On the Sum-Capacity of Degraded Gaussian Multiaccess Relay Channels

1 On the Sum-Capacity of De graded Gaussian Multiacc ess Relay Channels Lalitha Sankar Member , IEEE, , Narayan B. Mandayam, Senior Member , IEEE , and H. V incent Poor F ell ow , IEEE Abstract The sum-capacity is studied for a K -user degraded Gaussian multiaccess relay ch annel (MARC) where the mu ltiaccess signal r ecei ved at the destination from th e K sources and relay is a degraded version of the signal received at the relay fr om all sources, giv en the transmit signal at the relay . An outer bound on the capacity region is d e veloped using cu tset bou nds. An ach ie vable rate region is obtained for the de code-and- forward (DF) stra tegy . It is shown that for every ch oice of inpu t d istrib ution, th e rate regions f or th e inner (DF) and ou ter bo unds are given by the intersection of two K -dimensional polymatr oids, one re sulting from the m ultiaccess link at th e relay and th e other from that at the destination. Although the inner and outer boun d rate regions are not identical in g eneral, for bo th cases, a c lassical result on the inter section of two po lymatroids is u sed to show that the inter section belongs to either the set of a ctive cases or ina ctive cases , wh ere the two bou nds on the K -u ser sum- rate are active or inacti ve, respectively . It is shown that DF achieves the capacity region fo r a class of degraded Gaussian MARCs in which the relay h as a high SNR link to the destina tion r elati ve to the multiaccess link from the source s to the relay . Otherwise, DF is shown to ach ie ve the sum- capacity for an a ctive class of degraded Gau ssian MARCs f or which the DF sum-rate is maximized by a polymatr oid intersection belong ing to the set o f active cases. This class is shown to include the class o f symmetric Gaussian MARCs where all user s transmit at the same power . The w ork of L. Sankar and N. B. Mandayam was supported in part by the Nati onal Science Foundation under Grant No. IT R- 0205362 . The work of H. V . Poor was supported by the National Science Foundation under Grants ANI-03-38807 and CNS- 06-25632 . The material in this paper was presented in part at the Information Theory and Applications W orkshop, S an Diego, CA, January 2006. L . Sankar and H. V . Poor are with Princeton Univ ersit y . N. B. Mandayam is wit h the WINLAB, Rutgers Univ ersity . 2 Index T erms Multiple-access relay chann el (MARC), outer b ounds, achievable strategies, Gaussian and degraded Gaussian MARC. I . I N T RO D U C T I O N The mul tiaccess relay channel (MARC) is a network i n which s e veral us ers (sources) commu- nicate with a singl e destination i n the presence of a relay [1]. The coding strategies d e veloped for the relay channel [2], [3] extend readily to the MARC [4], [5]. For example, the strategy of [3, theorem 1], now often called decode-and-forwar d (DF), has a relay that decodes user messages before forwarding them to the destination [4], [5]. Simil arly , the strategy in [3, theorem 6], now often called compre ss-and-forwar d (CF), has the relay quanti ze its output sym bols and t ransmit the resulting quantized bit s to the destinati on [5]. Capacity result s for relay channels ar e kno wn only for a fe w special cases such as t he class of d egra ded relay channels [3] and its m ulti-relay generalization [6], [7], the class of semi- deterministic relay channels [8], the class of orthogonal relay channels [9], [10], t he class of Gaussian relay without delay channels [11], [12], and the cl ass of ergodic phase-fading relay channels [4]. For the class of degraded relay channels, the degradedness condition requires t hat the recei ved signal at the dest ination be independent of the source signal when conditi oned on th e transmi t and recei ve signals at the relay . For the Gaussian case, this sim pliﬁes to t he requirement that the si gnal recei ved at the destination be a noi sier version of that receiv ed at the relay conditi oned on the transmitted signal at the relay . This condit ion i mmediately suggests that requiring the relay to decode the source signals should be opti mal. In fa ct, for this class, applying this degradedness condition simpl iﬁes the cut-set outer b ounds to coincide with the DF bounds. For the MARC , we generalize this degradedness condition to requiring th at t he signal recei ved at the destin ation be independent of all source signals conditioned o n the transmit and recei ve si gnals at the relay . Applying this degradedness condition to t he cut set outer bounds for a MARC, howe ver , does not si mplify th e b ounds to those achiev ed b y DF . A K -user Gaussian MARC is degraded when the m ultiaccess signal received at the destin ation from the K sources and relay is a nois ier version of the signal receiv ed at the relay from all sources, given the transm it signal at the relay . For a K -user d egra ded Gaussi an MARC, we dev elop the DF rate region as an inner bo und on the capac ity region using Gaussian signaling 3 at the sources and relay . The outer b ounds on the capacity region are obtained by specializi ng the cut-set bounds of [13, Th. 14.10.1] t o the case of independent sources [14] and by applying the de gradedness condition. In f act, for each c hoice of input distribution, both the DF and the cutset rate regions are intersections of two mul tiaccess rate regions, one with the relay as the recei ver and the other with the d estination as the recei ver . In general, howe ver , th e inner and outer bounds dif fer in their input distributions as well as the rate bounds. The outer bounds allow a more general dependence between the source and relay signals relative to DF where we use auxiliary rando m variables, one for each s ource, to relate the transmitted signals at the sources and relay . For the Gaussian degraded M ARC, we show t hat using Gauss ian input si gnals at th e sources and relay maximizes the ou ter bounds. For the inner b ounds, we use Gaussian signaling at the sources and the relay via K Gaussian auxiliary random variables. As a result, for each choice of the appropriate Gaussian in put distri b ution, both th e D F and ou ter b ounds are then parametrized by K source-relay cross-correlation coef ﬁcients , i.e., a K -length correlation vector . Speciﬁcally , each DF coef ﬁcient is a product of the t wo power fractions allo cated for cooperation at the corresponding so urce and the relay , respectiv ely . W e show that the DF rate region over all feasible c orrelation vectors is a conv ex region. On t he other hand, for t he outer bounds, all the rate bounds at the relay except for the bound on the K -user sum-rate are no n- conca ve functions of the correlation coef ﬁcients, and t hus, t he outer bound rate region requires time-sharing. F inally , we also show that for e very feasible choi ce of the correlation vec tor , the multiaccess re gions achie ved by the inner and out er boun ds at each recei ver are polymat roids, and the resulting region is an i ntersection of two p olymatroids. W e use a well-known resul t on the i ntersection of two pol ymatroids [15, chap. 46] t o broadly classify poly matroid intersections into two categories, namely , the set of active and t he set of inactive cases , depending o n whether the cons traints on t he K -user sum -rate at b oth receiver s are active or in acti ve, respectively . In fact, we use [15, chap. 46 ] to show that th e K -user sum- rate for the inactiv e cases is always bou nded by the min imum of the (inactive) K -user sum-rate bounds at each receiv er , and thus, by the l ar gest such b ound. For both the i nner and out er bounds, the intersection o f the two rate pol ymatroids results in either an acti ve or a inactiv e case for e very choice of correlati on vectors. In fact, th e mini mum of th e K -user s um-rate boun ds at t he relay and destination is the ef fective sum-rate only i f the polymatroid intersection i s an acti ve case and is strictl y an upper bou nd for a n inactive case. 4 Irrespecti ve of the above mentioned distin ction, we ﬁrst consider the problem of m aximizing the minimum o f the K -user sum -rate bounds at the relay and destination ov er the set of all correlation coefﬁcients. W e solve this max-mi n optimization problem using techniques analogous to the classi cal minimax p roblem of detection theory [16, II.C]. W e refer to a sum-rate optim al correlation vector as a max-min rule . For bot h the inner and o uter bounds, we s ho w that the max-min optim ization described above has two unique s olutions. The ﬁrst solutio n is gi ven by the maxim um K -user sum-rate achiev able at t he relay and results wh en the multiaccess link between the sources and the relay is t he bottle-neck link. For this case, we show that the int ersection of the rate regions at the relay and destination belo ngs to the set of activ e cases and is in fact t he same as the region achiev ed at the relay . W e further show that this re g ion is the same for both the in ner and outer bounds and is the capacity region for a class of degraded Gaussian MARCs where the source and relay powers satisfy the bot tle-neck condit ion for th is case. The second soluti on pertains to the case i n which the bottle-neck condition described above is not satisﬁed, i.e., t he K -user sum-rate at th e relay is at least as large as that at th e d estination. For thi s case, we show that for both the inner and outer bounds the max-min optim ization solution requires the K -user sum-rate b ounds at the relay and destinati on to be equal. In fact, we show th at both the inner and outer b ounds achie ve the same m aximum su m-rate for this case. Further , for both sets of bounds, we show that this maximum is achie ved by a set of correlation vectors, i.e., the max-min rul e i s a s et rather than a si ngleton. Recall, howe ver , that the sum-rate computed th us i s achie vable for either bound onl y if there exists at least one max-min rule for which the polym atroid intersection belongs to the set of active cases; otherwise, the computed maximum is strictly an u pper bound on the m aximum sum-rate. Combi ning this with the fact that the maxim um inner and ou ter K -user sum rate bound s for t his case are the s ame, we est ablish that DF achieve s the su m-capacity of an active class of degraded Gaussian MARCs, i.e., a class for which the maximum sum-rate is achieve d because t here exists at least one max-min rule for which the polymatroid intersection is an activ e case. W e also sho w that th e class of symmetric Gaussian M ARCs, in which all sources have the same power , belongs t o this active class. Finally , for the remaini ng inacti ve class of de graded Gaussian MARCs in which no acti ve case results for any choice of the m ax-min rule, we p rovide a common upper bo und on bo th the DF and the cutset sum-rates. 5 This paper is organized as follows. In Section II we present a model for a degraded Gaussian MARC. In Section III we dev elop t he cut -set b ounds on the capacity region of a MARC. In Section IV we determine the maxim um K -user DF sum-rate. W e discuss our results and conclude in Section V. I I . C H A N N E L M O D E L A N D P R E L I M I NA R I E S A K -user degraded Gaussian MARC has K u ser (source) nodes, one relay node, and one destination node (see Fig. 1). The sources emit the messages W k , k = 1 , 2 , . . . , K , which are statistically independent and take on values uniformly in the sets { 1 , 2 , . . . , M k } . The channel is used n times s o that the rate of W k is R k = B k / n bits per chann el use where B k = log 2 M k bits. In each use of the channel, the input to the channel from source k is X k while th e relay’ s input is X r . The channel out puts Y r and Y d , respectiv ely , at the relay a nd the desti nation are Y r = K X k =1 X k ! + Z r (1) Y d = K X k =1 X k ! + X r + Z d (2) = Y r + X r + Z ∆ (3) where Z r and Z ∆ are i ndependent Gaussian random v ariables with zero means and variances N r and N ∆ , respective ly , such that the noise variance at the destination is N d = N r + N ∆ . (4) W e assume that the relay operates in a ful l-duplex manner , i.e., it can transmit and receiv e simultaneous ly in the s ame bandwi dth. Further , i ts input X r in each channel use is a causal function of its output s from previous channel uses. W e write K = { 1 , 2 , . . . , K } for the set of sources, T = K ∪ { r } for the set of transmi tters, R = { r, d } for t he set of recei vers, X S = { X k : k ∈ S } for all S ⊆ K , and S c to denote the complement of S in K . The transmitted sign als from sou rce k and the relay have a per symbol power constraint E  | X k | 2  ≤ P k k ∈ T . (5) 6 Rela y d Y r Y r X 1 X 2 X r Z d r Z Z Z ∆ = + Fig. 1. A two-user Gaussian degraded MARC. One can e quiv alently e x press the relationsh ip between the input and output si gnals in (3) as a Markov chain ( X 1 , X 2 , . . . , X K ) − ( Y r , X r ) − Y d . (6) For K = 1 , (6) s impliﬁes to the degradedness conditio n in [3, (10)] for th e classic (single source) relay channel. A degraded Gaussian MARC is s ymmetric if P k = P , for all k . Thus , a class of symmetric DG-MARCs is characterized by four parameters, namely , P , P r , N r , and N d . The capacity region C MARC is the closure of the set of rate tupl es ( R 1 , R 2 , . . . , R K ) for which the destination can, for suf ﬁciently lar ge n , decode the K source messages with an arbitrarily small positive error p robability . A s further notati on, we write R S = P k ∈S R k and Y R = ( Y r , Y d ) . W e write 0 and 1 to denot e vectors whose entries are all zero and one, respecti vely , and C ( x ) = log(1 + x ) / 2 to denote the capacity of an A WGN channel with signal-to-noise ratio (SNR ) x . W e us e t he usual notation for entropy and m utual i nformation [13], [17] and take all logarithms to the base 2 so that in each channel use our rate uni ts are b its. I I I . O U T E R B O U N D S An outer bound on the capacity re gion of a MARC is presented in [14] using the cut -set bounds in [13, Th. 14.10.1 ] as applied to the case of independent sources. W e summarize the bounds below . Pr opos ition 1: The capacity region C MARC is contained i n the union of the set of rate tuples 7 ( R 1 , R 2 , . . . , R K ) that satisfy , for all S ⊆ K , R S ≤ min { I ( X S ; Y r , Y d | X S c , X r , U ) , I ( X S , X r ; Y d | X S c , U ) } (7) where the unio n is over all di stributions that factor as p ( u ) ·  Y K k =1 p ( x k | u )  · p ( x r | x K , u ) · p ( y r , y d | x K , x r ) . (8) Remark 1: The ti me-sharing random variable U ensures t hat the region i n (7) i s con vex. One can a pply Caratheodory’ s theorem [18] to this K -dimensio nal con vex region t o bound the cardinality of U as |U | ≤ K + 1 . Consider the outer bounds in Proposition 1. For a d e graded Gaussian MA RC appl ying t he degradness deﬁniti on in (6) simpliﬁes (7) as R S ≤ min { I ( X S ; Y r | X r X S c U ) , I ( X S X r ; Y d | X S c U ) } for all S ⊆ K (9) for the same joint distri b ution in (8). In the following theorem, we dev elop the bounds in (9) with U as a constant . For notational con venience, for a const ant U , we write B r, S and B d, S to denote the ﬁrst and second terms, respectiv ely , of the m inimum on the ri ght-side of (9). The proof of th e following th eorem is detailed in Appendi x I. Theor em 1: For a de graded Gauss ian MARC, t he bounds B r, S and B d, S are given by B r, S =              C  P k ∈S P k N r  P k ∈S c γ k = 1 C    P k ∈S P k N r − P k ∈S √ γ k P k ! 2 N r γ S c    otherwise (10) and B d, S = C   P k ∈S P k + γ S c P r + 2 P k ∈S √ γ k P k P r N d   (11) where γ S c = 1 − P k ∈S c γ k and √ γ k P k P r △ = E ( X k X r ) for all k ∈ K . (12) Remark 2: For K = 1 , t he bounds in (10) and (11) simplify to t he ﬁrst and second bound, 8 respectiv ely , for the degraded relay channel in [3, theorem 5]. Remark 3: The source-relay cross-correlation va riables γ k , for all k , satisfy (105), i.e., they lie in the closed con vex region Γ O B giv en by Γ O B = ( γ K : X k ∈K γ k ≤ 1 ) . (13) The bound B r, S in (10), in general, i s not a conca ve function of γ K for any S ⊂ K . For a ﬁxed γ S c , in Ap pendix IV we show that B r, S is a concave function of γ S . This in turn implies that B r, K is a concave function of γ K . Further , in Ap pendix III we show that for all S , B d, S in (11) is a concav e function of γ K . Remark 4: In the expression for B d, S in (11), the terms in volving the cross-corre lation coef- ﬁcients quantify the coherent c ombinin g gains that result from cho osing correlated source and relay signal s. On the other hand, the expression for B r, S in (10) q uantiﬁes the upper bo unds on the rate achiev able at the relay when on e or more source signals are correlated with the transmitted signal at t he relay . The rate region R O B enclosed by the cut-set outer bou nds is obtained as follows. From (119) for any choice of γ K , the rate region is an intersection of the regions enclosed by the bounds B r, S and B d, S for all S . Since B r, S is not a concave function of γ K , one m ust also consi der all possible con ve x combinations of γ K to obtain R O B . For the K -dimensional con vex region R O B , one can app ly Caratheodory’ s theorem [18 ] t o express every rate tu ple ( R 1 , R 2 , . . . , R K ) in R O B as a con vex combinati on of at most K + 1 rate tuples, where each rate tuple is obtained for a speciﬁc choice o f γ K . Let Θ denote the col lection of all ve ctors η that satis fy P K +1 m =1 η m = 1 (14) and l et ζ ≡ ( { γ K } K +1 ,η ) ∈ Γ K +1 O B × Θ deno te a collection of K + 1 power fractions and weights such that the rate tuple achiev ed by the m th vector γ ( m ) K is weighted by th e m th non-negati ve entry of the weigh t vec tor η , for all m ∈ K ∪ { K + 1 } . Finally , since Γ O B in (13) is a closed con vex set, P K +1 m =1 η m γ ( m ) K ∈ Γ O B . The following theorem presents an outer bou nd on the capacity region of the degraded Gaussian MARC. Theor em 2: The capacity region C MARC of a degra ded Gauss ian MARC is contained i n the 9 R  R              R  R  + R                 ! " # $ % & ' ( ) R * R + R , R - . / 0 1 2 3 4 5 6 7 8 9 : ; < = > ? @ A B C D E F G H I R J R K L M N O P Q R S T U V W X Y R Z R [ R \ + R ] ^ _ ` a b c d e f g h i j k l m n o p q R r Fig. 2. Fiv e possible i ntersections of R r and R d for a two-user Gaussian MARC. region R O B giv en as R O B = [ ζ ∈ Γ O B  R ob r  ζ  ∩ R ob d  ζ  (15) where the rate region R ob j  ζ  , j = r , d , is R ob j  ζ  =  ( R 1 , R 2 , . . . , R K ) : 0 ≤ R S ≤ B j, S  ζ  (16) and the bound B j, S is give n by B j, S  ζ  = K +1 X m =1 η m B j, S  γ ( m ) K  . (17) Theor em 3: The regions R ob r  ζ  and R ob d  ζ  are polymatroids . Pr oof : In Appendix II we show t hat for each cho ice o f input distribution satisfying (8), th e bounds in (51) are submodu lar set functi ons, i.e., they enclose regions that are polym atroids. For the optim al Gaussian i nput distribution, this im plies t hat R ob r  ζ  and R ob d  ζ  are polym atroids for ev ery choi ce of ζ . 10 The region R O B in (61) is a unio n of t he in tersections of the regions R ob r and R ob d , where the uni on is taken ov er all con vex combinations of γ K . Since R O B is conv ex, we obtain the boundary of R O B by maximizing the weighted sum P k ∈K µ k R k over all Γ O B and for all µ k > 0 . Speciﬁcally , we determine the sum-rate R K when µ k = 1 for all k . In general, to determine th e intersecting po lytope, one has to consider all possible polytope shapes for the re gions R ob r and R ob d . Howe ver , since R ob r and R ob d are polymatroid s, we use the foll o wing lemma on pol ymatroid intersections [15, p. 796, Cor . 46.1 c] t o broadly classify t he intersectio n of two polym atroids into two categories. The ﬁrst inactive set category i ncludes all intersections for which t he constraints on the two K -user sum-rates are not active. This implies that no rate tuple on the sum-rate p lane achie ved at one of the recei vers l ies within or on t he boundary of the rate region achieved at the other recei ver . On the oth er hand, the intersection s for which there exists at least one such rate tuple, i .e., the constrain ts on the two K -user sum -rates are activ e in the ﬁnal intersection, belong to th e category of active set . In Fig. 2, for a t wo-user M ARC we illustrate the ﬁv e possib le choices for the sum-rate resul ting from an in tersection of R ob r ( γ K ) and R ob d ( γ K ) . Cases 1 and 2 b elong to the in acti ve set while cases 3 a, 3 b , and 3 c belong to the active set. W e henceforth refer to mem bers of th e activ e and the inactiv e sets as active and inactiv e cases, respectively . Note t hat F ig. 2 il lustrates two s peciﬁc R ob r and R ob d polymatroids for cases 3 a , 3 b , and 3 c . In general the active set includes all intersections that satisfy th e deﬁnition for this set in cluding cases such as R ob r ⊆ R ob d and vi ce-v ersa. Finally , note that the sum-rate is a m inimum of t he sum-rates at the two recei vers only for the active cases 3 a , 3 b , and 3 c . For t he inactive cases 1 and 2 , t he R 1 + R 2 constraints are no longer acti ve and the su m-rate is g i ven by the bounds B r, { 2 } + B d, { 1 } and B r, { 2 } + B d, { 1 } , respective ly . W e use the foll o wing lemma on polymatroid intersections to generalize this observation and develop an ou ter b ound on the K -user sum-rate. Lemma 1: Let R S ≤ f 1 ( S ) and R S ≤ f 2 ( S ) , for all S ⊆ K , be two polym atroids such t hat f 1 and f 2 are nondecreasing submodul ar set functions on K with f 1 ( ∅ ) = f 2 ( ∅ ) = 0 . Th en max R K = min S ⊆K ( f 1 ( S ) + f 2 ( K\S )) . (18) From Lemma 1 we see that the maximum K -user sum-rate R K that results from the intersection of two poly matroids, R S ≤ f 1 ( S ) and R S ≤ f 2 ( S ) i s given by the minim um of the two K -user sum-rate planes f 1 ( K ) and f 2 ( K ) onl y if both the sum-rates are at most as lar ge as the sum of the orthogonal rate planes f 1 ( S ) and f 2 ( K\S ) , for all ∅ 6 = S ⊂ K . Further , the resulting intersection 11 belongs to the set of active cases. Con versely , when there exists at least one ∅ 6 = S ⊂ K for which the above con dition is not true, an inactive case results. Physically , an inactive case result s when a subset S of all users achiev e better rates at one of t he receiv ers while the remaining subset of users achieve a better rate at the oth er receiver . For such i nacti ve cases, the maxim um sum-rate in (18) i s the sum of two ort hogonal rate planes achiev ed by the two complementary subsets of users. As a result, the K -user sum-rate bounds f 1 ( K ) and f 2 ( K ) are no longer active for t his case, and thus, the region of intersection is no lo nger a polym atroid with 2 K − 1 faces. In th e following theorem we us e Lemma 1 to de velop the upper bound on the K -user sum- rate. For a Gaussian input dis trib ution, the polymatroi ds R ob r and R ob d are parametrized by ζ , and thus, Lemm a 1 applies for each choi ce of ζ . Theor em 4: For each ζ ∈ Γ O B , the m aximum K -user sum-rate R K resulting from the inter- secting polymatroi ds R ob r and R ob d is R K =    B d, A + B r, A c condition 1 min  B r, K , B d, K  otherwise (19) where B d, S and B r, S for all S are functions of ζ K and condit ion 1 is deﬁned for any ∅ 6 = A ⊂ K as B d, A + B r, A c < min  B r, K , B d, K  . (20) Remark 5: The condi tion in (20) determines whether the i ntersection of two polymatroids belongs to eit her t he set of acti ve or the set of inactiv e cases with respect to the K -user sum- rate. Pr oof : The proof fol lo ws from applying Lemma 1 to the maximization of R K for each choice of ζ . For a ﬁxed t ransmit power P k , for all k ∈ T , and noise var iances N r and N d , the choice of ζ determines whether the int ersection of R ob r  ζ  and R ob d  ζ  belongs to the set of active or inactiv e cases. For each choice of ζ , from Theorem 4 an active case result s only if for all 2 K − 1 non-empty subsets A of K , the condi tion i n (20) does not hold . Further , for any ζ that results in an inactive case, from Theorem 4 , the sum-rate is b ounded as B d, A + B r, A c < min  B r, K , B d, K  < max ζ ∈ Γ O B min  B r, K , B d, K  . (21) 12 T o this end, we consi der the o ptimization problem R K = max ζ ∈ Γ O B min  B r, K  ζ  , B d, K  ζ  . (22) In general, optim izing non-con vex functions is not straightforward. Ho we ver , since B r, K and B d, K are concav e functions of γ K , the above max-min op timization simpliﬁes to R K = max γ K ∈ Γ O B min n B r, K  γ K  , B d, K  γ K o . (23) Note t hat the optimizati on is performed o ver the same set in (22) and (23) as Γ O B is a closed con vex set. In Appendix V, we show that the max-min problem in (23) is a dual of the classi cal minimax problem of detection theory , (see for e.g., [16, II.C]). This all o ws us to appl y the techniques used to obtain a m inimax solu tion to maximi ze the bounds in (23) over all γ K in Γ O B (see also [9]). W e write γ ∗ K to denote a sum-rate optimal al location, i .e., a max-min rule , and write G to denote the set of all γ ∗ K maximizing (23). A g eneral sol ution to t he m ax-min optimizatio n in (23) si mpliﬁes to three cases [16, II.C]. The ﬁrst two correspond to t hose in which the maxim um achiev ed by one of the two functions is smaller than the other , while th e third corre sponds to the ca se in which the maximum results when the two functi ons are equal (see Fig. 4). For B r, K and B d, K deﬁned in (10) and (11 ), respectively , we now show that the solution simpl iﬁes to the consideration of onl y two cases. The following theorem summarizes the solut ion to the max-min problem in (23). The proof is deve loped i n Appendix V. Theor em 5: The max-min optim ization R K = max γ K ∈ Γ O B min n B r, K  γ K  , B d, K  γ K o (24) simpliﬁes to the following t wo cases. Case 1 : R K = C   P k ∈K P k N r   , B r, K (0 ) < B d, K (0) Case 2 : R K = C  P k ∈K P k N r  − ( x ∗ ) 2 P max N r  ≡ B ∗ , B ∗ r, K  γ ∗ K  = B ∗ d, K  γ ∗ K  (25) where P max = max k ∈K P k , λ k △ = P k /P max , and x ∗ △ = P k ∈K p λ k γ ∗ k is the uni que solution 13 satisfying B r. K  γ ∗ K  = B r. K  γ ∗ K  and is given by x ∗ = − K 1 + p K 2 1 + ( K 3 − K 2 ) K 0 K 0 (26) with K 0 = P max / N r , K 1 = √ P max P r / N d K 2 = P k ∈K P k N d + P r N d , and K 3 = P k ∈K P k N r . (27) Remark 6: The maximizati on in (24) is independent of whether the optim al γ ∗ K results i n an activ e or an inactiv e case. Howev er , not all max-min rules γ ∗ K ∈ G will resul t in an activ e case. In general, activ e cases m ay be achieved only by a su bset G a ⊆ G . H o we ver , irrespecti ve of the kind of intersection, from Lemma 1, (25) is an upp er b ound o n the K -user sum -rate cut set bounds. In th e foll o wing theorem we sho w that it sufﬁc es to consider two condit ions in determining the largest out er bo und on t he K -user sum-capacity . W e enumerate the two condi tions as Condition 1: B r, K (0 ) ≤ B d, K (0) Condition 2: B r, K (0 ) > B d, K (0) . (28) The ﬁrst condition implies that the maximum K -user cutset bou nd at the relay is sm aller than the corresponding bound at the d estination; for this case, we show that B r, S (0 ) < B d, S (0) for all S ⊂ K , i.e., R O B = R ob r ⊂ R ob d . On the o ther hand, when con dition 2 occurs, i.e., when condition 1 does not hold in (28), we u se the monotone properties of B r, K and B d, K and Lemma 1 to show t hat R K ≤ max γ K ∈ Γ O B min n B r, K  γ K  , B d, K  γ K o (29) with equality achie ved in (29) when the polymatroid intersection is an active c ase. From Theorem 5 we have that a continuous set , G , of γ ∗ K maximizes the right -hand-side of (29). W e sh o w that the bound in (29) is achie ved wi th equality when th ere exists a γ ∗ K that results in an activ e case, i.e., i n a non-empty G a . Finall y , for the class of symmetric de graded G-MARCs, we pro ve the existence of a n active case that maxim izes the sum -rate. 14 Theor em 6: The largest outer bound R ob K on the K -user sum -rate i s R ob K = C  P k ∈K P k / N r  , if B r, K (0 ) < B d, K (0) R ob K ≤ C  P k ∈K P k N r  − ( x ∗ ) 2 P max N r  , otherwise (30) where x ∗ △ = P k ∈K p λ k γ ∗ k is the unique solu tion satisfying B r, K ( γ ∗ ) = B d, K ( γ ∗ ) and is given by (26) and (27). The bound in (30) is achie ved w ith equality only when t he intersection of R ob r ( γ ∗ K ) and R ob d ( γ ∗ K ) results in an activ e case. The bound is achiev ed with equality for the class of symm etric degraded G-MARCs. Pr oof : Let γ K = 0 . From (10) we see that B S ,r ( γ S 6 = 0 ) < B S ,r (0) , for all S ⊆ K , i.e, the region R ( ob ) r ( γ K ) is lar gest at γ K = 0 . Expanding B S ,r and B S ,d at γ K = 0 from (10) and (11), respectiv ely , we hav e B r, S (0 ) = C  P k ∈S P k N r  (31) B d, S (0 ) = C  P k ∈S P k N d + P r N d  . (32) The sum-rate resulting from t he i ntersection of R ob r (0) and R ob d (0) f alls into one of fol lo wing two cases. Case 1 : The ﬁrst case results when B K ,r (0) ≤ B K ,d (0) . From (31) and (32) this condi tion simpliﬁes to P k ∈K P k N r ≤ P k ∈K P k N d + P r N d . (33) Expanding (33), we have, for any S ⊂ K , P k ∈S P k N r ≤ P k ∈S P k + P r N d − P k ∈S c P k ( N d − N r ) N d N r (34) < P k ∈S P k + P r N d (35) where (35) follows from the degradedness conditi on in (4). Thus, B r, K (0 ) ≤ B d, K (0) impli es that B r, S (0 ) < B d, S (0) for all S ⊂ K , i.e., R ob r (0) ⊂ R ob d (0) , and R O B (0) = R ob r (0) . The maximum 15 K -user sum-rate upp er bound for t his acti ve case is then R ob K = B r, K = C ( P k ∈K P k / N r ) . (36) Case 2 : The second case results when B K ,r (0 ) > B K ,d (0) , i.e., when P k ∈K P k N r > P k ∈K P k N d + P r N d . (37) Unlike cas e 1 , (37) does no t im ply that B S ,r (0 ) > B S ,d (0) or v ice-ve rsa, for all S ⊂ K . From Theorem 4 , the intersection of R ob r (0 ) and R ob d (0) ca n result in either an acti ve or an inactive case and thus, from (20), we have R ob K ≤ min( B r, K (0 ) , B d, K (0)) = B d, K (0) (38) with equality f or the active case. Note that from symmetry an active case results for the symmetric G-MARC. W e now show that the sum-rate is increased for a γ ∗ K 6 = 0 such that B r, K  γ ∗ K  = B d, K  γ ∗ K  . T o simplify the expositio n, we write B r, K and B d, K in (10) and (11) as B r, K ( x ) = C   P k ∈K P k N r − x 2 P max N r   (39) B d, K ( x ) = C   P k ∈K P k N d + P r N d + 2 x √ P max P r N d   (40) where x △ = K X k =1 p γ k λ k (41) and λ k = P k /P max where P max = max k ∈K P k , for all k . F or all γ k ∈ [0 , 1] , w e h a ve ∂ x ∂ γ k = √ λ k 2 √ γ k k ∈ K (42) ∂ 2 x ∂ γ 2 k = − √ λ k 4 γ 3 / 2 k k ∈ K (43) ∂ 2 x ∂ γ k ∂ γ j = 0 k 6 = j . (44) 16 Thus, x is a concave function of γ K over the hyper-cube γ k ∈ [0 , 1] , for all k , and therefore, is conca ve for a ll γ k satisfying (13). Further , from (13), we see that x is maximized when the entries of γ K satisfy P K k =1 γ k = 1 . Using techniques sim ilar to those i n Appendix III, one can show that x achieves its maxim um for a γ ′ K with entries γ ′ k = λ k P K k =1 λ k for all k , (45) and thus, we ha ve x ∈ " 0 , r X K k =1 λ k # ⊆ [0 , √ K ] . (46) The function s B r, K ( x ) and B d, K ( x ) in (39) and (40 ) are monoton ically dec reasing and increasing functions of x , respecti vely . Substi tuting (45) in (10), we have B r, S ( γ ′ K ) = 0 for all S ⊆ K . Thus, for th e case in which B K ,r (0 ) > B K ,d (0) , one can shrink the region R ob r from R ob r (0) just sufﬁciently such that for so me γ ∗ K 6 = 0 , B r, K  γ ∗ K  = B d, K  γ ∗ K  > B K ,d (0 ) . In Theorem 6 we show that B K ,r = B K ,d is maximized by a set G of γ ∗ K satisfying G =  γ ∗ K : P k ∈K γ ∗ k λ k = ( x ∗ ) 2  (47) where x ∗ is the unique value satisfying the quadratic B K ,r ( x ) = B K ,d ( x ) . For γ ∗ K 6 = 0 , from (10) one can verify that B r, S  γ ∗ K  < B r, S (0 ) for all S , i.e., R ob r  γ ∗ K  ⊂ R ob r (0 ) . On the ot her hand, substitu ting γ ∗ K in (11), B d, S for all S ⊂ K simpliﬁes t o B d, S  γ ∗ K  = C    P k ∈S P k + (1 − P k ∈S c γ ∗ k ) P r + 2 P k ∈S p γ ∗ k P k P r N d    . (48) Comparing B d, S (0 ) in (32) with B d, S  γ ∗ K  in (48) abov e, one cannot in general show that R ob d  γ ∗ K  ⊇ R ob d (0 ) . In f act, the γ ∗ K chosen will determine t he relationship between B d, S ( γ ∗ K ) and B d, S (0 ) for any S . Thus, for any γ ∗ K that equalizes B r, K and B d, K the p olytope R ob r ∩ R ob d belongs to eit her the set of activ e or inactiv e cases. Recall that we write G a to denote the set of γ ∗ K that results i n an acti ve case, i.e., the set of γ ∗ K for whi ch the c ondition i n (20) does no t hold for all 2 K − 1 non-empty s ubsets A of K . From Th eorem 4 , we have that th e sum-rate for the inactive case i s always bou nded by the m aximum sum-rate developed in Theorem 5 . Thus, 17 the maxim um K -user sum -rate when B r, K (0 ) > B d, K (0) is R K =    B d, K ( γ ∗ K ) = B r, K ( γ ∗ K ) ≡ B ∗ γ ∗ K ∈ G a 6 = ∅ max ξ B d, A ( ξ ) + B r, A c ( ξ ) < B ∗ G a = ∅ (49) where B ∗ is deﬁned in Theorem 5. W e no w show that for the class of symmetric G-MARC channels the bound B ∗ is achieved, i.e., G a 6 = ∅ . For t his class since P k = P for all k ∈ K , from symmetry B ∗ can be achieve d by choosing γ ∗ k = γ ∗ for all k such th at from (41), we ha ve γ ∗ = ( x ∗ ) 2 /K 2 . (50) From (46), sin ce 0 ≤ x ∗ ≤ √ K , there exists an γ ∗ < 1 . From (13), w e also require γ ∗ < 1 /K . In Theorem 12 i n Section IV below , we prove the existence of a γ ∗ < 1 /K for symmetric channels. From symmetry , since no subset of users can achieve better rates at one receiver than the oth er , the resulti ng R r ( γ ∗ ) ∩ R d ( γ ∗ ) belongs to the s et of inactive cases. Th e K -user sum-rate cutset bound for this class i s giv en by the B ∗ in (25) wi th P max = P and λ k = 1 for all k ∈ K . Finally , from cont inuity , one can expect that for sm all perturbations of u ser powers from th e symmetric case, an active case will result. Howe ver , for arbitrary user powe rs, it is possibl e that G a = ∅ , i.e., the set of all feasible γ ∗ K results i n non -inacti ve cases. In general, howe ver , obtainin g a clos ed-form expression for the maximum sum -rate for the in acti ve cases is not straightforward. I V . D E C O D E - A N D - F O RW A R D A DF code construction for a discrete memoryless MARC using block Markov encoding and backward decoding is deve loped in [4, Appendix A] (s ee also [19]) and we extend i t here to the degraded Gaus sian M ARC. W e ﬁrst s ummarize t he rate re gion achie ved by DF b elo w . Pr opos ition 2: The DF rate region i s the uni on of the set of rate t uples ( R 1 , R 2 , . . . , R K ) that satisfy , for all S ⊆ K , R S ≤ min { I ( X S ; Y r | X S c V K X r U ) , I ( X S X r ; Y d | X S c V S c U ) } (51) where the unio n is over all di stributions that factor as p ( u ) ·  Q K k =1 p ( v k | u ) p ( x k | v k , u )  · p ( x r | v K , u ) · p ( y r , y d | x T ) . (52) 18 Pr oof : Se e [19]. Remark 7: The ti me-sharing random variable U ensures that the region of Th eorem 2 is con vex. Remark 8: The independent auxiliary random variables V k , k = 1 , 2 , . . . , K , help the sources cooperate with the relay . For the degraded Gaussian MARC, we employ the following code construction . W e generate zero-mean, unit var iance, i ndependent and identically distributed (i.i.d.) Gaussian random v ari- ables V k , V k , 0 , and V r, 0 , for all k ∈ K , such t hat the channel in puts from s ource k and t he relay are X k = √ α k P k V k , 0 + p (1 − α k ) P k V k , k ∈ K , (53) X r = K P k =1 √ β k P r V k + s  1 − K P k =1 β k  P r V r, 0 (54) where α k ∈ [0 , 1] and β k ∈ [0 , 1] are power fractions for all k . W e write α K =  α 1 , α 2 , . . . , α K  (55) β K =  β 1 , β 2 , . . . , β K  (56) and Γ = n α K , β K  : α k ∈ [0 , 1] , 0 ≤ P k ∈K β k ≤ 1 for all k . o (57) for the set of feasibl e power fractions α K and β K . Substitu ting (53) and (54) in (51), for any ( α K , β K ) ∈ Γ , we ob tain R S ≤ min  I r, S ( α K ) , I d, S  α K , β K  for all S ⊆ K (58) 19 where I r, S and I d, S , the bounds at the relay and destination respectively , are I r, S = C   P k ∈S α k P k N r   (59) I d, S = C      P k ∈S P k N d +  1 − P k ∈S c β k  P r N d + 2 P k ∈S r (1 − α k ) β k P k N d P r N d      . (60) From the conca vity of the log functio n i t follo ws that I r, S , for all S , is a conca ve function o f α K . In Appendix III we show that I d, S is a concave function of α K and β K . Th e D F rate region, R D F , achiev ed over all ( α K , β K ) ∈ Γ , is then gi ven by the following theorem. Theor em 7: The DF rate region R D F for a degraded Gaus sian M ARC is R D F = [ ( α K ,β K ) ∈ Γ  R r ( α K ) ∩ R d  α K , β K  (61) where the rate region R t , t = r, d , is R t  α K , β K  = n ( R 1 , R 2 , . . . , R K ) : 0 ≤ R S ≤ I t, S  α K , β K  , for all S ⊆ K o . (62) Pr oof : The rate region R D F follows directly from Propos ition 2, the code construction in (53)-(54), and the fact that I r, S and I d, S are concav e functions of ( α K , β K ) . Theor em 8: The rate region R D F is con ve x. Pr oof : T o show t hat R D F is con vex, it sufﬁces to sho w that I r, S and I d, S , for all S , are conca ve functions over the con vex set Γ of ( α K , β K ) . This is because the c oncavity of I r, S and I d, S , for all S , ensu res that a con ve x sum of two o r more rate tuples in R D F , each corresponding to a different value o f ( α K , β K ) tuple, also belon gs to R D F , i.e., satisﬁes (62) for t = r , d . Theor em 9: The rate regions R r and R d are polym atroids. Pr oof : In Appendix II we sh o w that for e very choi ce of i nput distribution satisfying (52) the bound s in (5 1) are submodul ar set functions, and thu s, enclose regions that are polym atroids. For the G aussian input distribution in (53 ) and (54), this i mplies that R r ( α ) and R d  α , β  are polymatroids for every choice of ( α , β ) , i.e., R r and R d are completely d eﬁned by the corner (verte x) points on their dominant K -user sum -rate face [15, Chap. 44]. 20 The region R D F in (61) is a u nion of the intersection of t he regions R r and R d achie ved at the relay and destination respectively , where the uni on is over all ( α K , β K ) ∈ Γ . S ince R D F is con vex, each poi nt on the bo undary of R D F is obtained by m aximizing the weig hted sum P k ∈K µ k R k over a ll Γ , and for all µ k > 0 . Speciﬁcally , we determine the opt imal policy ( α ∗ K , β ∗ K ) that maximizes the sum -rate R K when µ k = 1 for all k . From (61), we see that every point on the boundary of R D F results from the intersection of the polymat roids R r ( α K ) and R d ( α K , β K ) for some ( α K , β K ) . Since R r and R d are p olymatroids, as with the outer bound analys is, here too we use Lemm a 1 on p olymatroid in tersections to broadly classify the i ntersection of t wo polymatroids into th e categories o f active and i nacti ve sets . In the following theorem we use Lemma 1 to write the bound o n the K -user DF sum-rate. W e remark that R r and R d are polymatroids parametrized by ( α K , β K ) , and thus , Lemma 1 applies for each choi ce of ( α K , β K ) . Theor em 10: For any ( α K , β K ) , t he maximum K -user sum-rate R K resulting from the inter- secting polymatroi ds R r and R d is R K =    I d, A + I r, A c , condition 2 min ( I r, K , I d, K ) , otherwis e (63) where condition 2 is deﬁned for a ∅ 6 = A ⊂ K as I d, A + I r, A c < min ( I r, K , I d, K ) . (64) Remark 9: The condi tion in (64) determines whether the i ntersection of two polymatroids belongs to either t he set of activ e or inactive cases with respect t o the K -user sum -rate. Pr oof : The proof follows from applying Lem ma 1 to the maximization R K = P k ∈K R k for each choice of ( α K , β K ) . W e seek t o determine the maximum sum-rate R K over all ( α K , β K ) ∈ Γ . T o this end, we ﬁrst consider the opti mization probl em R K = max ( α K ,β K ) ∈ Γ min  I r, K ( α K ) , I d, K  α K , β K  . (65) W e writ e ( α ∗ K , β ∗ K ) to denote the max-min rule optim izing (65) and write P to denote the set of all ( α ∗ K , β ∗ K ) maximizing (23). A general soluti on to the max-mi n opti mization in (23) sim pliﬁes to three cases [16, II.C]. The ﬁrst two correspond to those in which th e m aximum achieved by one of the two functions is s maller t han the other , while the third corresponds to th e case in whi ch 21 the maximum results when the two functions are equal (see Fig. 4 ). For I r, K and I d, K deﬁned in (59) and (60), respectiv ely , we can s ho w that the so lution simpli ﬁes t o the consideration of only two cases. The following theorem sum marizes th e solut ion to the max-min prob lem i n (65). The proof is developed in Appendix V. Theor em 11: The max-min optim ization R K = max ( α K ,β K ) ∈ Γ min  I r, K ( α K ) , I d, K  α K , β K  (66) simpliﬁes to the following t wo cases. Case 1 : R K = C   P k ∈K P k N r   I r, K (1 ) < I d, K (1 , 0) Case 2 : R K = C  P k ∈K P k N r  − ( q ∗ ) 2 P max N r  ≡ I ∗ I ∗ r, K = I ∗ d, K (67) where I ∗ t, K = I t, K ( α ∗ K , β ∗ K ) , t = r, d , P max = max k P k with λ k = P k /P max , and q ∗ △ = X λ k (1 − α ∗ k ) (68) is the uniqu e value satis fying the q uadratic I r, K ( α ∗ K , β ∗ K ) = I d, K ( α ∗ K , β ∗ K ) and is g i ven by q ∗ = − K 1 + p K 2 1 + ( K 3 − K 2 ) K 0 K 0 (69) with K 0 = P max / N r , K 1 = √ P max P r / N d K 2 = P k ∈K P k N d + P r N d , and K 3 = P k ∈K P k N r . (70) The entries of the optimal β ∗ K are giv en by β ∗ k =    ( 1 − α ∗ k ) P k P K k =1 ( 1 − α ∗ k ) P k α ∗ K 6 = 1 0 α ∗ K = 1 for all k ∈ K . (71) Remark 10: The optimal q ∗ in (69) is the same as that for the opt imal x ∗ in (26). Thus, from (25) and (67), we see that for both cases, the maxim um cutset bound is equal to the maxi mum DF bound on R K . From Lemma 1 we see t hat the m aximum sum-rate can be achieve d by either an activ e or an inactiv e case. In th e following t heorem we show that it sufﬁces to consider two condi tions in 22 determining the maxim um K -user DF s um-rate. W e enumerate the two conditio ns as Condition 1: I r, K (1 ) ≤ I d, K (1 , 0) Condition 2: I r, K (1 ) > I d, K (1 , 0) . (72) The ﬁrst condition impli es that t he m aximum sum -rate at the relay is sm aller than th e corre- sponding rate at the destination; for th is case, we sho w that I r, S (1) < I d, S (1 , 0) for all S ⊂ K , i.e., R D F = R r ⊂ R d . Physically , this corresponds to t he case where the relay has a high SNR link to the destination and t he multiaccess link from the sources to the relay is the bottleneck link. Under th is condition, we sho w th at t he sum-capacity of a degraded G aussian MARC is achie ved by DF . On the other hand, when condition 2 occurs, i.e., when condit ion 1 does not hold in (72), we use the mon otone p roperties of I r, K and I d, K and Lemma 1 t o sh o w t hat R K ≤ max ( α K ,β K ) ∈ Γ min n I r, K ( α K ) , I d, K  α K , β K o (73) with equality when the int ersection of R r ( α K ) and R d ( α K , β K ) results in an acti ve case. From Theorem 11, a continuous set P of ( α ∗ K , β ∗ K ) with a uniq ue β ∗ K for each choice of α ∗ K maximizes the right-side of (73). Furthermore, we show that the bound in (73) is the sum-capacity when an acti ve case achie ves the maximum sum-rate. Finall y , for the class of sy mmetric degraded G-MARCs, we prove the existence of an active case that ac hiev es the sum-capacity . Theor em 12: The K -user DF s um-rate R K for a degraded Gaus sian M ARC is R K = C  P k ∈K P k / N r  , I r, K (1 ) < I d, K (1 , 0) R K ≤ C  P k ∈K P k N r  − ( q ∗ ) 2 P max N r  , otherwise . (74) For I r, K (1 ) < I d, K (1 , 0) , DF ac hiev es the capacity region and the sum -capacity of the degraded Gaussian M ARC. The upper bound on R K in (74) is achie ved wit h equality only for a class of active degraded Gaussian MARCs for wh ich there exists a ( α ∗ K , β ∗ K ) ∈ P such that R r ( α ∗ K ) ∩ R d ( α ∗ K , β ∗ K ) is an active case and i s the sum-capacity for t his class. This active class al so includes the class of sy mmetric degraded Gaussian MARCs. Pr oof : Let α K = 1 and β K = 0 . From (59 ) and (60), we see that I S ,r and I S ,d are monotonicall y increasing and dec reasing functions of α K , respectiv ely , for a ﬁxed β K , i.e., for any α (1) K and α (2) K satisfying (57), with entries α (1) k ≤ α (2) k for all k ∈ K , R r ( α (1) K ) ⊆ R r ( α (2) K ) 23 and R d ( α (1) K , β K ) ⊇ R d ( α (2) K , β K ) . Thus , R r ( α K ) achieves its largest region for α K = 1 . The bounds I r, S and I d, S can be expanded for this case using (59) and (60), respectively , as I r, S = C  P k ∈S P k N r  (75) I d, S = C  P k ∈S P k N d + P r N d  . (76) The resulting sum-rate s atisﬁes one of two conditio ns and we enu merate them below . Condition 1 : Th e ﬁrst conditio n is I r, K (1) ≤ I d, K (1 , 0) . From (75) and (76), t his case requires P k ∈K P k N r ≤ P k ∈K P k N d + P r N d . (77) Expanding (77), we have, for any S ⊂ K , P k ∈S P k N r ≤ P k ∈S P k + P r N d − P k ∈S c P k ( N d − N r ) N d N r < P k ∈S P k + P r N d (78) where (78) fol lo ws from (4). Thus, I r, K (1 ) ≤ I d, K (1 , 0) implies that I r, S (1) < I d, S (1 , 0) for all S ⊂ K , i.e., R r (1 ) ⊂ R d (1) and thus, R D F (1) = R r (1) . Further , since R r (1) ∩ R d (1 , 0) = R r (1 ) , the polymatroid i ntersection for thi s condition belo ngs to t he intersecting set. Finally , recall t hat we chose β K = 0 . From (59), we see that t he choice of β K does not affect R r . Further , a non-zero β K does not increase I d, K . Howe ver , it can decrease I d, S for some or all S ⊂ K as I d, S  1 , β K  = C       P k ∈S P k  + P r  1 − P k ∈S c β k  N d      ≤ I d, S (1 , 0) (79) thereby p otentially decreasing R D F (1 ) . Th us, for the condition in (77) and from Theorem 5, the K -user sum -capacity of a de graded G-M ARC for this case is R K = I r, K (1 ) = B r, K (0) = C ( P k ∈K P k / N r ) . (80) The max-min rule for t his conditi on is ( α ∗ K , β ∗ K ) = (1 , 0) . Finally , from condition 1 in Theorem 6 for a class of degraded Gaussian MARCs where th e so urce and relay powers satisfy (77), DF 24 achie ves the capac ity region since R D F = R r (1 ) = R ob r (0) . (81) Condition 2 : The second c ondition requires I K ,r (1 ) > I K ,d (1 , 0) , i .e., P k ∈K P k N r > P k ∈K P k N d + P r N d . (82) Unlike condition 1, one cannot show here that I S ,r > I S ,d for all S ⊂ K or vi ce-v ersa. Thus, from Theorem 10, the intersectio n o f R r (1 ) and R d (1 , 0) can result in either an active or an inactiv e case. Fr om (63) in Theorem 10, we t hen ha ve R K ≤ min { I r, K (1 ) , I d, K (1 , 0) } = I d, K (1 , 0) (83) with equality f or the active case. Note that from symmetry an active case results for the symmetric G-MARC. Howe ver , th e bound on t he sum-rate, and thus, the sum-rate t oo, can be increased using the fact that I r, K and I d, K are monotonicall y increasing and decreasing funct ions of α K , respectiv ely . In fact, from (59) and (60), we see t hat reducing some or all of the e ntries of α K from their maximum va lue of 1 reduces I r, K and either reduces or keeps unchanged s ome or all I r, S while in creasing I d, K . Further , si nce I r, S (0 ) = 0 for all S ⊆ K , on e can shri nk the region R r just sufﬁciently to ensu re t hat there exists some α ∗ K and β ∗ K such that I r, K ( α ∗ K ) = I d, K ( α ∗ K , β ∗ K ) . From Theorem 11 I r, K = I d, K is maximized by a set P of ( α ∗ K , β ∗ K ) where α ∗ K and β ∗ K satisfy (68) and (71), respectively . Evaluating I d, S at a max-mi n rule ( α ∗ K , β ∗ K ) , we have I d, S = C   P k ∈S P k N d + P k ∈S (1 − α ∗ k ) P k P r N d ( q ∗ K ) 2 + 2 s P k ∈S (1 − α ∗ k ) P k P r N 2 d   . (84) For α ∗ K 6 = 1 , since I r, S , for all S , is a monotoni cally decreasing function of α K we hav e R r ( α ∗ K ) ⊂ R r (1 ) . On the other hand, comparing (7 6) and (84) one cannot in g eneral show that R d  α ∗ K , β ∗ K  ⊇ R d (1 , 0) . In fact, the α ∗ K chosen will determine the relationshi p between I d, S ( α ∗ K , β ∗ K ) and I d, S (1 , 0) for any S . Thus, for any ( α ∗ K , β ∗ K ) that equalizes I r, K and I d, K , the polytope R r ∩ R d belongs to either the set of activ e or in acti ve cases. Let P a ⊆ P denote the set of ( α ∗ K , β ∗ K ) that result in active cases. From Theorem 10, we can writ e t he maximum K -user 25 DF sum-rate when I r, K (1 ) > I d, K (1 , 0) as R K =      I d, K ( α ∗ K , β ∗ K ) = I r, K ( α ∗ K ) = I ∗ , ( α ∗ K , β ∗ K ) ∈ P a 6 = ∅ max ( α ∗ K ,β ∗ K ) ∈P I d, A ( α ∗ K , β ∗ K ) + I r, A c ( α ∗ K ) < I ∗ , P a = ∅ (85) where I ∗ is gi ven by (67) in Theorem 11. Finally , as shown i n remark 10, I ∗ = B ∗ where B ∗ is the maxim um outer bound su m-rate. W e now show that for class of sym metric G-M ARC channels, when the cond ition in (82) holds, we achieve the K -user sum-capacity . For this class, since P k = P , from symmetry , I d, K = I r, K in (60) can be m aximized by choosing α ∗ k = α ∗ for all k in (68) such that (1 − α ∗ ) = ( q ∗ ) 2 /K . (86) From (68), since 0 < ( q ∗ ) 2 < P K k =1 λ k = K , there exists an 0 < α ∗ < 1 that achiev es I ∗ in (85). Further , from symmetry , no subset of users achie ves a larger rate at one of t he receiv er than any other subset, i.e., for α ∗ k = α ∗ and β k = 1 /K , for all k , R r ∩ R d belongs to the set of acti ve cases and the maximum K -us er sum-rate for t his class is I ∗ = B ∗ . Recall that for the outer bound in Theorem 6, we need to prove that γ ∗ K ∈ Γ O B where γ ∗ K has entries γ ∗ giv en by (50) for all k . Fr om (13) and (57), we can write γ k = (1 − α k ) β k where ( α K , β K ) ∈ Γ . (87) . W e then hav e X k ∈K γ k = X k ∈K (1 − α k ) β k < 1 (88) where (88) fol lo ws from (57) and t he fact that (1 − α k ) β k < β k for all ( α K , β K ) ∈ Γ . F or the symmetric case, t his i mplies that there exists a γ ∗ = (1 − α ∗ ) /K satisfying (88). In fact, for α ∗ in (86), we obtain γ ∗ = ( q ∗ ) 2 /K 2 = ( x ∗ ) 2 /K 2 < 1 , i.e., the symmetric γ ∗ in (50) is feasible and results i n an activ e case. Since an active case achieves the same maximum sum-rate for both the inner and out er bound, we see that DF achiev es the sum -capacity for the class of sy mmetric Gaussian MARCs. For the general case of arbitrary P k , from (85) and (49) we see th at DF achie ves t he maximum K -user sum-rate o uter bounds for an active class of degraded Gaussian MARCs for which 26 R r ( α ∗ K ) ∩ R d ( α ∗ K , β ∗ K ) belongs to the set of active cases. Further , DF achiev es the same maxim um value for all ( α ∗ K , β ∗ K ) ∈ P a 6 = ∅ . In Appendix VII, we show that for t he s ame choice of the K source-relay correlati on coefﬁ cients for both the inner and out er bounds, the outer cutset bounds are at least as lar ge as th e inner DF bounds for all S ⊆ K . This implies t hat for e very ( α ∗ K , β ∗ K ) ∈ P a , there exists a γ ∗ K with entries γ ∗ k = (1 − α ∗ k ) β ∗ k for all k (89) that results in an active case for the ou ter bounds, i.e., DF achieves the sum-capacity for the activ e class. Note that the outer bounds may also be maximized by other ( α K , β K ) t hat do not maximize the K -user DF sum-rate. Finally , as with the outer bounds, the optim ization in (85 ) for P a = ∅ is not straightforward. Further , comparing the DF and cutset bounds in (85) and (49), respecti vely for the inactive cases, we see that the expression for the outer b ounds in volves tim e-sharing and can in general be larger than the DF bound. It is straightforward to ﬁnd num erical examples for conditi on 1 in T heorem 12 where DF achie ves the capacity region. W e focus on condit ion 2 and present two examples where DF achie ves the sum-capacity of a two-user degraded Gaussian M ARCs, with P a = P for o ne and P a ⊂ P for the other . Example 1: Consider a two-user degraded Gaussian M ARC with P 1 / N r = 6 , P 2 / N r = 4 , P 1 / N d = 3 , P 2 / N d = 2 , and P r / N d = 2 . These SNR values satisfy the conditi on 2 giv en by (82) in Theorem 12 and thus, the DF sum-rate is maxim ized by a set of ( α ∗ K , β ∗ K ) where α ∗ K satisﬁes (1 − α ∗ 1 ) + 2 3 (1 − α ∗ 2 ) = ( q ∗ ) 2 = 0 . 40 8 , (90) and for e very choice of α ∗ K satisfying (90), β ∗ K is given by (71). The set of feasible α ∗ K has entries α ∗ 1 ∈ (0 . 83 , 1] wit h α ∗ 2 for each such α ∗ 1 satisfying (90) such that α ∗ 2 ∈ (0 . 75 , 1] . For these SNR parameters, the set P a = P and for each ( α ∗ K , β ∗ K ) ∈ P , the correlation values γ ∗ k = (1 − α ∗ k ) β ∗ k , for all k = 1 , 2 . result in the vector γ ∗ K ∈ G a . Example 2: W e next consider a two-user example with P 1 / N r = 6 , P 2 / N r = 0 . 4 , P 1 / N d = 3 , P 2 / N d = 0 . 2 , and P r / N d = 2 . These SNR values also satisfy the condition 2 given b y (82) in 27 Theorem 12 and t hus, t he DF su m-rate is maximized by a set of ( α ∗ K , β ∗ K ) where α ∗ K satisﬁes (1 − α ∗ 1 ) + 2 3 (1 − α ∗ 2 ) = ( q ∗ ) 2 = 0 . 19 7 . (91) The set of feasible α ∗ K has ent ries α ∗ 1 ∈ (0 . 96 , 1] with α ∗ 2 for each such α ∗ 1 satisfying (91) s uch that α ∗ 2 ∈ (0 . 416 , 1 ] . Not e that subject to (91), α 2 decreases as α 1 increases and vice-versa. For t hese SNR parameters, th e set P a consists of ( α ∗ K , β ∗ K ) where the entries α ∗ 1 and α ∗ 2 are restricted to th e set s (0 . 96 1 , 0 . 979] and (0 . 731 , 1] , respecti vely . The remaining values for α ∗ 1 and α ∗ 2 satisfying (91) result in a polymatroid intersection that belongs to the set of inacti ve cases. In fact, all such values resul t in t he inactiv e case 2 illust rated in Fig. 2 for K = 2 . Finally , for the t wo-user degraded Gaussian MARC, a numerical example illu strating P a = ∅ does n ot appear straight forwar d despit e using a wide range of ratios of P 1 to P 2 , i.e., not all rate-maximizing int ersections are such that one of the sources achieve bett er rates at one of the recei vers while the other source achieves a better rate at the oth er rec eiv er . A possible reason for thi s is because, at any receiver , the noise seen by both so urces is the same, and thus, the source w ith smaller power t ypically achiev es smaller rates at both receiv ers. It m ay be possible to increase th e rate achieve d at the dest ination by increasing the relay power ; howe ver , lar ge values of relay p o wer wil l result i n th e b ottle-neck case where con dition 1 in Theorem 12 hol ds. Thus, it appears that it may always be possib le to ﬁnd an active case, particularly , one that maximizes the sum-rate. If thi s is true for an y arbitrary K , then DF achieves the sum-capacity of the degraded Gaussian MARC. Remark 11: In the above analysis, we d etermined t he sum -capacity for a de graded Gaussi an MARC under a per symbol t ransmit power const raint at the sources and relay . One can also consider an a verage power constraint at every transm itter . The achiev able strategy remains unchanged; for the con verse we start with th e con vex sum s of the outer b ounds in (7) over n channel u ses. In the i th channel u se, the bounds at th e relay and destination are given by B r, S and B d, S in (10) and (11), respectively , for all S , except now th e correlation parameters and power parameters are index ed by i . Rec all that B d, S is a concav e function of the correlation coef ﬁcients and power . On the o ther hand, B r, S for all S ⊂ K is not a conca ve function of the power and cross-correlation parameters. Howe ver , we can use t he concavity of B r, K to sh o w that the maxi mum bounds on the sum-rate in Thereom 4 remain unchanged. This in conjunction 28 with the steps in Theorem 6, lead to the same sum-capacity results. Finally , we not e that as with the s ymbol power constraint, here too we require ti me-sharing t o develop t he outer bo und rate region. V . C O N C L U D I N G R E M A R K S In this paper , we ha ve studied th e sum-capacity o f degraded Gaussian M ARCs. In p articular , we h a ve deve loped the rate regions for th e achiev able strategy of DF and the cuts et outer bounds. The outer bo unds have been obt ained usi ng cut-s et bound s for the case of independent sou rces and have been shown to be maximized by Gauss ian si gnaling at the sources and re lay . W e hav e also shown that, i n general, the rate regions achieved by th e in ner and outer bounds are not the same. This dif ference is due to th e fact that the inpu t distributions and the rate expressions for the inner and o uter boun ds are n ot exactly the same. In fact, the input dist rib ution for the i nner bound uses auxi liary random variables to model the correlation between the i nputs at the s ources and the relay and i s more restrictive than the d istribution for the outer bound. Despite these differe nces, i n both cases the input distributions can b e quant iﬁed by a set of K source-relay cross-correlation coef ﬁcients. Further , in both cases, we hav e shown that the rate region for e very c hoice of the appropriate input distribution is an intersection of po lymatroids. W e h a ve used the properties of polymatroi d intersections to s ho w t hat for both the inner and outer b ounds the lar gest K -user sum-rate is at most the maximum o f the minimum of the two K -user sum rate boun ds, with equ ality only when the polymatroid intersections belongs to th e set of active cases in whi ch the K -user sum rate p lanes are acti ve. For both DF and the outer bounds, we ha ve shown that the lar gest K -user s um-rate c an be determined using max-min o ptimization t echniques. In fact, we ha ve shown t hat for both the inner and out er bounds the max-m in optimization problem results in one of t wo unique sol utions. The ﬁrst so lution result s when th e l ar gest sum-rate from t he K sources to the relay is the bo ttle- neck rate and for this case, we have shown that DF achieves the capacity region. W e have further shown that the s um-rate maximi zing polymatroi d intersection for this case belongs t o the set of acti ve cases. Speciﬁcally , the sum-capacity as well as the entire capacity r egion is achiev ed by a max-min rule where the sources and the relay do not allocate any power to cooperati vely achie ving coherent combining gains at the destination, i.e., the auxiliary random v ariabl es V k = 0 , for all k . Thus, under Gaussian signali ng, the capacity region is achieved by DF b ecause the 29 inner and ou ter bounds at the relay , for V K = 0 , are I ( X S ; Y r | X r X S c ) = I ( X S ; Y r | X r X S c V K ) for all S ⊆ K (see (7) and (51)). The second solut ion result s when the largest sum-rates at th e relay and the desti nation are equal. For this case, we ha ve sho wn that DF achieves the su m-capacity for a class of acti ve degraded Gaussian MARCs in which the sum -rate m aximizing polym atroid in tersection belongs to the s et of activ e cases. W e have also shown that this class of active degra ded Gaussi an MARCs contains t he class of symmetric Gaussian MA RCs. In general, for this class, we have shown that the max-min DF rule is such that V k 6 = 0 for all k , i.e., a non-empt y sub set of sources and t he relay d i vide their t ransmit power to achie ve cooperativ e combining gain s at th e destinat ion. W e hav e also s ho wn that the lar gest DF sum -rate is achiev ed by a relay power policy that maxim izes the cooperative gai ns ac hiev ed at the desti nation, i.e., X r is a uni que weighted sum of V k for all k where th e weight for each s ource k is proportional to the power allocated by source k to cooperating wit h the relay . Our analysis has also sho w n that the maximum su m-rate admits sev eral s olutions for the po wer fractions allocated at the sources for cooperation su bject to a constraint that results from the equating the two bound s on the s um-rate. For the outer bounds , we have shown that the K -us er sum-rate o uter bound is maxi mized by a set of cross -correlation coef ﬁcients that are subject to the sam e constraint as DF and th e m aximum sum-rate is the sam e as that for DF . Furthermore, for the class of activ e degraded Gaussi an M ARCs, we have shown that the set of DF max-mi n rules ( α ∗ K , β ∗ K ) also maximizes t he o uter bounds by us ing th e f act that the in ner and outer bound coefﬁ cients can be related as γ k = (1 − α k ) β k , for all k . Finall y , since a DF max-min rule requires a unique correlation between X r and V K , conditioni ng the outer bound th at uses Y r on X r alone sufﬁces to ob tain t he sum -capacity . V I . A C K N OW L E D G M E N T S L. Sankar i s grateful for numerous detail ed discussions on t he MARC with Gerhard Kramer of Bell Labs, Alcatel-Lucent and on pol ymatroid intersection s with Jan V ondrack of Princeton Univ ersity . 30 A P P E N D I X I O U T E R B O U N D S : P R O O F W e now de velop the proof for Theorem 1. Recall t hat we write B r, S and B d, S to denote, respectiv ely , the ﬁrst and second bound on R S in (9) for a const ant U . Expanding the bounds on R S in (9) for a constant U , we have R S ≤ min { h ( Y r | X r X S c ) − h ( Z r ) , h ( Y d | X S c ) − h ( Z d ) } . (92) For a ﬁxed cova riance m atrix of the input random variables X K and X r , o ne can apply a con di- tional entropy maxi mization theorem [20, Lemma 1] to show that h ( Y r | X r X S c ) and h ( Y d | X S c ) are maximized by choosin g the distribution i n (8) as jointly Gaussian. Cons ider the bound B r, S . Expanding Y r , we have R S ≤ C E  v ar  P k ∈S X k | X r X S c  N r ! . (93) For Gauss ian si gnals, using t he chain ru le, we hav e E  v ar  P k ∈S X k | X r X S c  = det( K A | C ) det( K B | C ) (94) where A = h P k ∈S X k X r i T (95) B = [ X r ] (96) C = [ X S c ] (97) and for random vectors X and Y , the conditi onal cova riance K X | Y is K X | Y = E h ( X − E [ X | Y ]) ( X − E [ X | Y ]) T i (98) where X T is the transpose of X . W e use the fa ct that X S and X S c are i ndependent to expand (94) as E  v ar  P k ∈S X k | X r X S c  = v ar  P k ∈S X k  − E 2  P k ∈S X k ˜ X r, S  P r, S (99) 31 where ˜ X r, S △ = ( X r − E ( X r | X S c )) is a Gauss ian random v ariable wit h variance P r, S = E h ˜ X 2 r, S i = E [ v ar ( X r | X S c )] . (100) Substitutin g (99) in (93) and us ing (5 ) t o bound v ar ( X k ) for all k , we o btain, R S ≤ C      P k ∈S v ar ( X k ) − 1 P r, S E 2  P k ∈S X k ˜ X r, S  N r      (101) ≤ C       P k ∈S P k  − 1 P r, S E 2  P k ∈S X k ˜ X r, S  N r      . (102) W e deﬁne γ k , for all k ∈ K , by E [ X k X r ] △ = p γ k P k P r . (103) Note that by deﬁ nition, γ k ∈ [0 , 1] for all k ∈ K (104) and K X k =1 γ k ≤ 1 . (105) Using the independence of X k for all k and (103), we write E  P k ∈S X k ˜ X r  = P k ∈S E [ X k X r ] = P k ∈S p γ k P k P r . (106) Next we u se (103) to ev alu ate P r, S . W e start by consi dering the random v ariable ˆ X r = X r − E [ X r | X K ] . (107) Using (103) and the independence of X k for all k , we ca n write t he var iance of ˆ X r as E h ˆ X 2 r i = E [ v ar ( X r | X K )] (108) = (1 − γ K ) P r . (109) 32 where we used (98) t o si mplify (108) to (109). Conti nuing thu s, we consi der the random variable ¯ X r = ˆ X r − E h ˆ X r | X K − 1 i . Using the ind ependence of X k for all k , we thus have E  ¯ X 2 r  = E h ˆ X 2 r i − E h E 2 h ˆ X r | X K − 1 ii (110) = E [ v ar ( X r | X K − 1 X K )] (111) = (1 − γ K − 1 − γ K ) P r . (112) Generalizing the above, we h a ve E [ v ar ( X r | X S c )] =  1 − P k ∈S c γ k  P r △ = γ S c P r for all S ⊆ K . (113) Finally , we substitute (113) and (10 6) in (101) to sim plify the ﬁ rst bound as R S ≤              C  P k ∈S P k N r  , if P k ∈S c γ k = 1 C    P k ∈S P k N r − P k ∈S √ γ k P k ! 2 N r γ S c    , otherwise. (114) Observe that for K = 1 , we hav e V 1 = X r and γ 1 = 1 , and t hus, (10) simpliﬁes to t he ﬁrst outer bound in [3, theorem 5] for the classic sin gle source degraded relay channel. Finally , from (113), observe that γ k , for all k , satisﬁes P k ∈K γ k ≤ 1 . (115) Consider the bound B d, S in (9) with U a constant. Expandin g Y d using (2), we h a ve R S ≤ C  E  v ar  P k ∈S X k + X r | X S c  N d  (116) = C     P k ∈S  P k + 2 E  X k ˜ X r, S  + E [ v ar ( X r | X S c )] N d     . (117) Using (5), (113,) and (106), we simplify (117) as R S ≤ C   P k ∈S P k + γ S c P r + 2 P k ∈S √ γ k P k P r N d   . (11 8) 33 Writing B r, S and B d, S to denote the bounds on the right-side o f (114) and (118), respectively , we hav e for a const ant U , R S ≤ min ( B r, S , B d, S ) for all S ⊆ K . (119) A P P E N D I X I I I N N E R A N D O U T E R B O U N D S : P O L Y M A T R O I D S W e ﬁrst prove that the rate regions R ob r and R ob d giv en by the cutset bound s are poly matroids. Using simil ar techniques, we then sho w t hat the DF re gions R r and R d are polym atroids. A. Outer Bound s Consider the set functions (see 51) f 1 ( S ) =    I ( X S X r ; Y d | X S c U ) S ⊆ K , S 6 = ∅ 0 S = ∅ (120) and f 2 ( S ) =    I ( X S ; Y r | X S c X r U ) S ⊆ K , S 6 = ∅ 0 S = ∅ (121) for some distribution satisfying (8). W e claim th at f 1 and f 2 are submodular [15, Ch. 44]. T o see this, we ﬁrst consider f 1 and k 1 , k 2 in K with k 1 6 = k 2 , k 1 / ∈ S , k 2 / ∈ S , and expand f 1 ( S ∪ { k 1 } ) + f 1 ( S ∪ { k 2 } ) = I ( X S X k 1 X r ; Y d | X ( S ∪{ k 1 } ) C U ) + I ( X S X k 2 X r ; Y d | X ( S ∪{ k 2 } ) C U ) (122) = I ( X k 1 ; Y d | X ( S ∪{ k 1 } ) C U ) + I ( X S X r ; Y d | X S C U ) (123) + I ( X S X k 2 X r ; Y d | X ( S ∪{ k 2 } ) C U ) (124) where (123 ) follows from th e chain rule for m utual informati on. W e lower bound the ﬁrst term in (123) as h ( X k 1 | X ( S ∪{ k 1 } ) C U ) − h ( X k 1 | X ( S ∪{ k 1 } ) C Y d U ) (125) = h ( X k 1 | X ( S ∪{ k 1 ,k 2 } ) C U ) − h ( X k 1 | X ( S ∪{ k 1 } ) C Y d U ) (126) ≥ I ( X k 1 ; Y d | X ( S ∪{ k 1 ,k 2 } ) C U ) (127) 34 where (125) follows from the Markov chain X k − U − X j for all k , j ∈ K , k 6 = j and (127) because conditioni ng cannot increase entropy . The expression (127) added to the ﬁnal term in (123) is I ( X S ∪{ k 1 ,k 2 } X r ; Y d | X ( S ∪{ k 1 ,k 2 } ) C U ) . Inserting (127) int o (123), we h a ve f 1 ( S ∪ { k 1 } ) + f 1 ( S ∪ { k 2 } ) ≥ f 1 ( S ) + f 1 ( S ∪ { k 1 , k 2 } ) for all S ⊆ K . The set function f 1 ( · ) is th erefore subm odular by [15 , Theorem 44.1, p. 767]. The above steps sho w that the rate re g ion R ob d deﬁned by the desti nation cutset bounds (see (7)) R S ≤ I ( X S X r ; Y d | X S c U ) , S ⊆ K (128) is a polymat roid associated wi th f 1 ( · ) (see [15, p. 767]). One can sim ilarly s ho w that f 2 ( · ) is sub modular . T o see this, consid er f 2 and k 1 , k 2 in K with k 1 6 = k 2 , k 1 / ∈ S , k 2 / ∈ S , and expand f 2 ( S ∪ { k 1 } ) + f 2 ( S ∪ { k 2 } ) = I ( X S X k 1 ; Y r | X ( S ∪{ k 1 } ) C X r U ) + I ( X S X k 2 ; Y r | X ( S ∪{ k 2 } ) C X r U ) (129) = I ( X k 1 ; Y r | X ( S ∪{ k 1 } ) C X r U ) + I ( X S ; Y r | X S C X r U ) + I ( X S X k 2 ; Y r | X ( S ∪{ k 2 } ) C X r U ) (130) where (130 ) follows from th e chain rule for m utual informati on. W e lower bound the ﬁrst term in (130) as h ( X k 1 | X ( S ∪{ k 1 } ) C X r U ) − h ( X k 1 | X ( S ∪{ k 1 } ) C Y r X r U ) = h ( X k 1 | X ( S ∪{ k 1 ,k 2 } ) C X r U ) − h ( X k 1 | X ( S ∪{ k 1 } ) C Y r X r U ) (131) ≥ I ( X k 1 ; Y r | X ( S ∪{ k 1 ,k 2 } ) C X r U ) (132) where (131) follows from the independence of X k and (132) because conditioning cannot increase entropy . The expression (132) added to t he ﬁnal t erm in (130) is I ( X S ∪{ k 1 ,k 2 } ; Y r | X ( S ∪{ k 1 ,k 2 } ) C X r U ) . 35 Inserting (127) int o (123), we h a ve f 2 ( S ∪ { k 1 } ) + f 2 ( S ∪ { k 2 } ) ≥ f 2 ( S ) + f 2 ( S ∪ { k 1 , k 2 } ) for all S ⊆ K . The set function f 2 ( · ) is th erefore submodular by [15, Theorem 44.1, p. 767]. This in turn i mplies th at the ra te region R ob r deﬁned by the relay cut set bou nds (see (7)) R S ≤ I ( X S ; Y r | X S c X r U ) , S ⊆ K (133) is a polymat roid associated wi th f 2 ( · ) (see [15, p. 767]). B. Inner Bounds For the in ner D F bo unds, we cons ider the set functions (see 5 1) f 3 ( S ) =    I ( X S X r ; Y d | X S c V S c U ) S ⊆ K , S 6 = ∅ 0 S = ∅ (134) and f 4 ( S ) =    I ( X S ; Y r | X S c V K X r U ) S ⊆ K , S 6 = ∅ 0 S = ∅ (135) for some d istribution satisfying (52). The functions f 1 ( · ) and f 2 ( · ) dif fer fr om f 3 ( · ) and f 4 ( · ) , respectiv ely , in the absence of the auxili ary random variables V K . Th e proof of sub-modularity of f 3 and f 4 follows along the same lines as t hose for the out er bounds except now w e have th e Markov chain ( X k , V k ) − U − ( X j , V j ) for all k 6 = j . W e thus have that th e rate region R r deﬁned by the DF relay bounds (see (51 )) R S ≤ I ( X S X r ; Y d | X S c V S c ) , S ⊆ K ( 136) is a polymatroid associated with f 3 ( · ) (see [15, p. 767]). Sim ilarly , the region R d deﬁned by the DF destination bounds (see (51 )) R S ≤ I ( X S ; Y r | X S c V K X r ) , S ⊆ K (137) 36 is a polymat roid associated wi th f 4 ( · ) (see [15, p. 767]). A P P E N D I X I I I C O N C A V I T Y O F B d, S A N D I d, S A. Outer Bound B d, S Recall that the cutset bou nd at the dest ination, B d, S , is given by B d, S = C     P k ∈S P k N d + 0 @ 1 − P k ∈S c γ k 1 A P r N d + 2 P k ∈S √ γ k P k P r N d     for all S ⊆ K . (138) W e show that B d, S is a conca ve function of γ K . T o prove concavity , one has t o sho w t hat the Hessian or second deriv ative of B d, S , ∇ 2 B d, S , is negativ e sem i-deﬁnite, i.e, x T ∇ 2 B d, S x ≤ 0 for all x ∈ R K [21, 3.1.4]. W e write B d, S = 1 2 log K 0 + 2 X k ∈S K k √ γ k ! (139) where K 0 = 1 + P k ∈S P k N d + P r (1 − c ) N d K k = q P k N d P r N d k ∈ S . (140) The gradient ∇ B d, S is give n by ∇ B d, S = [ ∂ B d, S /∂ γ k ] k ∈K (141) = 1 K s h v S v S c i T (142) = 1 K s (143) where v S is an |S | -leng th vector with entries v k = K k  √ γ k for all k ∈ S , v S c is an |S c | -length vector with entri es v m = − P r / N d for all m ∈ S c , and K s = 2 K 0 + 2 X k ∈S K k √ γ k ! . (144) 37 The Hessian of B d, S , ∇ 2 B d, S , is given by ∇ 2 B d, S =  ∂ 2 B d, S /∂ γ k ∂ γ m  ∀ k , m ∈K (145) = − 1 K s diag ( d ) − z z T (146) where z = √ 2 ( ∇ B d, S ) (147) d = h d S d S c i T (148) such that d S is an | S | -length ve ctor with entries d k = K k . 2 γ 3 / 2 k for all k ∈ S , and d S c is an |S c | -length vector with entries d k = − 2 P 2 r /( N 2 d K s ) for all k ∈ S c . Using the fact that K k and γ k are non-negative for all k , from (146), for an y x ∈ R K , we have x T ∇ 2 B d, S x = − 1 K s  P k ∈K x 2 k d k  −  x T · z  2 (149) ≤ 0 (150) with equality i f and only if x = 0 . In proving the concavity of B d, S , we ass ume on ly th at γ k > 0 , for all k . Thus, from continuity , the conca vity also holds for all n on-negati ve γ k satisfying (see (13)) X k ∈K γ k ≤ 1 . (151) For a ﬁxed γ S c , we now ﬁnd t he γ S that maximizes B d, S subject to (151) above. For a c ∈ [0 , 1) , we ﬁx γ S c such that its entries γ k , for all k ∈ S c , satisfy X k ∈S c γ k = 1 − c, (152) and thus, from (151) we have X k ∈S γ k ≤ c. (153) Since B d, S is a continuou s concav e function of γ S it achiev es i ts m aximum at a γ ∗ S where ∂ B d, S ∂ γ k    γ ∗ k = 0 for all k ∈ S . (154) 38 Using the method of Lagrange multipli ers, we ﬁnd that a γ ∗ S that m aximizes B d, S subject to (152) and (153) has entries γ ∗ k =  cP k P k ∈S P k k ∈ S . (155) B. Inner Bound I d, S Recall that the DF bo und, I d, S , at the desti nation is given as I d, S = C     P k ∈S P k N d + 0 @ 1 − P k ∈S c β k 1 A P r N d + 2 P k ∈S √ (1 − α k ) β k P k P r N d     for all S ⊆ K . (156) Comparing (138) and (156), for γ k △ = (1 − α k ) β k for all k ∈ S and γ k △ = β k for all k ∈ S c , the DF rate bound i n (156) simpliﬁes to that for the o uter bound in (138), and th us, one can use the same t echnique to sho w th at I d, S is a conca ve function of α K and β K . For the power fractions β k , we have X k ∈K β k ≤ 1 . ( 157) For a ﬁxed α K , we determine the optimal β S maximizing I d, S by ﬁxing the vector β S c such that X k ∈S c β k = 1 − c (158) X k ∈S β k ≤ c. (159) where c ∈ [0 , 1) . Since I d, S is independent of β S for α S = 1 , we ass ume that α S 6 = 1 . W e now c onsider th e special case in which α S 6 = 1 and β S c are ﬁxed. W e determine a β S that maximizes I d, S subject to (159) and (158). Since I d, S is a continuou s conca ve function of β S it achiev es i ts m aximum at a β ∗ S where ∂ I d, S ∂ β k    β ∗ k = 0 for all k ∈ S . (160) As before, using Lagrange m ultipliers, the o ptimal β ∗ S that maxi mizes I d, S , s ubject to (159), has entries β ∗ k =  c (1 − α k ) P k P k ∈S (1 − α k ) P k k ∈ S . (161) Rate r e gion for a ﬁxed α K : F or any choice of a non-zero α K and a β K satisfying (157 ), the 39 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 Rate R 2 (bits/ch. use) Rate R 1 (bits/ch. use) ( β 1 , β 2 )=(1,0) (.85,.15) (.5,.5) (.15,.85) (0,1) Sum Bound ( α 1 , α 2 ) = (.5,.5) P 1 / N r = P 2 / N r = 7 dB P 1 / N d = P 2 / N d = P r / N d = −10 dB Fig. 3. Rate region achiev ed at the destination for a two-user MARC and α 1 = α 2 = 1 / 2 . rate region give n by (156) for all S is a polymatroid. For α K = 1 , from (156) w e see that there are no gains achie ved from coherent com bining, i.e., it suf ﬁces to choose β K = 0 . Consider α K 6 = 1 . Since there is at least one k for which α k < 1 , gains from coherent combining at the destination are m aximized by choos ing β K to satisfy (157) wi th equalit y . For a ﬁxed α K , we then write th e rate region at the destinati on as a union over all polymatroid s, one for each choi ce of β K satisfying K X k =1 β k = 1 . (162) Observe that for β ∗ K with entries given by (161), the bound I d, S is m aximized. In Fig 3, we illustrate the rate region for a two-user degraded Gaussian M ARC with the SNR P 1 / N d = P 2 / N d chosen as − 10 dB, α = (1 / 2 , 1 / 2) , and ﬁ ve choices of β K . Observe t hat the maximum si ngle- user rate R 1 is achie ved by setting β 1 to 1 though this value does not maximi ze R 2 or R 1 + R 2 . For all other ( β 1 , β 2 ) such as (0 . 85 , 0 . 15) , as β 1 decreases and β 2 increases, R 1 decreases whil e R 2 increases achieving its maximum at β 2 = 1 . The bound on the sum rate R 1 + R 2 increases 40 from ( β 1 , β 2 ) = (1 , 0) , achieves its maximu m at ( β ∗ 1 , β ∗ 2 ) = (1 / 2 , 1 / 2) , and then decreases as β 2 approaches 1 . The resulting region at the destination is sho wn i n Fig. 3 as a u nion over all polymatroids , one for each choi ce o f β K . A P P E N D I X I V B r, S V S . γ K W e show that the function B r, S in (10) is a conca ve function o f γ S for a ﬁxed γ S c and for all S ⊆ K . Recall the expression for B r, S as B r, S = C      P k ∈S P k N r −  P k ∈S √ γ k P k  2 N r  1 − P k ∈S c γ k       (163) where we assume that X k ∈S c γ k = 1 − c < 1 . (164) Observe that B r, S is maximized when c = 1 , i.e., γ k = 0 for all k ∈ S , and mini mized for c = 0 . Further , comp aring B r, S and B d, S , one can see that for γ k =    P k  P k ∈S P k  , k ∈ S 0 , k ∈ S c (165) B r, S achie ves it s mi nimum, i.e., B r, S = 0 . W e write x △ = X k ∈S p γ k λ k ! (166) where P max = max k ∈K P k and λ k = P k / P max . (167) Substitutin g (166) in the expression for B r, S in (163), we have B r, S = C  P k ∈S P k N r − x 2 P max N r c  . (168) 41 Diffe rentiating B r, S with respect to x we ha ve dB r, S dx = − xP max N r c ·  1 + P k ∈S P k N r − x 2 P max N r c  − 1 (169) d 2 B r, S dx 2 = − P max N r c  1 + P k ∈S P k N r  −  xP max N r c  2  1 + P k ∈S P k N r − x 2 P max N r c  2 (170) < 0 (171) where the strict inequality i n (171) follows si nce all terms i n (170) are positive. Further , for any c > 0 , from (168) B r, S is m aximized at x = 0 , i.e., for γ k = 0 for all k ∈ S . Thus, we see that B r, S is a concave decreasing function of x . A P P E N D I X V P RO O F O F T H E O R E M 5 W e no w prove Theorem 5 and give the solutio n to th e max-min o ptimization R K = max γ K ∈ Γ O B min  B r, K  γ K  , B d, K  γ K  . (172) Consider the function J ( γ K , δ ) = δ B r, K  γ K  + (1 − δ ) B d, K  γ K  , δ ∈ [0 , 1 ] . (173) Observe that J is linear in δ ranging in v alue from I d, K for δ = 0 to I r, K for δ = 1 . Thus , the optimizatio n in (173) is equiv alent to m aximizing the minim um of the two end point s of the line J over Γ O B . Maximi zing J ( γ K , δ ) ov er γ K , we obtain a continuous con ve x function V ( δ ) = max γ K ∈ Γ O B J ( γ K , δ ) , δ ∈ [0 , 1 ] . (174) From (17 3) and (17 4), we see th at for any γ K , J ( γ K , δ ) either lies s trictly below or is tangential to V ( δ ) . The fol lo wing proposition sum marizes a well-known solution to the max-min probl em in (172) (see [9]). 42 δ 0 1 ( ) V δ 1 ( ) V δ 1 ( ) V δ * * ( , ) J γ δ K * , ( ) r B γ K K * , ( ) d B γ K K δ δ 0 0 * , ( ) d B γ K K * , ( ) r B γ K K * , ( ) d B γ K K * , ( ) r B γ K K * * ( , ) J γ δ K * * ( , ) J γ δ K Fig. 4. Illustration of Cases 1, 2, and 3. Pr opos ition 3: γ ∗ K ,δ ∗ is a max-min rule where δ ∗ = arg min δ ∈ [ 0 , 1] V ( δ ) . (175) The m aximum bound on R K , V ( δ ∗ ) , i s completely determined by the following t hree cases (see Fig. 4). Case 1: δ ∗ = 0 : V ( δ ∗ ) = B d, K ( γ ∗ K ,δ ∗ ) < B r, K ( γ ∗ K ,δ ∗ ) (176) Case 2: δ ∗ = 1 : V ( δ ∗ ) = B r, K ( γ ∗ K ,δ ∗ ) < B d, K ( γ ∗ K ,δ ∗ ) (177) Case 3: 0 < δ ∗ < 1 : V ( δ ∗ ) = B r, K ( γ ∗ K ,δ ∗ ) = B d, K ( γ ∗ K ,δ ∗ ) . (178) W e apply Propositio n 3 t o determine t he maximum bound on R K . W e study each case separately and determine the m ax-min rule γ ∗ K for each case. In general, t he max-min rul e γ ∗ K ,δ ∗ depends on an optimal δ ∗ . Howe ver , for n otational con venience we henceforth omi t the subscript δ ∗ in denoting th e m ax-min rule. W e dev elop the optimal γ ∗ K and the maximum sum-rate for each case. W e ﬁrst cons ider case 1 and sh o w that this case is n ot feasible. Case 1 : This ca se occurs when the maximum bound achie vable at the d estination is smaller than the bou nd at th e relay . In Appendi x III, we show that the bound B d, K ( γ K ) i s a concave function of γ K and achiev es a maximum at γ ∗ K whose entries γ ∗ k satisfy (115) and are gi ven as γ ∗ k = P k  P k ∈K P k  , for all k ∈ K . (179) Substitutin g (179) in (10), we have B r, K ( γ ∗ K ) = 0 whi ch contradicts the assumpti on in (176), 43 thus making thi s case i nfeasible. Case 2 : Consider the condition for case 2 in (177). This condition i mplies that t he case occurs when t he maximum boun d achie vable at the relay i s smaller than t he bound at the desti nation. From (59), we ob serve that B r, K decreases monotoni cally with γ k for all k and achiev es a maximum of B r, K ( γ ∗ K ) = C   P k ∈K P k N r   (180) at γ ∗ K = 0 . Comparing (10) and (11) at γ ∗ K = 0 , we obtain t he condit ion for this case as P k ∈K P k N r ≤ P k ∈K P k + P r N d . (181) Case 3 : Finally , con sider the cond ition for Case 3 in (178). This case occurs when the maximum rate bound achie vable at the relay and destination are equal. The m ax-min so lution for thi s case is obt ained by considering t wo sub-cases. The ﬁrst is t he relativ ely straightforward sub-case where γ ∗ K = 0 is the max-min rule. The resulting maximum sum-rate is the same as that for case 2 wit h the condition in (181) satisﬁed with equality . Cons ider t he s econd s ub-case where γ ∗ K 6 = 0 , i.e., when P k ∈K P k N r > P k ∈K P k + P r N d . (182) W e formulate the o ptimization problem for this case as maximize B r, K  γ  subject to B r, K  γ  = B d, K  γ  . (183) W e write P max = max k ∈K P k , λ k = P k / P max , (184) and deﬁne x △ = r P k ∈K λ k γ k . (185) 44 Substitutin g (184) and (185) i n (10) and (11), we h a ve B r, K ( x ) = C       P k ∈K P k  − x 2 P max N r      (186) B d, K ( x ) = C       P k ∈K P k  + P r + 2 x √ P max P r N d      . (187) Observe that B r, K ( x ) and B d, K ( x ) are monoto nically decreasing and increasing functions of x , respectiv ely , and thus, the maxi mization in (183 ) simpliﬁes to determini ng an x su ch that P k ∈K P k − x 2 P max N r = P k ∈K P k + P r + 2 x √ P max P r N d . (188) W e write K 0 = P max / N r , K 1 = √ P max P r / N d K 2 = P k ∈K P k N d + P r N d , and K 3 = P k ∈K P k N r . (189) From (82), sin ce K 3 > K 2 , the quadratic equat ion in (188) has onl y one positive solu tion g i ven by x ∗ = − K 1 + p K 2 1 + ( K 3 − K 2 ) K 0 K 0 . (190) The o ptimal powe r po licy for this case is then the s et G of γ ∗ K for which γ ∗ K satisﬁes (185) with x = x ∗ in (190). The m aximum achiev able sum-rate for this case is then obtained from (186) as C    P k ∈K P k − ( x ∗ ) 2 P max N r    . (191) A P P E N D I X V I P RO O F O F T H E O R E M 1 1 W e no w prove Theorem 11 and gi ve the soluti on to t he max-mi n opt imization R K = max ( α K ,β K ) ∈ Γ min  I r, K ( α K ) , I d, K  α K , β K  . (19 2) As in Appendix V, a solution to t he max-min optimization in (192) si mpliﬁes to three mutuall y 45 exclusi ve cases [16, II.C] such that the max-min rule ( α ∗ K , β ∗ K ) satisﬁes the conditions for o ne of three cases. Th e con ditions for the three cases are Case 1 : I d, K ( α ∗ K , β ∗ K ) < I r, K ( α ∗ K ) (193) Case 2 : I r, K ( α ∗ K ) < I d, K ( α ∗ K , β ∗ K ) (194) Case 3 : I r, K ( α ∗ K ) = I d, K ( α ∗ K , β ∗ K ) (195) W e deve lop t he conditions and determine the max-m in rule for each case. W e ﬁrst consider case 1 and show th at this case is no t feasible. Case 1 : This ca se occurs when the maximum bound achie vable at the d estination is smaller than the bound at t he relay . Observe t hat I d, K ( α K , β K ) i n (60) decreases monot onically with α k , for all k , and, for any β K , achiev es a maximum at α ∗ K = 0 of I d, K ( α ∗ K , β K ) = C   P k ∈K P k + P r + 2 P k ∈K √ β k P k P r N d   . (196) Howe ver , substituti ng α ∗ K = 0 in (59), we obt ain I r, K ( α ∗ K ) = 0 (197) which contradicts the assu mption i n (193), thus maki ng this case infeasible. Case 2 : Consider the condit ion for Case 2 i n (194). Thi s condition implies that t he case occurs wh en the maxim um bound achiev able at the relay is small er than the bound at the destination. From (59), we observe that I r, K increases monot onically with α k for all k and achie ves a m aximum o f I r, K ( α ∗ K ) = C   P k ∈K P k N r   (198) at α ∗ K = 1 . Comparing (59) and (60) at α ∗ K = 1 , we obtain t he condition for this case as P k ∈K P k N r ≤ P k ∈K P k + P r N d . (199) Case 3 : Finally , consider Case 3 in (195). This case occurs when the maximu m rate bou nd achie vable at the relay and destinati on are equal. The max-min s olution for this case is obtained 46 by consi dering two sub-cases. The ﬁrst is the relatively st raightforward sub-case where α ∗ K = 1 is the max-min rule. The resulting maximum sum-rate is the same as that for case 2 with the condition in (199) satis ﬁed wi th equal ity . Consider the second su b-case wh ere α ∗ K 6 = 1 , i.e., P k ∈K P k N r > P k ∈K P k + P r N d . (200) In Appendix III we sho w that, for a ﬁxed α K , I d, S , is a concave function of β K for all S ⊆ K . Furthermore, from (57), for α K 6 = 1 , I d, K in (60) i s maximized by a β ∗ K whose entries β ∗ k , for all k ∈ K , satisfy P k ∈K β ∗ k = 1 (201) and are given as β ∗ k =    (1 − α k ) P k P K k =1 (1 − α k ) P k α K 6 = 1 0 α K = 1 for all k ∈ K . (202) Observe that t he optimal po wer fraction β ∗ k that the relay allocates to cooperating with user k is proportional to the power allocated by user k to achiev e coherent combining gains at the destination. Thus, one can formulate the o ptimization probl em for t his case as maximize I r, K ( α ) subject to I r, K ( α ) = I d, K  α , β  , P k ∈K β k = 1 . (203) Using Lagrange mult ipliers we can s ho w that i t suf ﬁces to consider β k = β ∗ k in the maxim ization. Since the opti mal β ∗ k in (202) is a functi on of α K , I d, K ( α K , β ∗ K ) simpliﬁes to a functi on of α K as I d, K ( α K , β ∗ K ) = C     P k ∈K P k + P r + 2 r P k ∈K (1 − α k ) P k P r N d     . (204) W e further sim plify I d, K ( α K , β ∗ K ) and I r, K ( α K ) as follows. W e write P max = max k ∈K P k , λ k = P k / P max , (205) 47 and q △ = r P k ∈K (1 − α k ) λ k . (206) Substitutin g (205) and (206) i n (59) and (60), we h a ve I r, K ( q ) = C       P k ∈K P k  − q 2 P max N r      (207) I d, K ( q ) = C       P k ∈K P k  + P r + 2 q √ P max P r N d      . (208) Observe that I r, K ( q ) and I d, K ( q ) are mon otonically increasing and decreasing functions of q and thus, the maximi zation in (20 3) sim pliﬁes t o determi ning a q s uch that P k ∈K P k − q 2 P max N r = P k ∈K P k + P r + 2 q √ P max P r N d . (209) The condition in (209) has t he g eometric interpretation that the boun ds on R K are maximized when the K -user sum rate plane achiev ed at the relay is tang ential t o the concav e sum-rate surface achieved at the desti nation at its maximu m value. W e further simpl ify (209) by us ing the deﬁnitions in Appendix V for K 0 , K 1 , K 2 , and K 3 . From (200) , s ince K 3 > K 2 , the qu adratic equation in (209) has only one posi ti ve solutio n give n by q ∗ = − K 1 + p K 2 1 + ( K 3 − K 2 ) K 0 K 0 . (210) The opti mal power policy for this case is th en t he set P of ( α ∗ K , β ∗ K ( α ∗ K )) such t hat α ∗ K satisﬁes (206) for q = q ∗ and for each such cho ice of α ∗ K , β ∗ K is given by (202). The maximum achiev abl e sum-rate for this case is then given by C       P k ∈K P k  − ( q ∗ ) 2 P max N r      . (211) Remark 12: The op timal q ∗ in (210) is t he same as the opt imal x ∗ in (190). Further , the 48 maximum i nner (DF) and outer b ounds on the su m-rate are also the same for the equal-bounds case in (211) and (191), respectively . A P P E N D I X V II S U M - C A P A C I T Y P RO O F F O R T H E A C T I V E C L A S S In Theorem 12, we proved that DF achie ves the sum-capacity for an activ e class of degraded Gaussian MARCs. In the proo f we ar gue that since the maximum DF sum-rate is the same as the maximum outer bound sum-rate, ever y DF max-min rule ( α ∗ K , β ∗ K ) ∈ P a that achie ves this maximum su m-rate, i.e., for which R r ( α ∗ K ) ∩ R d ( α ∗ K , β ∗ K ) b elongs to the set of activ e cases, also achie ves the sum-capacity . W e now present a m ore detailed proof of the ar gument. W e begin by comparing the inner and outer bounds. As in the symmetric case, wit hout loss of generality , we write γ k = (1 − α k ) β k for all k (212) where ( α K , β K ) ∈ Γ . W e then ha ve, B r, K ( α K , β K ) = C      P k ∈K P k N r −  P k ∈K p (1 − α k ) β k P k  2 N r      (213) and B d, K ( α K , β K ) = C    P k ∈K P k + P r + 2 P k ∈K p (1 − α k ) β k P k P r N d    = I d, K ( α K , β K ) . (214) Choosing β K as the DF max-mi n rule β ∗ K in (71), simp liﬁes (213) to B r, K ( α K , β ∗ K ) = C  P k ∈K α k P k N r  = I r, K ( α K ) . (215 ) Using theorem 11, one can then ve rify that B r, K ( α ∗ K , β ∗ K ) = B d, K ( α ∗ K , β ∗ K ) is achie ved by all α ∗ K ∈ P . Consi der a α ∗ K ∈ P a and a corresponding β ∗ K such that the DF region R r ( α ∗ K ) ∩ R d ( α ∗ K , β ∗ K ) belongs to the s et of active cases. From Theorem 11, this implies th at I d, A ( α ∗ K , β ∗ K ) + I r, A c ( α ∗ K ) > I ∗ = B ∗ for all A ⊂ K . (216) 49 Using (212), we expand B d, S in (11) as a f unction of ( α ∗ K , β ∗ K ) as B d, S ( α ∗ K , β ∗ K ) = C      P k ∈K P k +  1 −  P k ∈S c (1 − α ∗ k ) β ∗ k  P r + 2 P k ∈S p (1 − α ∗ k ) β ∗ k P k P r N d      (217) ≥ I d, S ( α ∗ K , β ∗ K ) (218) where (218) follows from the fact that (1 − α ∗ k ) β ∗ k ≤ β ∗ k , for all k and for all ( α ∗ K , β ∗ K ) . It is , howe ver , not easy to compare B r, S ( α ∗ K , β ∗ K ) with I r, S ( α ∗ K ) . Note, howev er , that the choice of γ k in (212) requires the s ame source-relay c orrelation v alu es for bot h the inner and outer bou nds. Furthermore, for e very cho ice of Gauss ian inp ut distribution with the same K correlation values for both bou nds, comparing the degraded cutset and DF bounds in (9) and (51), respective ly , for a constant U , we ha ve I ( X S ; Y r | X S c X r ) ≥ I ( X S ; Y r | X S c V K X r ) for all S ⊆ K (219) where in (219) we use the fact th at condi tioning does not increase ent ropy to show that the cutset bounds at the relay are less restrictiv e than the corresponding DF bo unds. From (215), the inequality in (219 ) simpliﬁes to an equali ty for S = K and for ( α ∗ K , β ∗ K ) ∈ P a when γ k is giv en by (212 ). Combinin g these inequali ties with (216), we then have B d, A ( α ∗ K , β ∗ K ) + B r, A c ( α ∗ K ) > I ∗ = B ∗ for all A ⊂ K . 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On the Sum-Capacity of Degraded Gaussian Multiaccess Relay Channels

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