [START_REF] Hekstra | Dependence balance bounds for singleoutput two-way channels[END_REF] below. When S > I and C < ∆ th , the sum-capacity of the symmetric GCCIC is achievable to within 3.16 bits using schemes based on superposition coding whose achievable sum-rates are given by [START_REF] Tandon | Dependence balance based outer bounds for Gaussian networks with cooperation and feedback[END_REF], [START_REF] Tuninetti | An outer bound for the memoryless two-user interference channel with general cooperation[END_REF] and [START_REF] Telatar | Bounds on the capacity region of a class of interference channels[END_REF] below depending on the strength of the interference links compared to the cooperation link.

The result in Theorem 1 provides a capacity characterization to within a constant gap for the symmetric GCCIC for a set of channel parameters, which excludes the case of weak interference (S > I) when the cooperation link is not strong enough (C < ∆ th ). For this set of parameters, in fact, Theorem 1 provides only an approximate sum-capacity result. As we shall see later in more details, the main difficulty that arises in this regime to deal with approximate capacity characterization is due to the lack of outer bounds of the type 2R p + R c /R p + 2R c , which, to the best of our knowledge, are not available in the literature and whose derivation is outside the scope of this work.

The rest of the section is devoted to the proof of Theorem 1. In order to highlight the key steps in the proof, we use the gDoF as starting point for our discussion. The gDoF upper bound for the symmetric GCCIC is obtained by setting α p = α c = α in [START_REF] Duarte | Experiment-driven characterization of full-duplex wireless systems[END_REF]. Fig. 2 shows the gDoF and the gap (per user) for the symmetric GCCIC for the different regions in the (α, β) plane, where the whole set of parameters has been partitioned into multiple sub-regions depending upon different levels of cooperation (β) and interference (α) strengths. In regimes 1, 3, 4 and 5 of Fig. 2 the gDoF attained by the symmetric GCCIC is the same as that achieved by the noncooperative IC given by [START_REF] Etkin | Gaussian interference channel capacity to within one bit[END_REF] d IC (α) = min{max{1α, α}, max{1α/2, α/2}, 1}.

Thus, unilateral cooperation provides a strict gDoF gain over the noncooperative IC in regimes 2 and 6 of Fig. 2. For reference, the gDoF on the non-causal cognitive IC can be evaluated from [START_REF] Rini | Inner and outer bounds for the Gaussian cognitive interference channel and new capacity results[END_REF] as

d CIC (α) = max{1 -α/2, α/2}.
In general we have

d(α, 0) = d IC (α) ≤ d(α, β) ≤ d CIC (α) = lim β→+∞ d(α, β).
From Fig. 2, in regime 2 with β ≥ α -1, in regimes 3 and 4, and in regime 6 with β ≥ min{α, 1 -α}, causal unilateral source cooperation attains the ultimate gDoF limit of the noncausal cognitive IC.

At a high level, the approximately optimal coding schemes are as follows. In the strong interference and weak cooperation regime both users employ a noncooperative common message. In the strong interference and strong cooperation regime, PTx's common message becomes cooperative and is forwarded to PRx with the help of CTx. In the weak interference regime, each user splits its message into a common and a private part; for CTx the two message parts are noncooperative while for PTx are cooperative; PTx's cooperative common message is the 'cloud center' of a superposition coding scheme, and PTx's cooperative private message is the 'known interference' against which CTx's message is precoded in a DPC-based scheme. Binning/DPC is used in the weak interference and strong cooperation regime where CTx can easily decode the signal from PTx because of strong cooperation, but CRx cannot because of weak interference; therefore in this regime it makes sense that the best use of CTx's knowledge of PTx's message is to treat it as a 'known state' to precode its message against it.

α! β! 1 2 ! 2 3 ! 1! 2! 1 2 ! 1! 2! β = α! β!!=!2α!$!1! β =!1$ α! β = α!$!1! 1" 2" 2" 3" 4" 6a" 5" 6c" d(α,!β)!=!1! gap!=!1!bit! d(α,!β)!=! 1+#! 2 ! gap!=!1!bit! d(α,!β)!=! ! 2 ! gap!=!1!bit! d(α,!β)!=! ! 2 ! gap!=!1!bit! d(α,!β)!=!1!$! ! 2 ! gap!=!1.5!bits! d(α,!β)!=!1!$! ! 2 ! gap!=!1.5!bits! 6b" 6d" d(α,!β)!=!1!$! ! 2 ! gap!=!2!bits! d(α,!β)!=!1!$!α!+! ! 2 ! gap!=!2.5!bits! d(α,!β)!=!α!! gap!=!3.16!bits! d(α,!β)!=! 1+#! 2 ! gap!=!2.5!bits!
We shall now discuss each regime of Fig. 2 separately.

A. Regime 1 (strong interference): same gDoF as in the noncooperative IC, and capacity region to within 1 bit with a noncooperative scheme

Regime 1 corresponds to very strong interference (α ≥ 2) and weak cooperation (β ≤ 1). In the noncooperative IC with very strong interference it is exactly optimal to use only (noncooperative) common messages in order to achieve the whole capacity region; since the interference is very strong, it can be decoded by treating the intended signal as noise, after which each receiver is left with an interference-free point-topoint channel from its transmitter; this noncooperative strategy achieves

I III-A : R c ≤ log(1 + S), (7a) R p ≤ log(1 + S), (7b)
or d ≤ (1 + 1)/2 = 1.
Since the cooperation link is weak in regime 1, the amount of data PTx could communicate to CTx for cooperation is very limited. As a result in this regime unilateral cooperation does not improve the gDoF performance compared to the noncooperative case. In other words, in regime 1, cooperation provides a 'beam forming gain' but not a gDoF gain. To see this, the cut-set upper bounds on individual rates in (4a) and (4b), in the symmetric case for β ≤ 1 ⇐⇒ C ≤ S, give the following upper bounds on the individual rates

O III-A : R c ≤ log(1 + S), (8a) R p ≤ log(1 + S + C) ≤ log(1 + 2S) ≤ log(1 + S) + log(2). (8b)
From the upper bound on R p in (8b), we see that unilateral cooperation can at most double the SNR on the primary direct link, which can at most increase the rate by 1 bit compared to the noncooperative case. As a result, the gDoF with unilateral cooperation is d = 1 and the rate pair in ( 7) is optimal to within 1 bit, i.e., max{eq(8a)eq(7a), eq(8b)eq(7b)} ≤ max{0, log(2)} = 1 bit.

B. Regime 2 (strong interference): improved gDoF compared to the noncooperative IC, and capacity region to within 1 bit with a cooperative scheme

In regime 2 the interference is very strong (α ≥ 2) and the cooperation is strong (β > 1). Similarly to the noncooperative IC in very strong interference regime, the transmitters send a common message only. As opposed to regime 1, where both messages were sent noncooperatively, here the PTx takes advantage of the strong cooperation link and sends its message to PRx with the help of the CTx. In order to enable cooperation, a block Markov coding scheme is used as follows. Transmission is over a frame of B ≫ 1 slots. In slot t ∈ [1 : B], the PTx sends its old (cooperative common) message W p,t-1 and superposes to it the new (cooperative common) message W p,t , while the CTx forwards the primary old (cooperative common) message W p,t-1 and superposes to it its (noncooperative common) message W c,t . At the end of slot t, CTx decodes the new message W p,t after subtracting the contribution of the old message W p,t-1 . The destinations wait until the whole frame has been received and then proceed to jointly backward decode all messages. The details can be found in Appendix B-B and the achievable region is given in (52), which in the symmetric GCCIC in very strong interference reduces to

I III-B : R c ≤ log(1 + S), (9a) R p ≤ log(1 + C), (9b) R p + R c ≤ log(1 + S + I). (9c)
The region in ( 9) is strictly larger than the noncooperative capacity region in very strong interference given by ( 7) for S(1 + S) ≤ I, or α ≥ 2, and C > S, or β > 1, which is precisely the definition of regime 2. The sum-capacity from ( 9) can take two possible values, depending on which one among the MAC sum-rate bound in (9c) and the sum of the bounds on the individual rates in (9a)-(9b) is the most stringent. In particular, the following sum-rate is achievable

R p + R c ≤ log(1 + C) + log(1 + S) if C(1 + S) ≤ I log (1 + S + I) if C(1 + S) > I , that is, d ≤ (β + 1)/2 if β + 1 ≤ α and d ≤ α/2 otherwise (in both cases the gDoF is larger than d IC = 1).
From the outer bound region obtained from the cut-set upper bounds on the individual rates in (4a) and (4b) and the sumrate upper bound in (4e), under the condition β > 1 ⇐⇒ C > S, we have that any achievable rate pair must satisfy

O III-B : R c ≤ log(1 + S), (10a) R p ≤ log(1+S+C) ≤ log(1+C)+log(2), (10b) R p + R c ≤ log 1 + √ S + √ I 2 ≤ log (1 + S + I) + log(2), (10c) since
( √ x + √ y) 2 ≤ 2(x + y), ∀(x, y) ∈ R 2 + .
The upper bound in [START_REF] Yang | Interference channel with generalized feedback (a.k.a. with source cooperation): Part i: Achievable region[END_REF] and the achievable region in (9) are to within 1 bit of one another since GAP ≤ max {eq(10a)eq(9a), eq(10b)eq(9b), eq(10c)eq(9c) 2 ≤ log(2).

This shows that the whole capacity region, and therefore the gDoF d = min{β + 1, α}/2 too, is achievable to within 1 bit in regime 2.

C. Regime 3 (strong interference): same gDoF as in the noncooperative IC, and capacity region to within 1 bit with a cooperative scheme

Regime 3 corresponds to strong but not very strong interference (α ∈ [1, 2)). Note that there are no restrictions on the cooperation exponent β in this regime. Similarly to regimes 1 and 2, here we use only common messages -a strategy that is capacity achieving in the corresponding noncooperative IC. The difference between regime 1 and regime 3 is that stripping decoding is no longer optimal and the receivers must instead jointly decode the intended and non-intended messages as in a MAC. By taking the largest between the achievable region developed for regime 2 in ( 9) and the noncooperative achievable region for this regime (i.e., common messages only), which has R p ≤ log(1 + S) as a bound on the primary rate rather than R p ≤ log(1 + C), we obtain the following achievable region

I III-C : R c ≤ log(1 + S), (11a) R p ≤ log(1 + max{C, S}), (11b) R p + R c ≤ log(1 + S + I), (11c)
which implies d ≤ min{1+max{1, β}, max{1, α}}/2 = α/2, i.e., the sum-rate bound in (11c) is the tightest. In regime 3, no matter how strong the cooperation link is, cooperation does not increase the gDoF of the noncooperative IC.

From the outer bound region obtained from the cut-set upper bounds on the individual rates in (4a) and (4b) and the sumrate upper bound in (4e), we have that any achievable rate pair must satisfy

O III-C : R c ≤ log(1 + S), (12a) R p ≤ log(1 + S + C) ≤ log(1 + max{C, S}) + log(2), (12b)
R p + R c ≤ log 1 + √ S + √ I 2 ≤ log(1 + S + I) + log(2). (12c)
It is easy to see that the regions in ( 12) and ( 11) are to within 1 bit of one another.

D. Regime 4 (weak interference): same gDoF as in the noncooperative IC

Regime 4 corresponds to moderately weak interference (α ∈ [2/3, 1)). In this regime, rate splitting is needed to achieve the capacity to within 1 bit in the noncooperative IC [START_REF] Etkin | Gaussian interference channel capacity to within one bit[END_REF]. Therefore we propose to use here the noncooperative scheme that consists of two messages for each user: the noncooperative common and the noncooperative private. The power of the noncooperative private message (which is treated as noise at the non-intended receiver) is such that it is received at or below the receiver noise floor [START_REF] Etkin | Gaussian interference channel capacity to within one bit[END_REF]. As shown in [START_REF] Etkin | Gaussian interference channel capacity to within one bit[END_REF], in the moderately weak interference regime the sum-rate upper bound of [14, Theorem 1] can be achieved to within 1 bit per user, that is, the following sum-rate is achievable

R p + R c ≤ log (1 + S + I) + log(1 + S) -log(1 + I) -2 log(2), (13)
or d ≤ max{1,α}+(1-α) 2 = 1 -α/2.
The cooperative sum-rate upper bound in (4e) can be further upper bounded as

R p + R c ≤ log (1 + S + I) + log(1 + S) -log(1 + I) + log(2). (14)
Therefore, the gap is at most GAP ≤ eq(14)-eq (13) 2 ≤ 3/2 log(2) and is achieved by the noncooperative scheme with rate splitting as in [START_REF] Etkin | Gaussian interference channel capacity to within one bit[END_REF].

In order to claim capacity to within a constant gap in the weak interference regime, we must derive an upper bound that reduces to, or is to within a constant gap of, the capacity outer bound in [START_REF] Etkin | Gaussian interference channel capacity to within one bit[END_REF]Theorem 3] when C = 0. The outer bound region in [START_REF] Etkin | Gaussian interference channel capacity to within one bit[END_REF]Theorem 3] is characterized by bounds on the individual rates, bounds on the sum-rate, and by bounds on 2R c + R p and R c + 2R p . Therefore, unless outer bounds on 2R c + R p and R c + 2R p for the cooperative case are developed, it is not possible to claim optimality to within a finite gap of the upper bound in (4) for small C. Developing outer bounds on 2R c + R p and R c + 2R p for the general IC with source cooperation is an important open problem, which is outside the scope of this work. An interesting question that could be answered by such a line of research is as follows. In [START_REF] Suh | Feedback capacity of the Gaussian interference channel to within 2 bits[END_REF], the authors interpreted the bounds on 2R c + R p and R c + 2R p as a measure of the amount of 'resource holes', or inefficiency, due to the distributed nature of the noncooperative IC. In [START_REF] Suh | Feedback capacity of the Gaussian interference channel to within 2 bits[END_REF], the authors showed that with output feedback from a destination to its source, such 'resource holes' are no longer present; in other words, feedback enables coordination among the sources which results in a full utilization of the channel resources. An interesting open question is whether unilateral cooperation enables sufficient coordination among the sources for full utilization of the channel resources. In the limiting case where unilateral cooperation equals non-causal cognition, we know from [START_REF] Rini | Inner and outer bounds for the Gaussian cognitive interference channel and new capacity results[END_REF] that the capacity region does not have bounds on 2R c + R p and R c + 2R p , i.e., there are no 'resource holes'. Therefore the question can be rephrased as: is there a minimum strength of the cooperation link C above which unilateral causal cooperation results in no 'resource holes' in weak interference, i.e., bounds on 2R c + R p and R c + 2R p are not needed to (approximately) characterize the capacity region?

E. Regime 5 (weak interference): same gDoF as in the noncooperative IC

In regime 5 the interference is moderately weak (α ∈ [1/2, 2/3)) and the cooperation is fairly weak (0 ≤ β < 2α -1). The gDoF upper bound gives d = α as for the noncooperative IC. Hence in this regime we use the scheme that is approximately optimal for the sum-capacity of the noncooperative IC, with noncooperative common and private messages and with power splits as in [START_REF] Etkin | Gaussian interference channel capacity to within one bit[END_REF]. The noncooperative scheme achieves

R p + R c ≤ 2 log 1 + I + S max{1, I} -2 log(2). ( 15
)
The cooperative sum-rate upper bound in (4f) can be further upper bounded as

R p + R c ≤ 2 log 1 + I + S max{1, I} +2 log(2)+∆ ′ , ( 16
)
where ∆ ′ is the latest ∆ in (4g) in the regime β < 2α-1 ⇐⇒ C < I 2 /S ⇐⇒ S I < I C within the weak interference regime 1 ≤ S I , that is,

∆ ′ = max 1≤ S I < I C log       (1 + C) 1 + S I + I C 2 1 + S I + √ I 2       ≤ max 1≤ S I < I C log (1 + C) 1 + 2 S I + 2 I C 1 + S I + I = max 1≤ I C log (1 + C) 1 + 4 I C 1 + I C (1 + C) = log max (1 + C) 5 2 + C , (1 + C) 4 1 + C ≤ log(5),
where in the derivation we used 1 ≤ C (note that for C < 1 the outer bounds in (4) are to within a constant gap of the corresponding bounds for C = 0). Therefore, the gap (per user) is at most GAP ≤ eq(16)-eq (15) 2 [START_REF]On the capacity of full-duplex causal cognitive interference channels to within a constant gap[END_REF] and is achieved by the noncooperative scheme.

≤ (2+2) log(2)+log(5) 2 ≈ 3.16 log
The observations we made for regime 4, regarding possible extensions to the whole capacity region in the general case, apply to regime 5 as well.

F. Regime 6 (weak interference): improved gDoF compared to the noncooperative IC

In regime 6, the interference is quite weak (α < 2/3) and the cooperation exponent satisfies β ≥ [2α -1] + . Since the interference is weak, we split the messages into a common part and a private part, as for the noncooperative IC. For the CTx the two messages are noncooperative, but for the PTx the common message is cooperative and the private message is noncooperative. In other words, in regime 6 we extend the scheme used in regime 2 by adding a private message for each transmitter. The cooperation mechanism is based on decode-and-forward: at any given time slot of a block Markov coding scheme CTx decodes the primary common message, which PTx and CTx 'beam form' to the receivers in the next slot. The new common and private messages of each user are superposed to the old primary cooperative common message. The details of the achievable scheme are reported in Appendix B-C, where we show that the sum-rate in (55), namely

R p + R c ≤ min log 1 + S 2I + log S + I + 1 2 , log 1 + S 2I + log 1 + C I + C + log S + I 2 + I 2 , (17)
is achievable. Depending on which expression attains the minimum, we obtain the four subregions, indicated as from 6a to 6d, into which regime 6 is subdivided. In particular, for subregions 6a and 6b the tightest outer bound is the one in (4e), while for subregions 6c and 6d the tightest sum-rate outer bound is the one in (4f). Note that the outer bound in (4f) reduces to the more involved part of the W-curve of [START_REF] Etkin | Gaussian interference channel capacity to within one bit[END_REF] for α < 2/3 when β = 0. In Appendix B-D we show that this scheme is optimal to within 2.5 bits.

The achievable scheme used for regime 6 (defined as α < 2/3) is also optimal to within a constant gap for the most part of regime 4 (defined as α ∈ [2/3, 1)). In particular, as a consequence of the gap derivation in Appendix B-D, the achievable scheme for regime 6 and the outer bound in (4e) are to within a constant gap of one another when the interference is weak (α ≤ 1) and the cooperation satisfies β ≥ min{α, 1-α}.

The largest gap in regime 6 is of 2.5 bits in sub-regimes 6c and 6d, where the tightest sum-rate outer bound is the one in (4f). This gap may be decreased in several ways. For example, one can develop tighter bounds than the one in (4f), or develop more involved coding schemes. An example of the latter method can be found next, where we consider a DPCbased achievable scheme for the weak interference regime / regimes 4 and 6.

G. Regimes 4 and 6 (weak interference) with strong cooperation: capacity to within 1 bit with a cooperative scheme

We return on an observation made earlier, namely, that when the cooperation link gain C is sufficiently large, we expect the performance of the GCCIC to approach that of the noncausal cognitive IC. We next show that a DPC-based scheme is optimal to within 1 bit for the whole capacity region in the weak interference regime when the cooperation gain C is sufficiently strong, and we give a sufficient condition to quantify what 'sufficiently strong C' means.

In the DPC-based achievable scheme in Appendix C-C, the primary private message is cooperative, while in the scheme used previously for regime 6 in Appendix B-C it was noncooperative. Here we propose that CTx, with knowledge of PTx's primary private message, uses DPC to rid CRx of the interference due to the primary private message. In particular, PTx sends

X p = γ p S + 1 -|γ p | 2 U p , for some |γ p | 2 ≤ 1,
where S carries the PTx's old private cooperative message and U p carries the PTx's new private cooperative message in a block Markov coding scheme. CTx sends

X c = γ c S + 1 -|γ c | 2 U c , for some |γ c | 2 ≤ 1,
where U c carries the CTx's private noncooperative message. In a given time slot, CTx knows PTx's old private cooperative message S and decodes PTx's new private cooperative message U p from its channel output. CTx then precodes its private noncooperative message against the 'known interference' S; thanks to DPC, CRx decodes U c as if the interference S was not present [START_REF] Costa | Writing on dirty paper (corresp.)[END_REF], while treating U p as noise. PRx does backward decoding in order to recover its message while treating U c as noise. This DPC-based scheme is similar to the capacity achieving scheme for the non-causal cognitive IC in weak interference [START_REF] Wu | Capacity of a class of cognitive radio channels: Interference channels with degraded message sets[END_REF], [START_REF] Jovicic | Cognitive radio: An information-theoretic perspective[END_REF], except for the fact that now CTx must decode PTx's message in U p , and that CRx's equivalent noise variance includes the interference due to U p . To overcome this last problem, inspired by [START_REF] Etkin | Gaussian interference channel capacity to within one bit[END_REF], we choose the power split γ p in such a way that the interference created by U p at CRx is at the same level of the noise. With this choice of parameters the achievable region in (62), specialized to the symmetric case, becomes

I III-G : R p ≤ log 1 + C 1 + I , (18a)
R p ≤ log   1+ S+|γ c | 2 I+2|γ c | IS I 1+I 1 + (1 -|γ c | 2 )I   , (18b) R c ≤ log 1 + (1 -|γ c | 2 )S 1 + I 1+I , (18c)
for all |γ c | ≤ 1. Under the condition

C 1 + I ≥ max |γc|≤1 S + |γ c | 2 I + 2|γ c | IS I 1+I 1 + (1 -|γ c | 2 )I ⇐⇒ C ≥ S + I + 2 IS I 1 + I (1 + I) (⇐⇒ β ≥ 1 + α) (19)
the constraint in (18a) is redundant.

The achievable region under the condition in [START_REF] Gelfand | Coding for channel with random parameters[END_REF] must next be compared to an outer bound. We use here as an outer bound the capacity region of the non-causal cognitive IC given in [START_REF]LTE-A, 3rd Generation Partnership Project; Technical Specification Group Radio Access Network; Evolved Universal Terrestrial Radio Access (EUTRA)[END_REF]. By comparing (5a) with (18b), and (5b) with (18c), it is easy to see that for every value of |γ c | ≤ 1 the two regions are at most GAP ≤ log 1 + I 1+I ≤ log(2) = 1 bit away. This capacity result to within a constant gap holds for a sufficiently large C and it agrees with the intuition that the GCCIC should perform more and more as the non-causal cognitive IC as C increases.

If we only consider the sum-capacity, in Appendix C-D we show that the scheme in (62), of which the scheme in ( 18) is a special case, achieves the sum-capacity upper bound in (4e) to within 1 bit when the channel gains satisfy C ≥ S, that is, β ≥ 1, which is smaller than the gap of 1.5 bits we found with the superposition-based scheme. Note that the condition C ≥ S for sum-capacity approximate optimality is less restrictive than the one in [START_REF] Gelfand | Coding for channel with random parameters[END_REF] (which is approximately C ≥ 4S(1+I)) needed for the approximate optimality of the whole rate region.

We have now concluded the proof of Theorem 1. Before concluding this Section, we compare the gDoF performance of the symmetric GCCIC with that of other channel models so as to determine when unilateral cooperation may be worth implementing in practical systems.

H. Comparisons

When the gDoF, or high SNR throughput, is the desired performance metric, we can make the following observations:

• Causal unilateral source cooperation does not improve on the gDoF of the noncooperative IC when

α ∈ 2 3 , 2 or β ≤ min 1, [2α -1] +
as shown by the green and yellow-shaded regions in Fig. 3, that is, the regimes 1, 3, 4 and 5 in Fig. 2. For

α! β! 1 2 ! 2 3 ! 1! 2! 1! β =!2α -!1! β =!1! β =!α! β =!1!-!α! β =!α!-!1!
Fig. 3: Regions in which the gDoF of the symmetric GCCIC is equal to that of the noncooperative IC (green and yellow regions), of the relay channel (red and yellow regions), of the non-causal cognitive IC (region with horizontal lines), and of bilateral source cooperation (region with vertical lines). Note that the different regions can overlap.

this set of parameters, unilateral cooperation might not be worth implementing in practical systems since the same gDoF is achieved without explicit cooperation, i.e., unilateral cooperation only provides a power gain. • In the regime 1 ≤ α ≤ β, unilateral cooperation attains the gDoF of the classical relay channel given by d RC = max{1, min{α, β}} = α, as shown by the red and yellow-shaded regions in Fig. 3, i.e., parts of the regime 2 and regime 3 in Fig. 2 where d = α/2, which correspond to a subset of the strong interference where the cooperation link is greater than the interference link. For this set of parameters cognitive radio might not be worth implementing in practical systems since the rate R c = 0 for the cognitive pair is approximately sum-capacity optimal. There are however other rate pairs (R c , R p ) attaining the optimal sum-rate with R c > 0. • The gDoF of the GCCIC is equal to that of the noncausal cognitive IC, given by d = max{1α/2, α/2}, everywhere except in the regimes 5, 6c and 6d in Fig. 2, and for α ≥ max{2, β + 1}, as shown by the horizontalline-shaded region in Fig. 3. For this set of parameters unilateral cooperation attains the ultimate performance limits of non-causal cognitive radio and therefore represents the ideal channel condition for cognitive radio. • The gDoF of unilateral cooperation equals that of bilateral cooperation, with cooperation links of the same strength as considered in [START_REF] Prabhakaran | Interference channels with source cooperation[END_REF], when β ≤ 1 or β ∈ [α -1] + , α except in the regimes 6c and 6d in Fig. 2 as shown by the vertical-line-shaded region in Fig. 3. For this set of parameters unilateral cooperation attains the same gDoF of bilateral cooperation but with less resources and therefore represents a better trade-off in practical systems. • For the symmetric case, our analysis suggests that superposition coding is approximately optimal if either the interference is strong or the cooperation is strong; when both interference and cooperation are weak, then cooperation based on DPC coding is approximately optimal. Even when superposition coding is approximately optimal in weak interference, DPC coding might lead to a smaller gap. The DPC-based scheme is more complex to implement in practice than superposition coding; hence there might be an interesting practical trade-off between complexity and constant gap.

IV. THE CAPACITY REGION TO WITHIN A CONSTANT GAP

FOR THE GENERAL GCCIC We now focus on the general GCCIC, which is more complex to analyze due to the fact that one has to deal with 5 different channel parameters. Following the naming convention of the noncooperative IC, we say that the general GCCIC has strong interference if {S p ≤ I p , S c ≤ I c }, weak interference if {S p > I p , S c > I c }, and mixed interference otherwise. Moreover, we say that the general GCCIC has strong cooperation if C > S p and weak cooperation otherwise. As we shall see later in more details, this section provides a capacity characterization to within a constant gap for the general GCCIC when, roughly speaking, the two receivers do not experience weak interference simultaneously. As highlighted for the symmetric GCCIC, for this regime, outer bounds of the type 2R p + R c /R p + 2R c seem to be needed, whose derivation is outside the scope of this work. In particular, our main result for the general GCCIC is as follows:

Theorem 2 When C ≤ S p and S c S p ≤ (1+I p )(1+I c ) (denoted as "Case A" in Fig. 4), the capacity region of the general GCCIC is achievable to within 2 bits using a noncooperative scheme based on superposition coding, whose achievable rate region is 1 bit away from the region in (21) below. When S p < C ≤ I p (denoted as "Case B" in Fig. 4), the capacity region of the general GCCIC is achievable to within 1 bit using a cooperative scheme based on superposition coding, whose achievable rate region is given in [START_REF] Rini | Inner and outer bounds for the Gaussian cognitive interference channel and new capacity results[END_REF] below. In this strategy only the common message of PTx is sent cooperatively. When max{S p , I p } < C and S c ≤ αI c with α = min 1, 1+2Ip 1+Ip+Sp

(denoted as "Case C" in Fig. 4), the capacity region of the general GCCIC is achievable to within 1.8 bits using a cooperative scheme based on superposition coding, whose achievable rate region is given in [START_REF] Wu | Capacity of a class of cognitive radio channels: Interference channels with degraded message sets[END_REF] below. When S c > I c and C ≥ S p + I c + 2 S p I c Ip 1+Ip (1 + I p ), the capacity region of the GCCIC is achievable to within 1 bit with a cooperative scheme based on DPC and superposition coding, whose achievable rate region is the generalization of the region in [START_REF] Hekstra | Dependence balance bounds for singleoutput two-way channels[END_REF]. This strategy involves private messages only.

The rest of the section is devoted to the proof of Theorem 2. We divide the whole set of parameters depending on the strength of the cooperation link C compared to the direct link S p and the interference link I p . Fig. 4 shows the regimes of Theorem 2 for which we have an approximate capacity result (indicated as "Case A", "Case B" and "Case C" as in Theorem 2). As it can be noted from Fig. 4, our capacity characterization to within a constant gap roughly excludes the weak interference regime. The case when S c > I c and C ≥ S p + I c + 2 S p I c Ip 1+Ip (1 + I p ) is a straightforward generalization of the condition in [START_REF] Gelfand | Coding for channel with random parameters[END_REF] for the symmetric case studied in Section III-G and shall therefore not be further discussed. We shall now discuss each case separately.

A. The case C ≤ S p : when unilateral cooperation may not be useful

We start our discussion with a simple observation. Under the condition C ≤ S p we can further bound (4) as

O IV-A : R c ≤ log(1+S c ), (20a) R p ≤ log(1+S p ) + log(2), (20b)
R p +R c ≤ log + 1+S c 1+I c +log(1+S p +I c )+log(2), (20c)
R p +R c ≤ log + 1+S p 1+I p +log(1+S c +I p )+log(4). (20d)
The bounds in [START_REF] Cover | Capacity theorems for the relay channel[END_REF] are to within 1 bit of

I IV-A : R c ≤ log(1 + S c ), (21a) R p ≤ log(1 + S p ), (21b)
R p + R c ≤ log(1 + S p + I c ) + log + 1 + S c 1 + I c , (21c)
R p + R c ≤ log(1 + S c + I p ) + log + 1 + S p 1 + I p , (21d)
which is achievable to within 1 bit for the noncooperative IC when the 'R 1 + 2R 2 , 2R 1 + R 2 '-type of bounds in [16, Theorem 3] are redundant 2 ; with the notation adopted in this paper, one can easily show that these bounds are redundant if

S c S p ≤ (1 + I p )(1 + I c ). (22)
Hence we can immediately conclude that the noncooperative scheme of [START_REF] Etkin | Gaussian interference channel capacity to within one bit[END_REF] is optimal to within 2 bits in the regime 2 By using the 'worst noise covariance argument' as in [START_REF] Tuninetti | An outer bound region for interference channels with generalized feedback[END_REF], one can show that the upper bound in [16, Theorem 3], which was derived for the noncooperative IC in weak interference, is actually valid for all channel parameters if one replaces log 1+SNR i 1+INR j with log + 1+SNR i 1+INR j , i = j, i = 1, 2. By using the notation of [START_REF] Etkin | Gaussian interference channel capacity to within one bit[END_REF], the steps of the proof are as follows

n(R 1 + 2R 2 -3ǫ) ≤ I(X n 1 ; Y n 1 ) + I(X n 2 ; Y n 2 ) + I(X n 2 ; Y n 2 ) ≤ I(X n 1 ; Y n 1 , S n 1 ) + I(X n 2 ; Y n 2 , Y n 1 , X n 1 ) + I(X n 1 , X n 2 ; Y n 2 ) -I(X n 1 ; Y n 2 |X n 2 ) = I(X n 1 ; Y n 1 , S n 1 ) + I(X n 2 ; Y n 1 |X n 1 ) -I(X n 1 ; Y n 2 |X n 2 ) =h(Y n 1 |S n 1 )-h(Z n 1 ) + I(X n 2 ; Y n 2 |X n 1 , Y n 1 ) use worst noise covariance +I(X n 1 , X n 2 ; Y n 2 ).
identified by [START_REF] Wu | Partial decode-forward binning schemes for the causal cognitive relay channels[END_REF] when the cooperation link gain satisfies C ≤ S p . Notice that the regime in [START_REF] Wu | Partial decode-forward binning schemes for the causal cognitive relay channels[END_REF], depicted in Fig. 4 on the left, includes the strong interference regime and most of the mixed interference regime; in other words, it roughly excludes the weak interference regime.

The capacity result that we just proved is the generalization of the symmetric capacity result of Theorem 1 in Regime 1 and part of Regime 3 of Fig. 2 (i.e., in the symmetric case the condition in [START_REF] Wu | Partial decode-forward binning schemes for the causal cognitive relay channels[END_REF] simplifies to S ≤ 1 + I, which at high SNR corresponds to 1 ≤ α, and the condition C ≤ S at high SNR corresponds to β ≤ 1). As for Theorem 1 in the corresponding regime, a noncooperative scheme is approximately optimal.

When S c S p > (1 + I p )(1 + I c ) and C ≤ S p (which in the symmetric case corresponds to 1 > α and β ≤ 1 and for which we could only show a sum-capacity result to within a constant gap in Theorem 1) we expect that, in order to show an approximate capacity result, upper bounds on R p + 2R c and 2R p + R c must be derived.

B. The case S p < C ≤ I p : when unilateral cooperation is useful

For S p < C ≤ I p we further bound (4) as

O IV-B : R c ≤ log(1 + S c ), (23a) R p ≤ log(1 + C) + log(2), (23b)
R p +R c ≤ log + 1+S c 1+I c +log(1+S p +I c )+log(2), (23c)
R p + R c ≤ log (1 + S c + I p ) + 2 log(2). (23d)
In this regime, unilateral cooperation helps increasing the rate of the primary user. In the symmetric case, the upper bound in [START_REF] Mirmohseni | On the capacity of interference channel with causal and noncausal generalized feedback at the cognitive transmitter[END_REF] reduces to the part of Regime 2 and 3 of Fig. 2 for 1 < β ≤ α; we therefore consider the generalization of the achievable scheme we used for Regime 2 of Fig. 2

I IV-B : R c ≤ log(1 + S c ), (24a) R p ≤ log(1 + C), (24b)
R p + R c ≤ log(1 + S p + I c ) + log + 1 + S c 1 + I c , (24c)
R p + R c ≤ log(1 + S c + I p ). (24d)
By comparing the upper bound in [START_REF] Mirmohseni | On the capacity of interference channel with causal and noncausal generalized feedback at the cognitive transmitter[END_REF] with the achievable region in [START_REF] Rini | Inner and outer bounds for the Gaussian cognitive interference channel and new capacity results[END_REF] we conclude that the capacity region is known to within 1 bit for a general GCCIC where the channel gains satisfy S p < C ≤ I p . Notice that we did not impose any condition on the strength of I c compared to S c , i.e., in other words this gap result holds regardless of whether the interference at PRx is strong (I c ≥ S c ) or weak (I c < S c ).

C. The case max{S p , I p } < C and S c ≤ I c : when unilateral cooperation is useful For this case we further bound (4) as

O IV-C : R c ≤ log(1 + S c ), (26a) R p ≤ log(1 + C) + log(2), (26b) R p + R c ≤ log (1 + S p + I c ) + log(2), (26c)
R p +R c ≤ log 1+2C 1+I p +log (1+S c +I p )+log(2). (26d)
In this regime, unilateral cooperation helps increasing both the rate of the primary user and the sum-capacity. In the symmetric case, the upper bound in [START_REF] Suh | Feedback capacity of the Gaussian interference channel to within 2 bits[END_REF] reduces to the part of Regime 2 and 3 of Fig. 2 this is so because here we do not specify which one among S p and I p is the largest, and therefore the interference at CRx could be either strong or weak. This is exactly the strategy described in Appendix C-E, which is based on superposition coding only (as the cognitive common message is not precoded against the interference of the primary private message); both the common and the private messages of PTx are cooperative; this scheme can be thought of as the extension of the scheme used in Section IV-B so as to include a private message for PTx in case the interference at CRx is weak. The achievable region is given in (64). With the possible suboptimal choices |γ p | 2 = 1 1+Ip , |γ c | 2 = 1 1+Sc inspired by [START_REF] Etkin | Gaussian interference channel capacity to within one bit[END_REF], the achievable region in (64) becomes

I IV-C : R c ≤ log   1 + Ip 1+Ip + S c 1 + Ip 1+Ip + Sc 1+Sc   , (27a)
R p ≤ log (1 + C) , (27b) R p + R c ≤ log (1 + S p + I c ) , (27c)
R p +R c ≤ log 1+ C 1+I p +log   1+S c +I p 1+ Ip 1+Ip + Sc 1+Sc   , (27d) R p + R c ≤ log 1 + S p 1 + I p + I c 1 + S c + log   1 + S c + I p 1 + Ip 1+Ip + Sc 1+Sc   , (27e)
R p + 2R c ≤ log 1 + S p 1 + I p + I c + log   1 + S c + I p 1 + Ip 1+Ip + Sc 1+Sc   . (27f)
eq.(26c)eq.(27e

) ≤ log (1 + S p + I c ) + log(2) -log 1 + S p 1 + I p + I c 1 + S c -log   1 + S c + I p 1 + Ip 1+Ip + Sc 1+Sc   ≤ log (1 + S p + I c ) -log 1 + S p + I c 1 + max{I p , S c } -log (1 + S c + I p ) + log(6) = log 1 + S p + I c 1 + max{I p , S c } + S p + I c 1 + max{I p , S c } 1 + max{I p , S c } + min{I p , S c } + log(6) ≤ log(6) (25)
By comparing the upper bounds in [START_REF] Suh | Feedback capacity of the Gaussian interference channel to within 2 bits[END_REF] with the inner bounds in [START_REF] Wu | Capacity of a class of cognitive radio channels: Interference channels with degraded message sets[END_REF] it can be shown that they are at most

GAP ≤ max log(3), log(2), log(2) 2 , log(12) 2 , log (6) 2
= log(12) 2 ≈ 1.8 bits, bits away when the condition in (66) holds for the considered choice of parameters, namely

S c 1 + I p + S p 1 + 2I p ≤ I c (28)
so that the bound on R p + 2R c in ( 27) can be dropped. Notice that the sum-rate bound in (26c) and the one in (27e) are the same up to a constant gap, which is given by ( 25) at the top of the page.

The condition in ( 28) is similar to the condition in ( 22), which we derived in order to claim that bounds of the form R p + 2R c /2R p + R c were redundant in the noncooperative achievable region in the weak interference regime. In general, as can be noticed from the analysis so far, the weak interference regime is more challenging than the other regimes. In the next sections we concentrate on two special GCCIC where one of the interfering links is absent: the case where CRx does not experience interference (i.e., the so-called Zchannel for which I p = 0), and the case where PRx does not experience interference (i.e., the so-called S-channel for which I c = 0), for which we shall prove a constant gap result also in the weak interference regime. As we shall see, DPC-based schemes appear to be needed for approximate optimality in weak interference.

V. THE CAPACITY REGION TO WITHIN A CONSTANT GAP

FOR THE Z-CHANNEL Our main result for the Z-channel is as follows:

Theorem 3 The capacity region of the Z-channel (i.e., the link PTx→CRx is non-existent) is characterized to within 2 bits as follows. When C ≤ S p , the capacity region of the Zchannel is achievable to within 2 bits using a noncooperative scheme based on superposition coding, whose achievable rate region is 1 bit away from the region in (31) below. When C > S p and S c ≤ I c , the capacity region of the Z-channel is achievable to within 1 bit using a cooperative scheme based on superposition coding whose achievable rate region is given in (33) below. This scheme uses a private message for PTx and a common message for CTx. Finally, when C > S p and S c > I c , the capacity region of the Z-channel is achievable to within 1 bit using a cooperative scheme based on DPC and superposition coding whose achievable rate region is given in (35) below. This scheme uses only private messages for both PTx and CTx.

The rest of the section is devoted to the proof of Theorem 3, that is, the upper bound

R c ≤ log (1 + S c ) , (29a)
R p ≤ log 1 + S p + I c 2 , (29b)
R p ≤ log (1 + C + S p ) , (29c)
R p + R c ≤ log + 1+S c 1+I c +log 1+ S p + I c 2 , (29d)
from ( 4) by setting I p = 0, can be achieved to within a constant gap. The region in [START_REF] Sason | On achievable rate regions for the Gaussian interference channel[END_REF] without the bound in (29c) (i.e., the only one that depends on C) is the capacity upper bound for the non-causal cognitive IC in [24, Theorem III.1], which unifies previously known outer bounds for the weak (S c > I c ) and strong (S c ≤ I c ) interference regimes and is achievable to within 1 bit. Hence, we interpret the bound in (29c) as the 'cost' of causal cooperation on the Z-channel.

Moreover, as we shall see later in more details, the capacity region of the Z-channel, differently from that of the general GCCIC, does not have bounds of the type 2R p +R c /R p +2R c . This important feature allows for the characterization to within a constant gap of the whole capacity region for any value of the channel parameters.

For the proof of Theorem 3, we consider separately different parameter regimes. Given the result in Theorem 2, we only need to consider the case

I c ≤ S c (1 + S p ) (since S c S p -1 < S c (1 + S p )).
In the symmetric case, the regime I c ≤ S c (1 + S p ) is equivalent to I ≤ S(1 + S), or α ≤ 2 at high SNR, that is, we need to focus on the case where the Z-channel does not exhibit very strong interference.

A. Case C ≤ S p : when unilateral cooperation might not be useful

For the case C ≤ S p we further outer bound the capacity upper bound in [START_REF] Sason | On achievable rate regions for the Gaussian interference channel[END_REF] as

O V-A : R c ≤ log (1 + S c ) , (30a) R p ≤ log (1 + S p ) + log(2), (30b)
R p +R c ≤ log + 1+S c 1+I c +log (1+S p +I c )+log(2). (30c)
The region in (30) is at most 1 bit away from

I V-A : R c ≤ log (1 + S c ) , (31a) R p ≤ log (1 + S p ) , (31b)
R p + R c ≤ log + 1 + S c 1 + I c + log (1 + S p + I c ) , (31c)
which is achievable to within 1 bit by a noncooperative scheme [START_REF] Etkin | Gaussian interference channel capacity to within one bit[END_REF]. Therefore, for this set of parameters we have that the outer bound in (30) is achievable to within 2 bits.

The difference between the case C ≤ S p for the Zchannel and the corresponding case for the general channel in Theorem 2 in Section IV-A is that here we do not need to impose the condition in [START_REF] Wu | Partial decode-forward binning schemes for the causal cognitive relay channels[END_REF] to claim the redundancy of the bounds on R p + 2R c /2R p + R c in the noncooperative achievable region. This is so because those bounds do not matter, up to a constant gap of 1 bit, in the corresponding noncooperative IC [START_REF] Etkin | Gaussian interference channel capacity to within one bit[END_REF].

B. Case C > S p , S c ≤ I c (i.

e., strong interference at PRx): when unilateral cooperation is useful

In this case, we further outer bound the region in [START_REF] Sason | On achievable rate regions for the Gaussian interference channel[END_REF] as

O V-B : R c ≤ log (1 + S c ) , (32a) R p ≤ log (1 + C) + log(2), (32b) R p + R c ≤ log (1 + S p + I c ) + log(2). (32c)
In this regime, we use the same strategy employed for the general GCCIC in the same regime, i.e., for C > S p and I c ≥ S c in Fig. 4 Case C, by setting I p = 0. Here PTx takes advantage of the strong cooperation link and sends its message with the help of the CTx. Moreover, since the PTx does not create interference at the CRx (I p = 0), it sends a (cooperative) private message only. On the other hand, since the interference at the PRx is strong, the CTx sends a (noncooperative) common message only. This is exactly the strategy described in Appendix C-E and the resulting achievable region is given by (64) (this is the same achievable region we used in Section IV-C). In (64), we further set I p = 0 and |γ p | = 1 so that the PTx sends a private message only. With the possible suboptimal choice |γ c | 2 = 1 1+Sc , the achievable region in (64) becomes

I V-B : R c ≤ log 1 + S c 1 + Sc 1+Sc , (33a)
R p ≤ log (1 + C) , (33b) R p + R c ≤ log (1 + S p + I c ) , (33c)
R p +R c ≤ log 1+S p + I c 1+S c +log 1+S c 1+ Sc 1+Sc . (33d)
Notice that the bound on R p + 2R c in (64f) is always redundant because of the condition in (65) since here we set |γ p | = 1; this implies that the difference between this case for the Z-channel and the corresponding case for the general channel in Theorem 2 in Section IV-C is that here we do not need to impose the condition in [START_REF] Jovicic | Cognitive radio: An information-theoretic perspective[END_REF] to claim the redundancy of the bound on R p + 2R c in the achievable region.

It is not difficult to see that the outer bound in (32) and the inner bound in (33) are at most 1 bit away.

C. Case C > S p , S c > I c (i.

e., weak interference at PRx): when unilateral cooperation is useful

For this case, an outer bound for the Z-channel is given by the capacity of the non-causal cognitive IC in weak interference in [START_REF]LTE-A, 3rd Generation Partnership Project; Technical Specification Group Radio Access Network; Evolved Universal Terrestrial Radio Access (EUTRA)[END_REF] together with the cut-set bound in (4b), i.e.,

O V-C : R c ≤ log 1 + (1 -|γ c | 2 )S c , (34a)
R p ≤ log 1+ S p +|γ c | 2 I c +2|γ c | S p I c 1+(1 -|γ c | 2 )I c , (34b)
R p ≤ log (1 + C) + log(2), (34c)
union over all |γ c | ≤ 1. Since C > S p , PTx takes advantage of the strong cooperation link and sends its message with the help of the CTx. Moreover, since the PTx does not create interference at the CRx (I p = 0), it sends a (cooperative) private message only. The outer bound in (34b) suggests that the PRx should treat as noise the message of the CTx, while the bound in (34a) tells us that the CRx should decode its own message without experiencing interference. In order to model this last observation, we use a DPC-based scheme. In this strategy the CTx precodes its message against the 'known interference' so that the CRx decodes its own message as if the interference was not present [START_REF] Costa | Writing on dirty paper (corresp.)[END_REF]. This is exactly the strategy described in Appendix C-C and the resulting achievable region is given by (62) with I p = 0. We further set |γ p | = 0 in (62) and we obtain

I V-C : R c ≤ log 1 + (1 -|γ c | 2 )S c , (35a)
R p ≤ log 1 + S p + |γ c | 2 I c 1 + (1 -|γ c | 2 )I c , (35b)
R p ≤ log (1 + C) , (35c)
for all |γ c | ≤ 1. By simple computations, the achievable region in (35) can be shown to be at most 1 bit away from the upper bound in (34).

Note that here we used a DPC-based scheme in order to determine the capacity to within a constant gap in weak interference, while in Section IV-C for the general GCCIC we only used superposition coding.

D. Comparisons

We conclude this section by comparing the performance of unilateral cooperation on the Z-channel with other forms of cooperation. Moreover, we also consider whether the absence of an interfering link is beneficial in the GCCIC. We shall use as performance metric the gDoF, or high SNR throughput. In order to reduce the number of parameters, we restrict our attention to the case where the direct links have the same strength. For future reference, the gDoF of the noncooperative Z-channel is given by [START_REF] Sason | On achievable rate regions for the Gaussian interference channel[END_REF]

d IC-Z = min{max{1 -α/2, α/2}, 1}
and that of the non-causal cognitive Z-channel, which can be evaluated from [START_REF] Rini | Inner and outer bounds for the Gaussian cognitive interference channel and new capacity results[END_REF], is When comparing unilateral cooperation with other channel models in terms of gDoF we observe:

d CIC-Z = max{1 -α/2, α/2}.
• For the noncooperative IC, it is well known that removing an interference link cannot degrade the performance and the sum-capacity is known exactly for all channel parameters [START_REF] Sason | On achievable rate regions for the Gaussian interference channel[END_REF]. The same cannot be said in full generality for the cooperative channel because "useful cooperative information" can flow through the interference link. Thus for the Z-channel, cooperation only improves the gDoF with respect to the noncooperative case in the regime α ≥ 2 and β ≥ 1, i.e., in very strong interference and strong cooperation (the gDoF achieved with and without cooperation is the same in the green and yellow regions in Fig. 6). • For the Z-channel, unilateral cooperation attains the gDoF of the classical relay channel when 1 ≤ α ≤ β, as shown by the red and yellow-shaded regions in Fig. 6. • The Z-channel achieves the same gDoF of the non-causal cognitive channel everywhere except in α > max{2, β + 1} (region with horizontal lines in Fig. 6). • The gDoF of unilateral cooperation equals the gDoF upper bound of bilateral cooperation [START_REF] Prabhakaran | Interference channels with source cooperation[END_REF] when β ≤ max{1, α} (region with vertical lines in Fig. 6) that corresponds to the case where the cooperation link is weaker than the best between the direct link and the interference link. In this case bilateral cooperation might not be worth implementing in practice. Notice that here we compare the (provably achievable) gDoF for the case of unilateral cooperation to an upper bound for bilateral cooperation. To the best of our knowledge, it has not been shown that the gDoF upper bound for the Z-channel with bilateral source cooperation is achievable, which we expect to be. • By comparing Fig. 2 and Fig. 5 we observe that the gDoF of the Z-channel is always greater or equal than that of the interference-symmetric GCCIC. This is due to the fact that the PTx does not cooperate in sending the cognitive signal. Therefore by removing the link between PTx and CRx we rid CRx of only an interfering signal and this leads to an improvement in gDoF.

The regimes where the Z-channel strictly outperforms the interference-symmetric GCCIC are when 0 ≤ α ≤ 2 3 and β ≤ min{α, 1 -α} (region with vertical lines in Fig. 9), i.e., weak interference and fairly weak cooperation. This regime can be thought of as the one where interference is the most harmful for the interference-symmetric GCCIC.

VI. THE CAPACITY REGION TO WITHIN A CONSTANT GAP FOR THE S-CHANNEL

Our main result for the S-channel is as follows:

Theorem 4 The capacity region of the S-channel (i.e., the link CTx→PRx is non-existent) is characterized to within 2 bits as follows. When C ≤ max{I p , S p }, the capacity region of the Schannel is achievable to within 2 bits using a noncooperative

α! β! 1! 2! 1! β =!α! β =!α!!-!1!
Fig. 6: Regions in which the gDoF of the Z-channel is equal to that of the noncooperative IC (green and yellow regions), of the relay channel (red and yellow regions), of the non-causal cognitive IC (region with horizontal lines), and of bilateral source cooperation (region with vertical lines). Note that the different regions can overlap.

scheme based on superposition coding, whose achievable rate region is 1 bit away from the region in (38) below. When C > max{I p , S p }, the capacity region of the S-channel is achievable to within 2 bits using a cooperative scheme based on DPC and superposition coding. In particular, if C ≥ ∆ th with ∆ th = min{I p , S p }(1 + max{I p , S p }), the proposed scheme uses only private messages for both PTx and CTx and the achievable rate region is given in (40) below, while if C < ∆ th the scheme uses common and private messages for PTx and the achievable rate region is given in (42) below.

As for the Z-channel, also the capacity region of the Schannel, differently from that of the general GCCIC, does not have bounds of the type 2R p + R c /R p + 2R c . This feature allows for the characterization to within a constant gap of the whole capacity region for any value of the channel parameters.

The rest of the section is devoted to the proof of Theorem 4. We distinguish two cases, depending on whether the following upper bound

R c ≤ log (1 + S c ) , (36a) R p ≤ log (1 + S p ) , (36b)
R p + R c ≤ log 1 + ( S c + I p ) 2 + log 1 + C + max{I p , S p } 1 + I p , (36c)
from ( 4) with I c = 0, can be achieved with a noncooperative scheme or not. Note that the bounds on R p and R c in (36) are the capacity region of the corresponding non-causal cognitive IC; therefore we interpret the sum-rate bound in (36) as the 'cost' for causally learning the primary message at the CTx through a noisy channel.

For the proof of Theorem 4, we consider separately different parameter regimes. Given the result in Theorem 2, we should only consider the case I p ≤ S c S p -1 when C ≤ S p , and I p ≤ C when C > S p . However, here we will use a DPCbased scheme for the case max{S p , I p } < C for which we only used superposition coding in Section IV-C.

A. Case C ≤ max{I p , S p }: when unilateral cooperation might not be useful For the case C ≤ max{I p , S p } we can further bound (36) as

O VI-A : R c ≤ log (1 + S c ) , (37a) R p ≤ log (1 + S p ) , (37b) R p + R c ≤ log (1 + S c + I p ) + log + 1 + S p 1 + I p + 2 log(2). (37c)
The region in (37) is at most 1 bit away from

I VI-A : R c ≤ log (1 + S c ) , (38a) R p ≤ log (1 + S p ) , (38b)
R p + R c ≤ log (1+S c +I p )+log + 1+S p 1+I p , (38c) α! β
! 1 2 ! 2 3 ! 1! 2! 1 2 ! 1! 2! β = α ! d(α,!β)!=!1! gap!=!2!bits! d(α,!β)!=! ! 2 ! gap!=!2!bits! d(α,!β)!=! ! 2 ! gap!=!2!bits! d(α,!β)!=!1 - ! 2 ! gap!=!2!bits! d(α,!β)!=! !!!!! 2 ! gap!=!2!bits! d(α,!β)!=!1! gap!=!2!bits! β = α + ! 1 ! d(α,!β)!=!1! gap!=!2!bits! 4" 1" 1" 3" 3" 2" 2"
Fig. 7: Optimal gDoF and constant gap for the S-channel in the different regimes of (α, β).

which is achievable to within 1 bit by a noncooperative scheme [START_REF] Etkin | Gaussian interference channel capacity to within one bit[END_REF]. Therefore we conclude that for C ≤ max{I p , S p } a noncooperative scheme is optimal to within 2 bits.

As for the Z-channel, the difference between this case and the corresponding case for the general GCCIC in Theorem 2 is that here we do not need to impose extra conditions to claim the redundancy of the bounds on R p + 2R c /2R p + R c in the noncooperative achievable region since those bounds do not matter, up to a constant gap, in the noncooperative IC [START_REF] Etkin | Gaussian interference channel capacity to within one bit[END_REF].

B. Case C > max{I p , S p }: when unilateral cooperation is useful

When C > max{I p , S p }, a sufficient condition for the sumrate upper bound in (36) to be redundant is that

1 + S p ≤ 1 + C + max{I p , S p } 1 + I p ⇐⇒ C ≥ min{I p , S p }(1 + max{I p , S p }). (39)
For the set of parameters in (39), we use the achievable region in (62) from Appendix C-C, adapted to the S-channel case by setting I c = 0, and with |γ c | = 0, C(1 -|γ p | 2 ) = S p , to obtain the following achievable region

I VI-B : R c ≤ log 1 + S c 1 + SpIp C , (40a)
R p ≤ log (1 + S p ) . ( 40b
)
By comparing the rate bounds in (40) with those in (36), we see that when (39) holds the gap is at most 1 bit since

log (1 + S c ) -log 1 + S c 1 + SpIp C ≤ log 1 + S p I p C ≤ log 1 + min{I p , S p } max{I p , S p } min{I p , S p }(1 + max{I p , S p }) ≤ log(2).
This shows that, when the condition in (39) holds, not only the upper bound is achievable to within 1 bit but we can also achieve to within 1 bit the ultimate capacity of the corresponding non-causal cognitive channel. This result agrees with the intuition that, as the strength of the cooperation link increases, the performance of the causal cognitive channel should approach that of the corresponding non-causal model. The condition in (39) can thus be interpreted as a sufficient condition on the strength of the cooperation link to achieve the capacity region of the corresponding non-causal model to within a constant gap.

We are now left with the case

max{I p , S p } < C, C < min{I p , S p }(1 + max{I p , S p }) ⊆ {S p < C < S p (1 + I p )} . (41)
In the regime S p < C < S p (1 + I p ) we use the DPC-based achievable scheme in Appendix C-F. In this scheme CTx sends a private message only since X c is not received at PRx; PTx sends a private and a common message, both with the help of CTx. The PTx's common message is forwarded by CTx

α! β! 1! 2! 1! 2! β =!α +!1! β =!α!
Fig. 8: Regions in which the gDoF of the S-channel is equal to that of the noncooperative IC (green region), of the non-causal cognitive IC (region with horizontal lines), and of bilateral source cooperation (region with vertical lines). Note that the different regions can overlap.

to facilitate decoding at both receivers. The PTx's private message is decoded at CTx and its effect is 'pre-canceled' at CRx thanks to DPC. The achievable region is given by (68) in Appendix C-F, namely

I VI-B : R p ≤ log(1 + S p ), (42a)
R c ≤ log   1 + S c 1 + Ip 1+Ip   , (42b)
R p + R c ≤ log   1 + S c + I p 1 + S c + Ip 1+Ip C Sp   +log 1 + C 1 + I p + log   1 + S c 1 + Ip 1+Ip   . (42c)
In Appendix C-F we show that the achievable region in (42) is optimal to within 2 bits when S p < C < S p (1 + I p ).

Note that here we used a DPC-based scheme in order to determine the capacity to within a constant gap in weak interference, while for the general GCCIC we only used superposition coding. Also, the choice of parameters in Appendix C-F is unconventional, i.e., not inspired by [START_REF] Etkin | Gaussian interference channel capacity to within one bit[END_REF], and might be necessary to show an approximate capacity result in weak interference for the general GCCIC.

C. Comparisons

We conclude this section by comparing the performance of unilateral cooperation on the S-channel with other forms of cooperation. In order to reduce the number of parameters, we restrict our attention to the case where the direct links have the same strength. For future reference, the gDoF of the noncooperative S-channel is given by [START_REF] Sason | On achievable rate regions for the Gaussian interference channel[END_REF] d IC-S = min{max{1α/2, α/2}, 1} and that of the non-causal cognitive S-channel is given by [START_REF] Rini | Inner and outer bounds for the Gaussian cognitive interference channel and new capacity results[END_REF] d CIC-S = 1. Fig. 7 shows the gDoF and the gap for the S-channel in the (α, β) plane. The whole set of parameters has been partitioned into multiple sub-regions depending upon different levels of cooperation (β) and interference (α) strengths. We observe:

• Unilateral cooperation achieves the same gDoF of the noncooperative IC when α ≥ 2 or β ≤ max{1, α} (green region in Fig. 8). In other words, unilateral cooperation is worth implementing in practice when the interference is not very strong and the cooperation link is the strongest among all links. • The gDoF of unilateral cooperation never equals the gDoF of the relay channel. Actually when the link CTx→CRx is not present, the channel achieves d = 1 2 (since R c = 0) that is always smaller than the gDoF achieved when the link CTx→CRx exists, i.e., R c = 0.

α! β! 1 2 ! 2 3 ! 1! 2! 1 2 ! 1! 2! β =!2α!-!1! β =!α! β =!1!-!α! β =!α!!-!1!
Fig. 9: Regions in which the S-channel outperforms the symmetric GCCIC (green region), the symmetric GCCIC outperforms the S-channel (red region), the Z-channel outperforms the symmetric GCCIC (region with vertical lines). Note that the different regions can overlap.

• The S-channel achieves the same gDoF of the noncausal cognitive IC everywhere except in α ≤ 2 and β ≤ min{2, α+1} (region with horizontal lines in Fig. 8). • The gDoF of unilateral cooperation equals the gDoF upper bound of bilateral cooperation when α ≥ 2 and β ≤ 1 or when α ≤ 2 and β ≤ min{2, α + 1} (region with vertical lines in Fig. 8). • The S-channel outperforms the interference-symmetric GCCIC when either 0 ≤ α ≤ 2 3 and β ≤ min{α, 1 -α} or when α ≤ 2 and β ≥ max{1, α} (green region in Fig. 9). On the other hand, the interference-symmetric GCCIC outperforms the S-channel in very strong interference and strong cooperation, i.e., α ≥ 2 and β ≥ 1. This is due to the fact that the information for the PRx can no longer be routed through the CTx since √ I c e jθc = 0 (red region in Fig. 9).

VII. CONCLUSIONS

In this work we considered the CCIC, a network with two source-destination pairs sharing the same channel. In contrast to the noncooperative IC, in the CCIC the CTx exploits information about the PTx from its own channel observations. This scenario represents a more practically relevant model for cognitive radio than the non-causal cognitive IC, where the CTx is assumed to have a priori knowledge of the PTx's message. In particular, we believe that it is applicable in some practical heterogeneous deployments for 4G cellular networks.

We proposed achievable schemes that match known outer bounds to within a constant gap if, roughly speaking, the chan-nel does not exhibit weak interference at both destinations. We characterized the capacity region to within a constant gap for the case where one interfering link is absent, which includes cases of weak interference. From our analysis a practical guideline for system design is that superposition coding is approximately optimal when the interference at the primary receiver is strong and that binning / dirty paper coding is approximately optimal when the interference at the primary receiver is weak. We identified the set of parameters where causal cooperation achieves the same gDoF of the noncooperative IC and of the relay channel. We also highlighted under which channel conditions the gDoF achieved with bilateral source cooperation and with non-causal cognition equals that achieved with only unilateral causal cooperation.

APPENDIX A CAPACITY REGION UPPER BOUND AND GDOF UPPER

BOUND

In this work we use known outer bounds from [START_REF] Host-Madsen | Capacity bounds for cooperative diversity[END_REF], [START_REF] Tuninetti | An outer bound region for interference channels with generalized feedback[END_REF], [START_REF] Prabhakaran | Interference channels with source cooperation[END_REF]. These bounds were developed for the case of bilateral source cooperation. Here we adapt them to the case of unilateral source cooperation.

A. Cut-set upper bounds

The cut-set upper bound for a relay channel with gain S on the link from the source to the destination, gain C on the link from the source to the relay, and gain I on the link from the relay to the destination is upper bounded by [START_REF] Gamal | Network Information Theory[END_REF]

The behavior of the rate r (RC) (S, I, C) in (43) at high SNR, with I = S α , C = S β , is given by (6d).

For an IC with cooperative sources, the rate of a given source cannot be larger than the rate that this source can achieve when the other source acts as a pure relay. Therefore, for the GCCIC, we have

R p ≤ r (RC) (S p , I c , C) (44)
R c ≤ r (RC) (S c , I p , 0) (45)
which are the upper bounds on the individual rates in (4a) and (4b), which imply the sum-rate upper bound in (4d).

B. Sum-rate bounds from [START_REF] Tuninetti | An outer bound region for interference channels with generalized feedback[END_REF] From [START_REF] Tuninetti | An outer bound region for interference channels with generalized feedback[END_REF] we have

R p + R c ≤ max |ρ|≤1 log 1 + 1 -|ρ| 2 (C + max{I p , S p }) 1 + (1 -|ρ| 2 ) I p + log 1 + I p + S c + 2|ρ| S c I p ≤ log 1 + C + max{I p , S p } 1 + I p + log 1 + ( I p + S c ) 2 .
By swapping the role of the users, we obtain a similar sumrate upper bound, and the combination of the two gives the sum-rate upper bound in (4e). The function

r (DT) (S, I, C) := log 1 + C + max{S, I} 1 + I + log 1 + ( √ I + √ S) 2
with I = S α , C = S β , has the high SNR behavior given by (6e).

C. Sum-rate bound from [START_REF] Prabhakaran | Interference channels with source cooperation[END_REF] From [START_REF] Prabhakaran | Interference channels with source cooperation[END_REF] we have the sum-rate upper bound reported in (4f), whose behavior at high SNR, with the parameterization in (2), gives (6f).

APPENDIX B ACHIEVABLE SCHEMES BASED ON SUPERPOSITION CODING ONLY

A. Superposition-only Achievable Scheme

We specialize the 'superposition only' achievable scheme in [START_REF] Yang | Interference channel with generalized feedback (a.k.a. with source cooperation): Part i: Achievable region[END_REF]Thereom IV.1] to the case of unilateral cooperation. In [10, Thereom IV.1], the network comprises four nodes numbered from 1 to 4; nodes 1 and 2 are sources and nodes 3 and 4 destinations; source node j ∈ [1 : 2], with input to the channel X j and output from the channel Y j , has a message W j for node j + 2; destination node j ∈ [3 : 4] has channel output Y j from which it decodes the message W j-2 .

Both users do rate splitting, where only the common message of user 1 is cooperative, while all other messages are noncooperative. We set

Q = V 2 , Y 1 = ∅, T 2 = X 2 , U 1 = ∅, T 1 = X 1 in [10, Thereom IV.1], i.e., then R 1 = R 11n + R 10c , R 2 = R 22n + R 20n
, to obtain a scheme that comprises: a cooperative common message (carried by the pair (Q, V 1 ) at rate R 10c ) for user 1, a noncooperative private message (carried by X 1 at rate R 11n ) for user 1, a noncooperative common message (carried by U 2 at rate R 20n ) for user 2 and a noncooperative private message (carried by X 2 at rate R 22n ) for user 2. Here Q carries the 'past cooperative common message', and V 1 the 'new cooperative common message' in a block Markov encoding scheme.

The set of possible input distributions is

P Q,V1,X1,U2,X2 = P Q P V1,X1|Q P U2,X2|Q . (46)
A schematic representation of the achievable scheme is given in Fig. 10, where an arrow indicates superposition coding.

Regarding encoding. Source 2 cooperates with source 1 by using decode-and-forward in a block Markov coding scheme. In a given slot the old cooperative common message of source 1 is carried by Q, to which the new cooperative common message of source 1 is superposed and carried by V 1 , to which the noncooperative private message of source 1 is superposed and carried by X 1 . After source 2 decodes the new cooperative common message of source 1 carried by V 1 , with knowledge of Q and by treating the noncooperative private message of source 1 in X 1 as noise, it superposes its noncooperative common message carried by U 2 to the old cooperative common message of source 1 carried by Q, and then it superposes its noncooperative private message carried by X 2 . In this scheme the common messages are jointly (backward) decoded at all destinations while treating the nonintended private massage as noise.

Regarding decoding. There are three decoding nodes in the network and therefore three groups of rate constraints. These are:

• Node 2/CTx decodes V 1 from its channel output with knowledge of (Q, U 2 , X 2 ). Successful decoding is possible if (see [10, eq(6a)]

R 10c ≤ I(Y 2 ; V 1 |U 2 , X 2 , Q). (47a)
• Node 3/PRx jointly decodes (Q, V 1 , X 1 , U 2 ) from its channel output, with knowledge of some message indices in V 1 , by treating X 2 as noise. Successful decoding is possible if (see [10, eq(6b)-(6f)]

R 10c + R 20n + R 11n ≤ I(Y 3 ; Q, V 1 , X 1 , U 2 ) (47b) R 20n + R 11n ≤ I(Y 3 ; X 1 , U 2 |Q, V 1 ) (47c) R 11n ≤ I(Y 3 ; X 1 |Q, V 1 , U 2 ). (47d) • Node 4/CRx jointly decodes (Q, V 1 , U 2 , X 2 ) from its
channel output, with knowledge of some message index

Q(W 1c,t-1 ) X 1 (W 1c,t-1 ,W 1c,t ,W 1p,t ) U 2 (W 1c,t-1 ,W 2c,t ) X 2 (W 1c,t-1 ,W 2c,t ,W 2p,t ) V 1 (W 1c,t-1 ,W 1c,t )
Fig. 10: Achievable scheme based on superposition coding.

in V 1 , by treating X 1 as noise. Successful decoding is possible if (see [10, eq(7b)-(7f)]

R 10c + R 20n + R 22n ≤ I(Y 4 ; Q, V 1 , X 2 , U 2 ) (47e) R 20n + R 22n ≤ I(Y 4 ; X 2 , U 2 |Q, V 1 ) (47f) R 22n ≤ I(Y 4 ; X 2 |Q, V 1 , U 2 ). (47g)
The achievable region, after Fourier-Motzkin elimination, is given by [10, Thereom IV.1] R 1 ≤ eq(47b)

(48a) R 1 ≤ eq(47a) + eq(47d)

(48b) R 2 ≤ eq(47f)

(48c) R 1 + R 2 ≤ eq(47b) + eq(47g)

(48d) R 1 + R 2 ≤ eq(47e) + eq(47d)

(48e) R 1 + R 2 ≤ eq(47a) + eq(47c) + eq(47g)

(48f) R 1 + 2R 2 ≤ eq(47c) + eq(47g) + eq(47e)

for all distributions that factor as (46).

Remark 1. The rate bound in (48g) is redundant if min{eq(48d), eq(48e), eq(48f)} + eq(48c) ≤ eq(48g)

that is, if for the considered input distribution we have either eq(47b) + eq(47f) ≤ eq(47c) + eq(47e

) ⇐⇒ I(Y 3 ; Q, V 1 ) ≤ I(Y 4 ; Q, V 1 ), (49a)
or eq(47d) + eq(47f) ≤ eq(47c) + eq(47g

) ⇐⇒ I(Y 4 ; U 2 |Q, V 1 ) ≤ I(Y 3 ; U 2 |Q, V 1 ), (49b)
or eq(47a) + eq(47f) ≤ eq(47e)

⇐⇒ I(Y 2 ; V 1 |U 2 , X 2 , Q) ≤ I(Y 4 ; Q, V 1 ). ( 49c
)
Remark 2. If the private message of user 1 carried by X 1 is also decoded at Node 2 (a strategy that could be leading to a larger region than the one in (48) when the link between PTx and CTx is very large), then successful decoding at the cooperating source is possible if

R 1 = R 10c + R 11n ≤ I(Y 2 ; V 1 , X 1 |U 2 , X 2 , Q), (50a) R 11n ≤ I(Y 2 ; X 1 |V 1 , U 2 , X 2 , Q). (50b)
If we now do Fourier-Motzkin elimination of the region in (47), by replacing the constraint in (47a) with those in (50), we obtain a new achievable region where the bounds that depend on (47a) in (48) change as follows: the bound in (48b) is replaced by (50a), and the one in (48f) by R 1 + R 2 ≤ eq(47e)+eq(50b). In Appendix C we shall further improve on this scheme by using DPC to cancel the 'known interference' due to the private message decoded at the cooperating source.

B. Achievable Scheme 1: message 1 is common, and message 2 is split

By identifying Node1 with the PTx (i.e., X p = X 1 ), Node2 with the CTx (i.e., X c = X 2 , Y f = Y 2 ), Node3 with the PRx (i.e., Y p = Y 3 ) and Node4 with the CRx (i.e., Y c = Y 4 ), by further setting

Q = ∅, V 1 = X 1 (i.e., R 11n = 0, R 1 = R 10c )
in the scheme in (48) in Appendix B-A, the following region is achievable

R p ≤ I(Y f ; X p |U 2 , X c ) (51a) R c ≤ I(Y c ; U 2 , X c |X p ) (51b) R p + R c ≤ I(Y p ; U 2 , X p ) + I(Y c ; X c |U 2 , X p ) (51c) R p + R c ≤ I(Y c ; X p , U 2 , X c ) (51d)
for all input distributions that factor as P Xp,U2,Xc = P Xp P Xc,U2 .

In Gaussian noise, we choose X p , U 2 , L 2 to be i.i.d. N (0, 1), and

X c = γ c U 2 + 1 -|γ c | 2 L 2 for |γ c | ≤ 1.
With this choice of inputs, the channel outputs are

Y f = √ CX p + Z f Y p = S p X p + I c e jθc γ c U 2 + 1 -|γ c | 2 L 2 + Z p Y c = S c γ c U 2 + 1 -|γ c | 2 L 2 + I p e jθp X p + Z c
and the achievable region in (51) reduces to

R p ≤ log(1 + C) R c ≤ log(1 + S c ) R p + R c ≤ log(1 + S p + I c ) + log 1 + (1 -|γ c | 2 )S c 1 + (1 -|γ c | 2 )I c R p + R c ≤ log(1 + S c + I p ) for all |γ c | ≤ 1. If S c ≤ I c we choose |γ c | = 1 otherwise |γ c | = 0 to obtain R p ≤ log(1 + C) (52a) R c ≤ log(1 + S c ) (52b) R p + R c ≤ log(1 + S p + I c ) + log + 1 + S c 1 + I c (52c) R p + R c ≤ log(1 + S c + I p ). (52d)

C. Achievable Scheme 2: both messages are split

For the GCCIC we identify Node1 with the PTx (i.e., X p = X 1 ), Node2 with the CTx (i.e., X c = X 2 , Y f = Y 2 ), Node3 with the PRx (i.e., Y p = Y 3 ) and Node4 with the CRx (i.e., Y c = Y 4 ) in the scheme in (48) in Appendix B-A.

In Gaussian noise, in order to comply with (46), we choose

Q = ∅, V 1 , L 1 , U 2 , L 2 i.i.d. N (0, 1) and we let X c = γ c U 2 + 1 -|γ c | 2 L 2 : |γ c | 2 ≤ 1 X p = γ p V 1 + 1 -|γ p | 2 L 1 : |γ p | 2 ≤ 1.
With this choice of inputs the channel outputs are given by

Y f = √ C γ p V 1 + 1 -|γ p | 2 L 1 + Z f Y p = S p γ p V 1 + 1 -|γ p | 2 L 1 + I c e jθc γ c U 2 + 1 -|γ c | 2 L 2 + Z p Y c = S c γ c U 2 + 1 -|γ c | 2 L 2 + I p e jθp γ p V 1 + 1 -|γ p | 2 L 1 + Z c .
Inspired by [START_REF] Etkin | Gaussian interference channel capacity to within one bit[END_REF] for the noncooperative IC in weak interference, we set (1 -|γ c | 2 )I c = (1 -|γ p | 2 )I p = 1 (here we assume 1 ≤ min{I p , I c }) so that the scheme in (48) in Appendix B-A results in the following achievable region

R p ≤ log 1 + S p + I c 2 (53a) R p ≤ log 1 + C 1 + C/I p + log 1 + S p /I p 2 (53b)
R c ≤ log 1 + S c 2 (53c) R p + R c ≤ log 1 + S p + I c 2 + log 1 + S c /I c 2 (53d) R p + R c ≤ log 1 + S c + I p 2 + log 1 + S p /I p 2 (53e) R p + R c ≤ log 1 + C 1 + C/I p + log 1 + I c + S p /I p 2 + log 1 + S c /I c 2 (53f) R p + 2R c ≤ log 1 + I c + S p /I p 2 + log 1 + S c /I c 2 + log 1 + S c + I p 2 . ( 53g
)
Note that the sum-rate in (53d) and the first upper bound in (4e) differ by at most 3 bits, and the sum-rate in (53e) and the second upper bound in (4e) by at most 4 bits when C ≤ max{S p , I p }.

For the symmetric case, i.e., S c = S p = S, I c = I p = I, the following sum-rate is achievable from (53) R p + R c ≤ max min{ min{eq(48a), eq(48b)} + eq(48c), (54a) eq(48d), eq(48e), eq(48f), (54b) min{eq(48a), eq(48b)} + eq(48g)

2 We next show that the sum-rate in (54) is equal to the term in (54b). In order to show that the term in (54a) is redundant, consider the following facts:

• eq(48a) + eq(48c) is always greater than eq(48d) because S ≥ S I , since we assume I ≥ 1; • eq(48b) + eq(48c) is always greater than eq(48f) since 2I + SI ≥ S + I 2 + I ⇐⇒ S ≥ I, which is always satisfied since we are in the weak interference regime. In order to show that the term in (54c) is redundant, consider the following facts:

• the bound eq(48a)+eq(48g) 2 is always bigger than eq(48d) since we assume I ≥ 1 and it is therefore redundant;

• the bound eq(48b)+eq(48g) 2 is equal to eq(48e)+eq(48f) 2 and hence it is redundant. Therefore we conclude that in the weak interference regime 1 ≤ I ≤ S the sum-rate in (54) is equal to (54b) and, since eq(48e) is equal to eq(48d), is given by

R p + R c ≤ min log 1 + S 2I + log S + I + 1 2 , (55a) log 1 + S 2I +log 1 + C I + C +log S + I 2 + I 2 . (55b)
For future use, the term in (55b) is the smallest term if

(S + I + 1)(I + C) ≥ S + I 2 + I + SC + CI 2 + CI ⇐⇒ S ≥ C(I + 1).
D. Constant gap result for the sum-capacity of the symmetric GCCIC in Regime 6 of Fig. 2 We analyze the regime I p = I c = I < S p = S c = S. Parameter Range: S(S + I) > I 2 (I + 1) and C ≥ I 2 S . In order to find the tightest upper bound we need to split this region in different subregions, namely:

• Regime 6a) S < C (I + 1): here the tightest gDoF upper bound gives d(α, β) ≤ 1 -α 2 ;

• Regime 6b) S ≥ C (I + 1) and C ≥ I: here the tightest gDoF upper bound gives d(α, β) ≤ 1 -α 2 ;

• Regime 6c) S ≥ C (I + 1), I 2 ≤ S and C < I: here the tightest gDoF upper bound gives

d(α, β) ≤ 1 -α + β 2 ;
• Regime 6d) S ≥ C (I + 1), I 2 > S, C < I and S(S + I) > I 2 (I + 1): here the tightest gDoF upper bound gives

d(α, β) ≤ 1 + β 2 .
Inner Bound: We use the achievable scheme developed in Appendix B-C whose achievable sum-rate is given in (55), which in the weak interference regime (i.e., α ≤ 1) implies that the following gDoF is achievable

d(α, β) ≥ 1 2 min{[1 -α] + + max{1, α}, [1 -α] + + β -max{α, β} + max{1, 2α}} =    1 -α/2 for β ≥ min{α, 1 -α} 1 -α + β/2 for β < α, α ∈ [0, 1/2] (1 + β)/2 for β < 1 -α, α ∈ [1/2, 1] . ( 56
)
This shows the achievability of the gDoF upper bound in Regime 6 of Fig. 2. Actually, the proposed scheme is gDoF optimal in the whole weak interference regime α ≤ 1 except for β ≤ min{1α, [2α -1] + }, where a noncooperative scheme is gDoF optimal.

Outer Bound: For the regime β ≥ min{α, 1 -α}, where d(α, β) ≤ 1α/2 (regimes 6a and 6b), we use the upper bound in (4e); otherwise (regimes 6c and 6d) we use the upper bound in (4f).

Gap: We analyze separately the different sub regimes:

• Regime 6a) For the regime S < C(1 + I) within I ≤ S GAP ≤ eq(4e)eq(55a)

≤ log 1 + S 1 + I + log 1 + ( √ S + √ I) 2 -log 1 + S 2I -log S + I + 1 2 ≤ 2 log(2) + max 1≤I≤S log 1 1 + I 1 + S 1 + S 2I = 2 log(2) + max 1≤I log 2I 1 + I = 3 log(2).
• Regime 6b) For the regime S ≥ C(I + 1) and C ≥ I GAP ≤ eq(4e)eq(55b)

≤ GAP ≤ eq(4f)eq(55b) ≤ 5 log(2), by following exactly the same steps as done for Regime 6c) above. This shows the achievability of the sum-capacity upper bound to within a constant gap of 2.5 bits (per user) in this regime.

I 2 ≤ S GAP ≤ eq(4f) -eq(55b) ≤ log   1 + √ S √ I + √ I 2   + log (1 + C) + log   1 + √ S √ I + √ I √ C 2   -log 1 + S 2I -log 1 + C I + C -log S + I 2 + I 2

APPENDIX C ACHIEVABLE SCHEMES BASED ON SUPERPOSITION CODING AND DPC

A. DPC-based Achievable Scheme

We specialize the 'binning+superposition' achievable scheme in [10, Section V]. In [10, Thereom V.1], the network redundant constraints)

R 10c + R 20n + R 22n ≤ I(Y 4 ; U 2 , T 2 , V 1 , Q)+ -(R ′ 20n + R ′ 22n ) (58h) R 20n + R 22n ≤ I(Y 4 ; U 2 , T 2 |V 1 , Q)+ -(R ′ 20n + R ′ 22n ) (58i) R 22n ≤ I(Y 4 ; T 2 |V 1 , Q, U 2 )-R ′ 22n . (58j)
From Remark 2 in Appendix B-A, after Fourier-Motzkin elimination of the achievable region in (58) where we take the constraints in (58a) and (58b) to hold with equality (i.e., R

′ 20n = I(U 2 ; S 1 |Q), R ′ 22n = I(S 1 ; T 2 |Q, U 2 )), we get R 1 ≤ eq(58e) (59a) R 1 ≤ eq(58c) (59b) R 2 ≤ eq(58i) (59c) R 1 + R 2 ≤ eq(58e) + eq(58j) (59d) R 1 + R 2 ≤ eq(58h) + eq(58g) (59e) R 1 + R 2 ≤ eq(58h) + eq(58d) (59f) R 1 + 2R 2 ≤ eq(58f) + eq(58j) + eq(58h) (59g)
for all distributions that factor as (57).

Remark 3. As done in Remark 1 in Appendix B-A, the rate bound in (59g) is redundant if min{eq(59d), eq(59e)} + eq(59c) ≤ eq(59g)

that is, if for the considered input distribution we have either eq(58e) + eq(58i) ≤ eq(58f) + eq(58h)

⇐⇒

I(Y 3 ; Q, V 1 ) ≤ I(Y 4 ; Q, V 1 ), (60a)
or eq(58g) + eq(58i) ≤ eq(58f) + eq(58j)

⇐⇒

I(Y 4 ; U 2 |Q, V 1 ) -I(U 2 ; S 1 |Q) ≤ I(Y 3 ; U 2 |Q, V 1 ). (60b)

B. DPC region for the Gaussian noise channel

We identify Node1 with the PTx (i.e., X p = X 1 ), Node2 with the CTx (i.e., X c = X 2 , Y f = Y 2 ), Node3 with the PRx (i.e., Y p = Y 3 ) and Node4 with the CRx (i.e., Y c = Y 4 ). For the Gaussian noise channel, in the achievable region in (59), we choose Q = ∅, we let S 1 , V 1 , Z 1 , U 2 , T ′ 2 to be i.i.d. N (0, 1), and

X p = |a 1 |e jθc S 1 + b 1 V 1 + c 1 Z 1 : |a 1 | 2 + |b 1 | 2 + |c 1 | 2 = 1, X c = |a 2 |S 1 + b 2 U 2 + c 2 T ′ 2 : |a 2 | 2 + |b 2 | 2 + |c 2 | 2 = 1, T 2 = T ′ 2 + λS 1 λ = S c |c 2 | 2 S c |c 2 | 2 + 1 + I p |c 1 | 2 I p e jθp e jθc |a 1 | + √ S c |a 2 | √ S c c 2 ,
where the choice of λ is so as to "pre-cancel" S 1 from Y c in decoding T 2 , i.e., so as to have

I(Y c ; T 2 |V 1 , Q, U 2 ) - I(S 1 ; T 2 |Q, U 2 ) = I(Y c ; T 2 |V 1 , Q, U 2 , S 1
). With these choices, the channel outputs are

Y f = √ C |a 1 |e jθc S 1 + b 1 V 1 + c 1 Z 1 + Z f , Y p = ( S p |a 1 | + I c |a 2 |)e jθc S 1 + S p (b 1 V 1 + c 1 Z 1 ) + I c e jθc (b 2 U 2 + c 2 T ′ 2 ) + Z p , Y c = ( I p e jθp e jθc |a 1 | + S c |a 2 |)S 1 + I p e jθp (b 1 V 1 + c 1 Z 1 ) + S c (b 2 U 2 + c 2 T ′ 2 ) + Z c ,
and the achievable region in (59) (notice that we have I(S 1 ; U 2 |Q) = 0 since U 2 is not precoded against S 1 ) becomes Remark 4. Motivated by the observation in [START_REF] Etkin | Gaussian interference channel capacity to within one bit[END_REF] that all terms that appear as noise should be at most at the level of the noise, we set

R p ≤ I(Y f ; Z 1 , V 1 |U 2 , T 2 , X c , S 1 , Q) = log 1 + C(|b 1 | 2 + |c 1 | 2 ) , R p ≤ I(Y p ; Q, V 1 , S 1 ,
|a 1 | = 0, |b 1 | 2 = I p 1 + I p , |c 1 | 2 = 1 1 + I p , |a 2 | 2 = I c 1 + I c 1 1 + S c , |b 2 | 2 = I c 1 + I c S c 1 + S c , |c 2 | 2 = 1 1 + I c
, so that the achievable region derived in this section is included into for either

R p ≤ log (1 + C) (61a) R p ≤ log (1 + S p + I c ) -log (2) (61b) R c ≤ log (1 + S c ) -2 log (2) (61c)
I(Y p ; V 1 ) ≤ I(Y c ; V 1 ) ⇐⇒ S p |b 1 | 2 1+S p |c 1 | 2 +I c ≤ I p |b 1 | 2 1+I p |c 1 | 2 +S c ⇐⇒ S p (1 + S c ) ≤ I p (1 + I c ), (61g)
or

I(Y c ; U 2 |V 1 ) ≤ I(Y p ; U 2 |V 1 ) ⇐⇒ S c |b 2 | 2 1+S c (1-|b 2 | 2 )+I p |c 1 | 2 ≤ I c |b 2 | 2 1+I c (1-|b 2 | 2 )+S p |c 1 | 2 ⇐⇒ S c 1 + I p + S p 1 + 2I p ≤ I c , (61h)
so that the bound on R p + 2R c is redundant (see conditions in (60)). In the regime C > max{S p , I p } (see Fig. 4 on the right) the gap would be 2 bits if one could neglect the sum-rate bound in (61e).

C. Achievable Scheme 3: both messages are private

From the general region in Section C-B, we set

= 1 2 [1 -α] + + 1 -[1 -β] + α < 1 ≤ β = 2 -α 2 .
This shows the achievability of the gDoF upper bound by means of (63a). By using the sum-capacity upper bound in (4e) under the condition S ≥ I and the achievable sum-rate in (63a) we obtain the following gap using S ≤ C. This example shows that an achievable scheme more complex than simple superposition coding, like a DPCbased one, can achieve a smaller gap.

E. Achievable Scheme 4: message 1 is split, and message 2 is common but not precoded

From the general region in Section C-B, we set

In the rate region in (64), the constraint on R p + 2R c becomes redundant if one of the conditions in (60) holds; in particular, if

I(Y p ; V 1 ) ≤ I(Y c ; V 1 ) ⇐⇒ (

Fig. 2 :

 2 Fig. 2: Optimal gDoF and constant gap for the symmetric GCCIC in the different regimes of (α, β).

S c S p - 1 S c S p - 1 S c ( 1 +Fig. 4 :

 1114 Fig.4: The regimes identified by Theorem 2 where capacity is known to within a constant gap (indicated as "Case A", "Case B" and "Case C").

Fig. 5 :

 5 Fig. 5: Optimal gDoF and constant gap for the Z-channel in the different regimes of (α, β).

Fig. 5

 5 Fig.5shows the gDoF and the gap for the Z-channel for different regions in the (α, β) plane. The whole set of parameters has been partitioned into multiple sub-regions depending upon different level of cooperation (β) and interference (α) strengths.When comparing unilateral cooperation with other channel models in terms of gDoF we observe:

2 ,

 2 log (1 + C + S) =: r (RC) (S, I, C).

≤ log 1

 1 S + I 2 + I +3 log(2) ≤ 5 log(2), where we upper bounded the gap by evaluating it for C = I, i.e., the maximum possible value for C, since the function is increasing in C. • Regime 6d) For the regime S ≥ C(I + 1), C < I, I 2 > S and S(S + I) ≥ I 2 (I + 1)

2 + log 1 + S c |c 2 | 2 1 + 2 + log 1 +S c |a 2 || 2 + log 1 + 2 + log 1 + S c |c 2 | 2 1 + 2 + 2 log 1 + S c |c 2 | 2 1 + 2 + log 1 +

 211212121121121 Z 1 , U 2 ) = log 1 + S p + I c + 2 S p I c |a 1 | 2 |a 2 | 2 1 + I c |c 2 | 2 , R c ≤ I(Y c ; U 2 , T 2 |V 1 , Q) -I(S 1 ; T 2 |Q, U 2 ) = I(Y c ; U 2 |V 1 , Q) + I(Y c ; T 2 |V 1 , Q, U 2 , S 1 ) = log 1+ S c |b 2 | 2 1+I p |c 1 | 2 +S c |c 2 | 2 +| I p e jθp e jθc |a 1 |+ √ S c |a 2 || I p |c 1 | 2 , R p + R c ≤ I(Y c ; T 2 |V 1 , Q, U 2 ) -I(S 1 ; T 2 |Q, U 2 ) + I(Y p ; Q, V 1 , S 1 , Z 1 , U 2 ) = log 1 + S p + I c + 2 S p I c |a 1 | 2 |a 2 | 2 1 + I c |c 2 | S c |c 2 | 2 1 + I p |c 1 | 2 , R p + R c ≤ I(Y c ; U 2 , T 2 , V 1 , Q) -I(S 1 ; T 2 |Q, U 2 ) + I(Y p ; S 1 , Z 1 |V 1 , Q, U 2 ) = log 1+ S c |b 2 | 2 +I p |b 1 | 2 1+I p |c 1 | 2 +S c |c 2 | 2 +| I p e jθp e jθc |a 1 |+ √ | S p |a 1 | + √ I c |a 2 || 2 + S p |c 1 | 2 1 + I c |c 2 | I p |c 1 | 2 , R p + R c ≤ I(Y c ; U 2 , T 2 , V 1 , Q) -I(S 1 ; T 2 |Q, U 2 ) + I(Y f ; Z 1 |U 2 , T 2 , X c , S 1 , Q, V 1 ) = log 1+ S c |b 2 | 2 +I p |b 1 | 2 1+I p |c 1 | 2 +S c |c 2 | 2 +| I p e jθp e jθc |a 1 |+ √ S c |a 2 || 2 + log 1 + S c |c 2 | 2 1 + I p |c 1 | 2 + log 1 + C|c 1 | 2 ,andR p + 2R c ≤ I(Y c ; T 2 |V 1 , Q, U 2 ) -I(S 1 ; T 2 |Q, U 2 ) + I(Y c ; U 2 , T 2 , V 1 , Q) -I(S 1 ; T 2 |Q, U 2 ) + I(Y p ; S 1 , Z 1 , U 2 |V 1 , Q) = log 1+ S c |b 2 | 2 +I p |b 1 | 2 1+I p |c 1 | 2 +S c |c 2 | 2 +| I p e jθp e jθc|a 1 |+ √ S c |a 2 || I p |c 1 | | S p |a 1 | + √ I c |a 2 || 2 + S p |c 1 | 2 + I c |b 2 | 2 1 + I c |c 2 | 2 .

RR

  p + R c ≤ log(1+S p +I c )+log 1+ S c 1+I c -2 log(2) (61d) R p + R c ≤ log (1 + I p + S c ) -3 log (p + R c ≤ log (1+I p +S c )+log 1+ C 1+I p -2log (2) (61f)

a 1 =

 1 γ p , b 1 = 0, c 1 = 1 -|γ p | 2 , |γ p | ≤ 1, a 2 = γ c , b 2 = 0, c 2 = 1 -|γ c | 2 , |γ c | ≤ 1, to obtain R p ≤ log 1 + C(1 -|γ p | 2 ) (62a) R p ≤ log 1 + S p + I c + 2|γ c ||γ p | S p I c 1 + (1 -|γ c | 2 )I c (62b) R c ≤ log 1 + (1 -|γ c | 2 )S c 1 + (1 -|γ p | 2 )I p (62c) for all (|γ c |, |γ p |) ∈ [0, 1] 2 .From (62) the following sum-rate is achievableR p + R c ≤ max (|γc|,|γp|)∈[0,1] 2 log 1 + (1 -|γ c | 2 )S c 1 + (1 -|γ p | 2 )I p + + min log 1 + C(1 -|γ p | 2 ) , log 1 + S p + |γ c | 2 I c + 2|γ c ||γ p | S p I c 1 + (1 -|γ c | 2 )I c .For the symmetric case, i.e., S c = S p = S, I c = I p = I, instead of solving analytically the optimization involved in the sumrate maximization, which does not seem to lead to a closedform expression, we choose to set|γ c | = 0 and (1-|γ p | 2 ) = 1 if C < S 1+I and (1 -|γ p | 2 ) = S C(1+I) otherwise (i.e., these values are not necessarily optimal). With these choices the following sum-rate is achievable R p + R c ≤ log 1 gap result for the sum-capacity of the symmetric GCCIC in Regimes 4 and 6 of Fig.2for α < 1 ≤ β With the DPC-based achievable scheme in Appendix C-C an achievable sum-rate is given by (63a), which we use to derive a smaller gap than those in Section III-D and Appendix B-D in the regime I < S and C ≥ S (parts of regimes 4 and 6 of Fig.2). The achievable sum-rate in (63a) implies d(α, β) ≤ lim S→∞

  GAP ≤ log 1 + S 1 + I + log (1 + S + I) + log(2)

a 1 =

 1 0, b 1 = 1 -|γ p | 2 , c 1 = γ p , |γ p | ≤ 1, a 2 = γ c , b 2 = 1 -|γ c | 2 , c 2 = 0, |γ c | ≤ 1, to obtain R p ≤ log (1 + C) (64a) R c ≤ log 1 + S c 1 -|γ c | 2 1 + |γ p | 2 I p + |γ c | 2 S c (64b) R p + R c ≤ log (1 + S p + I c ) (64c) R p + R c ≤ log 1 + |γ p | 2 C + log 1 + S c 1 -|γ c | 2 + I p 1 -|γ p | 2 1 + |γ p | 2 I p + |γ c | 2 S c (64d) R p + R c ≤ log 1 + |γ p | 2 S p + |γ c | 2 I c + log 1 + S c 1 -|γ c | 2 + I p 1 -|γ p | 2 1 + |γ p | 2 I p + |γ c | 2 S c (64e) R p + 2R c ≤ log 1 + |γ p | 2 S p + I c + log 1 + S c 1 -|γ c | 2 + I p 1 -|γ p | 2 1 + |γ p | 2 I p + |γ c | 2 S c .

1 -R 1 + 1 ⇐⇒ 1 + 1 +

 11111 |γ p | 2 )S p 1 + |γ p | 2 S p + I c ≤ (1 -|γ p | 2 )I p 1 + |γ p | 2 I p + S c ⇐⇒ either |γ p | = 1, or S p (1 + S c ) ≤ I p (1 + I c ),(65)or ifI(Y c ; U 2 |V 1 ) ≤ I(Y p ; U 2 |V 1 ) ⇐⇒ (1 -|γ c | 2 )S c 1 + |γ p | 2 I p + |γ c | 2 S c ≤ (1 -|γ c | 2 )I c 1 + |γ p | 2 S p + |γ c | 2 I c ⇐⇒ either |γ c | = 1, or S c 1 + |γ p | 2 S p 1 + |γ p | 2 I p ≤ I c . (66)F. Achievable Scheme 5: message 1 is split, and message 2 is private; gap for the S-channel From the region in Section C-B, we set c 2 = 1 to obtainR p ≤ log(1 + C(|c 1 | 2 + |b 1 | 2 )) (67a) R p ≤ log 1 + S p 1 + I c (67b) R c ≤ log 1 + S c 1 + I p |c 1 | 2 (67c) R p + R c ≤ log 1 + S c + I p 1 + I p (|a 1 | 2 + |c 1 | 2 ) + S c +log 1+ S c 1+I p |c 1 | 2 +log(1+C|c 1 | 2 ) (67d) R p + R c ≤ log 1 + S c + I p 1 + I p (|a 1 | 2 + |c 1 | 2 ) + S c + log 1 + S c 1 + I p |c 1 | 2 + log 1 + S p (|a 1 | 2 + |c 1 | 2 ) 1 + I c . (67e)An achievable region for the S-channel is obtained by setting I c = 0 in (67). Here we concentrate on the regime S p ≤ C ≤ (1 + I p )S p and evaluate the region in (67) for|a 1 | 2 = C -S p (1 + I p )S p , |b 1 | 2 = (1 + I p )S p -C (1 + I p )S p , |c 1 | 2 = 1 1 + I p .With these choices the region in (67) reduces toR p ≤ log(1 + S p ) p + R c ≤ log  S c + I p 1 + S c + on R p in (67a) would give R p ≤ log 1 S p ≤ C ≤ (1 + I p )S p ;notice also that the two sum-rate bounds in (67d) and in (67e) are the same.We next match the achievable region in (68) to the outer boundR p ≤ log (1 + S p ) (69a) R c ≤ log (1 + S c )(69b)R p + R c ≤ log 1 + ( S c + I p ) 2 + log 1 + C + max{S p , I p } 1 + I p(69c) from (4) with I c = 0. The bounds on R p in (68) and (69) are the same, and the bounds on R c in (68) and (69) are at most 1 bit apart. For the sum-rate, if C/S p ≤ S c (and recall that we focus on S p ≤ C) then GAP ≤ log 1 + ( S c + I p ) 2 C + max{S p , I p } 1 + C + I p ≤ log(2) + log max{C, I p } 1 + C + I p ≤ log(2) + 2 log(2) + log(2) = 4 log(2), while if C/S p > S c then GAP ≤ log(1 + S p ) + log(1 + S c )+ log   1 + S c + I p 1 + S c + S p )(1 + 2C/S p ) 1 + I p + C

  for 1 < α < β. Here PTx takes advantage of the strong cooperation link and sends its message with the help of the CTx. The sum-rate upper bound in (26c) suggests that PRx should decode the CTx's message in addition to its intended message, that is, CTx should use a (noncooperative) common message only; this is so because the condition S

c ≤ I c corresponds to strong interference at the PRx. The sum-rate upper bound in (26d), suggests that PTx should use both a (cooperative) common and a (cooperative) private message;

  S + I < S + I 2 + I, 1 ≤ I, and where we upper bounded the gap by evaluating it for C = I, i.e., minimum possible value for C, since the function is decreasing in C.

	log	1 + S 1 + I	+ log 1 + (	√	S +	√	I) 2 +
	-log 1 +	S 2I		-log	1 + C I + C	-log	S + I 2 + I 2
	≤ log	1 + S 1 + I	+ log	1 + S + I S + I 2 + I	+ 2 log(2)
	+ log	2I 2I + S	+ log	2I 1 + I
	= 4 log(2) + log	1 + S 2I + S	+ 2 log	I 1 + I
	+ log	1 + S + I S + I 2 + I	≤ 4 log(2)
	since 1 +

• Regime 6c) For the regime S ≥ C(I + 1), C < I and

In principle the system performance also depends on the phases of the interfering links (θc, θp). However, as far as gDoF and sum-capacity to within a constant gap are concerned, the phases (θc, θp) only matter if the IC channel matrix Sp √ Ice jθc Ipe jθp √ Sc is rank deficient, in which case one received signal is a noisier version of the other and the overall channels behave, sumcapacity-wise, as a Multiple Access Channel (MAC).

The work of D. Tuninetti was partially funded by NSF under award number 0643954; the contents of this article are solely the responsibility of the author and do not necessarily represent the official views of the NSF. The work of D. Tuninetti was possible thanks to the generous support of Telecom-ParisTech, Paris France, while the author was on a sabbatical leave at the same institution. Eurecom's research is partially supported by its industrial partners: BMW Group Research & Technology, IABG, Monaco Telecom, Orange, SAP, SFR, ST Microelectronics, Swisscom and Symantec. The research carried out at Eurecom leading to these results has received funding from the EU Celtic+ Framework Program Project SPECTRA. The research work carried out at Intel by U. Salim has received funding from the European Community's Seventh Framework Program grants SACRA (n • 249060) and iJOIN (n • 317941).

Fig. 11: Achievable scheme based on binning and superposition coding.

P Q,S1,V1,Z1,X1,U2,T2,X2 = P Q P V1|Q P S1|Q P Z1|Q,S1,V1 P U2,T2|S1,Q P X1|Q,S1,V1,Z1 P X2|Q,S1,U2,T2 .

comprises four nodes numbered from 1 to 4; nodes 1 and 2 are sources and nodes 3 and 4 destinations; source node j ∈ [1 : 2], with input to the channel X j and output from the channel Y j , has a message W j for node j + 2; destination node j ∈ [3 : 4] has channel output Y j from which it decodes message W j-2 .

Both users do rate splitting, where the messages of user 1 are cooperative while the messages of user 2 are noncooperative. In [10, Section V], we set

to obtain a scheme that comprises: a cooperative common message (carried by the pair (Q, V 1 ) at rate R 10c ) for user 1, a cooperative private message (carried by the pair (S 1 , Z 1 ) at rate R 11c ) for user 1, a noncooperative common message (carried by U 2 at rate R 20n ) for user 2 and a noncooperative private message (carried by T 2 at rate R 22n ) for user 2. Here the pair (Q, S 1 ) carries the 'past cooperative messages', and the pair (V 1 , Z 1 ) the 'new cooperative messages' in a block Markov encoding scheme. The channel inputs are functions of the auxiliary random variables, where X 1 is a function of

The set of possible input distributions is given by (57) at the top of the page. A schematic representation of the achievable scheme is given in Fig. 11, where a black arrow indicates superposition coding and a red arrow indicates binning.

Regarding encoding. The codebooks are generated as follows: first the codebook

. With this random coding codebook generation, the pair (U 2 , T 2 ) is independent of S 1 conditioned on Q. [10, Theorem V.1] involves several binning steps to allow for a large set of input distributions. Here, in order to simplify the scheme, we do not bin V 1 against S 1 ; the only binning steps are for (U 2 , T 2 ) against S 1 . We use a block Markov coding scheme to convey the message of user 1 to user 2. In particular, at the end of any given time slot in a block Markov coding scheme, encoder 2 knows (Q, S 1 , U 2 , T 2 ) and decodes (V 1 , Z 1 ) from its channel output; the decoded pair (V 1 , Z 1 ) becomes the pair (Q, S 1 ) of the next time slot; then, at the beginning of each time slot, encoder 2, by binning, finds the new pair (U 2 , T 2 ) that is jointly typical with (Q, S 1 ); for this to be possible, we must generate several (U 2 , T 2 ) sequences for each message of user 2 so as to be able to find one pair to send with the correct joint distribution with (Q, S 1 ); this entails the rate penalties in [10, eq(20)] for user 1 and then again [10, eq(20)] for user 2 by swapping the role of the subscripts 1 and 2, with

Regarding decoding. There are three decoding nodes in the network and therefore three groups of rate constraints. These are: [10, eq(21)] by swapping the role of the subscripts 1 and 2, with

• Node 3/PRx jointly decodes (Q, S 1 , U 2 ) from its channel output, with knowledge of some message indices in

, by treating T 2 as noise. Successful decoding is possible if (see [10, eq(22)] where only the bounds in [10, eq(22a)], [10, eq(22f)], and [10, eq(22g)] remain after setting several auxiliary random variables to zero and removing the redundant constraints)

• Node 4/CRx jointly decodes (Q, U 2 , T 2 ) from its channel output, with knowledge of some message index in V 1 , by treating Z 1 as noise (recall that the pair (U 2 , T 2 ) has been precoded/binned against S 1 ). Successful decoding is possible if (see [10, eq(22)], with the role of the users swapped, where only the bounds in [10, eq(22a)], [10, eq(22i)], and [10, eq(22k)] remain after setting several auxiliary random variables to zero and removing the