On the Capacity Region for Index Coding
Fatemeh Arbabjolfaei, Bernd Bandemer, Young-Han Kim, Eren Sasoglu, Lele Wang
aa r X i v : . [ c s . I T ] J un On the Capacity Region for Index Coding
Fatemeh Arbabjolfaei, Bernd Bandemer, Young-Han Kim, Eren S¸ as¸o ˘glu, and Lele Wang
Department of Electrical and Computer EngineeringUniversity of California, San DiegoLa Jolla, CA 92093, USAEmail: { farbabjo, bandemer, yhk, esasoglu, lew001 } @ucsd.edu Abstract —A new inner bound on the capacity region of thegeneral index coding problem is established. Unlike most existingbounds that are based on graph theoretic or algebraic tools, thebound relies on a random coding scheme and optimal decoding,and has a simple polymatroidal single-letter expression. Theutility of the inner bound is demonstrated by examples thatinclude the capacity region for all index coding problems withup to five messages (there are 9846 nonisomorphic ones).
I. I
NTRODUCTION
Consider the simple communication problem in Figure 1,which is often referred to as the index coding problem. Thesender wishes to communicate N messages M j ∈ [1 : 2 nR j ] , j ∈ [1 : N ] , to their respective receivers over a commonnoiseless link that carries n bits X n . Each receiver j ∈ [1 : N ] has prior knowledge of M A j , i.e., a subset A j ⊆ [1 : N ] \ { j } of the messages. Based on this side information M A j andthe received bits X n , receiver j finds the estimate ˆ M j of themessage M j . A nontrivial tradeoff arises between the rates R j , j ∈ [1 : N ] , of the messages since receivers with incompatibleknowledge compete for the shared broadcast medium. M , . . . , M N X n Encoder Decoder Decoder Decoder N ˆ M ˆ M ˆ M N M A M A M A N Fig. 1. The index coding problem.
We define a (2 nR , . . . , nR N , n ) code for index coding byan encoder x n ( m , . . . , m N ) and N decoders ˆ m j ( x n , m A j ) , j ∈ [1 : N ] . We assume that the message tuple ( M , . . . , M N ) is uniformly distributed over [1 : 2 nR ] × · · · × [1 : 2 nR N ] ,that is, the messages are uniformly distributed and indepen-dent of each other. The average probability of error is thendefined as P ( n ) e = P { ( ˆ M , . . . , ˆ M N ) = ( M , . . . , M N ) } .A rate tuple ( R , . . . , R N ) is said to be achievable if there exists a sequence of (2 nR , . . . , nR N , n ) codes such that lim n →∞ P ( n ) e = 0 . The capacity region C of the indexcoding problem is the closure of the set of achievable ratetuples ( R , . . . , R N ) . (Similarly, one can define the zero-errorcapacity region, which is shown in [1] to be identical to thecapacity region.) The goal is to find the capacity region andthe optimal coding scheme that achieves it.Note that an index coding problem is fully characterizedby the side information sets A j , j ∈ [1 : N ] . As an example,consider the 3-message index coding problem with A = { } , A = { , } , and A = { } . We represent this problemcompactly as (1 | , (2 | , , (3 | , (1)or as a directed graph (see Figure 2(a)), where nodes representindices of the messages/receivers and edges represent avail-ability of side information (e.g., the edge → means thatside information M is available at receiver 2). In general, anindex coding problem ( j |A j ) , j ∈ [1 : N ] , can be representedby a directed graph G = ( V , E ) , where V = [1 : N ] and ( j, k ) ∈ E iff j ∈ A k .Note that the 3-message index coding problem in (1) can berepresented as an instance of the network coding problem [2]as illustrated in Figure 2(b). The same observation can bemade for any index coding problem; thus, index coding is aspecial case of network coding.
12 3 (a) M M M ˆ M ˆ M ˆ M (b)Fig. 2. (a) Directed graph representation. (b) The equivalent network codingproblem. Here every edge of the graph can carry up to 1 bit per transmission. irst introduced by Birk and Kol [3] in the context ofsatellite broadcast communication, the index coding problemhas been studied extensively over the past six years in thetheoretical computer science and network coding communitieswith many contributions of combinatorial and algebraic flavors(see, for example, [4]–[16] and the references therein). OurShannon-theoretic formulation of the problem closely followsthat of Maleki, Cadambe, and Jafar [17], who established thecapacity region for several interesting classes of index codingproblems using interference alignment [18]. Despite all thesedevelopments, the capacity region of a general index codingproblem is not known.Confirming Maslow’s axiom [19] “if all you have is ahammer, everything looks like a nail,” we propose a randomcoding approach, in contrast to more advanced coding schemesof an algebraic nature. This approach is more in the spirit ofthe original paper by Ahlswede, Cai, Li, and Yeung [2], whererandom coding (binning) was used to establish the networkcoding theorem. In particular, we develop a composite coding scheme based on random coding and establish a correspondingsingle-letter inner bound on the capacity region.Instead of mechanical proofs, this paper focuses on basicintuitions behind our coding scheme, which we develop gradu-ally from simpler coding schemes—“flat coding” in Section IIIand “dual index coding” in Section IV. The composite codingscheme is explained in Section V. The next section discussesknown outer bounds on the capacity region.II. O UTER B OUNDS
We first recall the following outer bound on the capacityregion (see, for example, [10] or [20] for a similar bound inthe context of a general network coding problem).
Theorem 1:
Let B j = [1 : N ] \ ( { j } ∪ A j ) be the indexset of interfering messages at receiver j . If ( R , . . . , R N ) isachievable, then it must satisfy R j ≤ T { j }∪B j − T B j , j ∈ [1 : N ] , for some T J , J ⊆ [1 : N ] , such that1) T ∅ = 0 ,2) T [1: N ] = 1 ,3) T J ≤ T K for all J ⊆ K ⊆ [1 : N ] , and4) T J ∩K + T J ∪K ≤ T J + T K for all J , K ⊆ [1 : N ] .The upper bound is established by using Fano’s inequalityand setting T J = (1 /n ) H ( X n | M J c ) . Properties 1–4 are dueto the submodularity of entropy.Recent results by Sun and Jafar [21] indicate that this outerbound is not tight in general. Nevertheless, a relaxed versionof the bound is sometimes useful. Corollary 1: If ( R , . . . , R N ) is achievable for an indexcoding problem represented by the directed graph G , then itmust satisfy X j ∈J R j ≤ for all J ⊆ [1 : N ] for which the subgraph of G over J doesnot contain a directed cycle. Fig. 3. A graph representation of the 5-message index coding problem.
The following example, due to [15], [17], illustrates that thetwo outer bounds do not coincide in general.
Example 1:
Consider the symmetric five-message indexcoding problem ( j | j − , j + 1) , j ∈ [1 : 5] , namely, (1 | , , (2 | , , (3 | , , (4 | , , (5 | , . The corresponding graph representation is depicted in Fig-ure 3. Applying Corollary 1, we obtain R + R ≤ , R + R ≤ , R + R ≤ ,R + R ≤ , R + R ≤ . (2)In comparison, Theorem 1 leads to the inequality R + R + R + R + R ≤ , (3)in addition to the five inequalities above. As we discuss inSection V, (2) and (3) characterize the capacity region of thisindex coding problem.III. F LAT C ODING
Consider the following simple random coding scheme. Foreach ( m , . . . , m N ) ∈ [1 : 2 nR ] × · · · × [1 : 2 nR N ] , generatea codeword x n ( m , . . . , m N ) randomly and independently asa Bern( / ) sequence. To communicate ( m , . . . , m N ) , thesender transmits x n = x n ( m , . . . , m N ) . Receiver j usessimultaneous nonunique decoding [22] and finds the unique ˆ m j ∈ [1 : 2 nR j ] such that x n ( ˆ m j , m A j , m B j ) is jointlytypical with (i.e., identical to) the received sequence x n for some m B j , where B j = [1 : N ] \ ( { j } ∪ A j ) . Sincethe codebook generation is “flat” (compared with “layered”superposition coding), simultaneous nonunique decoding isessentially identical to uniquely decoding ( ˆ m j , ˆ m B j ) and thendiscarding the unnecessary part ˆ m B j . This flat coding schemeyields the following inner bound. Proposition 1:
A rate tuple ( R , . . . , R N ) is achievable forthe index coding problem ( j |A j ) , j ∈ [1 : N ] , if R j + X k ∈B j R k < , j ∈ [1 : N ] . As an example, consider the 3-message problem in (1).Under flat coding, receiver 1 finds the unique ˆ m such that x n ( ˆ m , m , m ) = x n for some m ∈ [1 : 2 nR ] and the givenside information m . By the packing lemma [23, Sec. 3.4], itcan be readily shown that the probability of decoding error forreceiver 1 tends to zero as n → ∞ if R + R < . (4)imilarly, we obtain R < (an inactive bound) and R + R < . (5)By comparing with Theorem 1 (or Corollary 1), it can beeasily checked that the rate region characterized by (4) and (5)is indeed the capacity region.It can be easily verified that for all index coding problemswith 1, 2, and 3 messages—there are 1, 3, and 16 noni-somorphic problems [24]—this flat coding scheme (or moregenerally, time sharing of flat coding over different subsets ofmessages) achieves the capacity region. Among the 218 four-message index coding problems, time sharing of flat codingover subsets of messages achieves the capacity region for allbut three. The following is one of the three exceptions. Example 2:
Consider the 4-message index coding problem (1 | , (2 | , , (3 | , , (4 | , . On the one hand, flat coding yields an inner bound onthe capacity region that consists of the rate quadruples ( R , R , R , R ) such that R + R + R < ,R + R < ,R + R < . It can be verified that this inner bound cannot be improvedupon by time sharing over subsets. On the other hand, Theo-rem 1 (or Corollary 1) yields an outer bound that consists ofthe rate quadruples ( R , R , R , R ) such that R + R ≤ , R + R ≤ ,R + R ≤ , R + R ≤ . (6)We will see in Section V that this outer bound is tight.While flat coding is suboptimal in general, the analysis (i.e.,the proof of Proposition 1) is trivial and does not rely on anygraph theoretic machinery. This observation will be crucialwhen we generalize the coding scheme subsequently.IV. D UAL I NDEX C ODING
Before we move on to a more powerful random codingscheme, we introduce a communication problem (depicted inFigure 4) that is, in some sense, dual to the index codingproblem. Here a set of (2 N − senders wish to communicatea message tuple ( M , . . . , M N ) to a common receiver througha noiseless channel, each encoding a subtuple M J into aseparate index W J ∈ [1 : 2 nS J ] for all nonempty J ⊆ [1 : N ] .What is the capacity region (as a function of the rates S J )?This problem is a special case of the general multipleaccess channel (MAC) with correlated messages studied byHan [25]. For the general MAC, superposition coding achievesthe capacity region that is characterized by independent aux-iliary random variables U , . . . , U N , each corresponding to amessage. However, for the dual index coding problem, we cancharacterize the capacity region explicitly. M M J M , . . . , M N ˆ M , . . . , ˆ M N W W J W [1: N ] Encoder Encoder J Encoder [1 : N ] Decoder
Fig. 4. The dual index coding problem.
Proposition 2:
The capacity region of the dual index codingproblem is the set of rate tuples ( R , . . . , R N ) that satisfy X j ∈J R j ≤ X J ′ ⊆ [1: N ]: J ′ ∩J 6 = ∅ S J ′ (7)for all J ⊆ [1 : N ] .What is perhaps more important than this explicit character-ization of the capacity region is the fact that it can be achievedby flat coding, which we will utilize later.As an example, consider the three-message three-senderdual index coding problem in Figure 5, where S , = 1 , S , = S , , = 2 , and S = S = S = S , = 0 . By (7),the capacity region is the set of rate triples ( R , R , R ) suchthat R + R + R ≤ , R ≤ , R ≤ . This can be achieved via flat coding of ( M , M ) , ( M , M ) ,and ( M , M , M ) , respectively, and simultaneous decodingat the receiver. M , M M , M M , M , M ˆ M , ˆ M , ˆ M { , } Encoder { , } Encoder { , , } Decoder
Fig. 5. An instance of dual index coding.
V. C
OMPOSITE CODING
Equipped with the results in the previous two subsections,we now introduce a layered random coding scheme, whichwe refer to as composite coding . This is best described by anexample.Consider again the 5-message problem in Example 1. In thefirst step of composite coding, the sender encodes ( M , M ) into an index W , at rate S , using random coding, and simi-larly encodes ( M , M ) , ( M , M ) , ( M , M ) , and ( M , M ) ,respectively, into indices W , , W , , W , , and W , . Equiv-alently, the sender is decomposed into 5 “virtual” senders,each encoding one of the above pairs of messages (as inhe dual index coding problem). In the second step, thesender uses flat coding to encode the “composite” indices W , , W , , W , , W , , W , . As with encoding, decodingalso takes two steps. Each receiver first recovers all compositeindices, and then recovers the desired message from the com-posite indices. For example, receiver 1 recovers W , , W , (along with other composite indices). Since receiver 1 has sideinformation ( M , M ) , it can recover M from ( W , , W , ) if R ≤ S , + S , . Following similar steps for other receiversand incorporating the flat coding rate condition, it can beeasily verified that a rate quintuple ( R , R , R , R , R ) isachievable if R < S , + S , ,R < S , + S , ,R < S , + S , ,R < S , + S , ,R < S , + S , for some ( S , , S , , S , , S , , S , ) satisfying S , + S , + S , + S , + S , ≤ . Fourier–Motzkin elimination [23, Ap-pendix D] of the composite index rates yields the inequalities(2) and (3) that define the outer bound, thus establishing thecapacity region.Now consider the four-message problem in Example 2. Inthis case, we only use the composite indices W , and W , , , with rates S , and S , , , , respectively, and set the rates ofthe remaining indices to zero. It can be easily verified fromProposition 2 that receiver 1 can recover M if R < S , ; re-ceiver 2 can recover M (and M ) if R + R < S , + S , , , and R < S , , , ; receiver 3 can recover M (and M ) if R + R < S , + S , , , and R < S , , , ; and receiver 4can recover M (and M ) if R + R < S , + S , , , . Addingthe constraint S , + S , , , ≤ and eliminating S , and S , , , , we obtain the inequalities (6) that define the outerbound, thus establishing the capacity region.In general, we can utilize (2 N − virtual senders to encode N messages. Moreover, the receivers can employ simultaneousnonunique decoding in the second step (or equivalently, ignore some of the composite indices, as in the examples above). Thiscoding scheme is illustrated in Figure 6. It easily follows fromthe arguments above that if we allow decoder j to decode asubset K j of the messages, then the rates of the compositemessages need to belong to the polymatroidal rate region R ( K j | A j ) defined by X j ∈J R j < X J ′ ⊆K j ∪A j : J ′ ∩J 6 = ∅ S J ′ (8)for all J ⊆ K j \ A j . Note that this is the capacity region ofthe dual index coding problem (Proposition 2) with messageset K j and side information A j . Taking the union over allchoices of decoding sets K j yields the following inner bound,which is the main result of the paper. Theorem 2 (Composite-coding inner bound):
A rate tuple ( R , . . . , R N ) is achievable for the index coding problem ( j |A j ) , j ∈ [1 : N ] , if ( R , . . . , R N ) ∈ \ j ∈ [1: N ] [ K j ⊆ [1: N ]: j ∈K j R ( K j | A j ) (9)for some ( S J : J ⊆ [1 : N ]) such that P J : J 6⊆A j S J ≤ forall j ∈ [1 : N ] .At first glance, composite coding seems to be time sharingof flat coding over all subsets of [1 : N ] . However, it employsthe optimal decoding rule that utilizes all composite indices(subsets) that are relevant to the desired message. As such,the corresponding rate region has a very similar form as theoptimal rate region for interference networks with randomcoding [26].Using the polco tool for polyhedral computations [27], wehave computed the composite-coding inner bound and theouter bound in Theorem 1 for all 9608 nonisomorphic five-message index coding problems [24]. In all cases, inner andouter bounds agree, establishing the capacity region.To further demonstrate the utility of composite coding, werevisit the following example in [17]. Example 3:
Consider the N -message symmetric index cod-ing problem ( j | j − U, j − U + 1 , . . . , j − , j + 1 , . . . , j + D ) X n Encoder Decoder Decoder Decoder Decoder Decoder N Decoder N ˆ M ˆ M ˆ M N M A M A M A M A M A N M A N M M J M , . . . , M N ˆ W , ˆ W , . . . , ˆ W [1: N ] ˆ W , ˆ W , . . . , ˆ W [1: N ] ˆ W , ˆ W , . . . , ˆ W [1: N ] W W J W [1: N ] Encoder Encoder J Encoder [1 : N ] Fig. 6. Composite coding scheme. or j ∈ [1 : N ] , where all message indices are understoodmodulo N . For instance, the 5-message problem in Exam-ple 1 is a special case of this problem with N = 5 and D = U = 1 . We assume without loss of generality that ≤ U ≤ D ≤ N − U − . Let S J = 1 / ( N − ( D − U )) if J is of the form [ k : k + U ] , and let S J = 0 oth-erwise. Since receiver j ∈ [1 : N ] has M j +1 , . . . , M j + D as side information, it already knows the D − U compositeindices W [ j +1: j +1+ U ] , . . . , W [ j + D − U : j + D ] . Thus, there areonly N − ( D − U ) composite indices that need to be recov-ered from x n , which is feasible since P J : J 6⊆A j S J = 1 .Now receiver j can recover M j from the composite indices W [ j − U : j ] , . . . , W [ j : j + U ] , provided that R j < S [ j − U : j ] + · · · + S [ j : j + U ] . Hence, the symmetric rate of ( U + 1) / ( N − D + U ) isachievable. In [17] it is shown that this symmetric rate is infact optimal, which can be also verified directly by the outerbound in Theorem 1. For N = 6 , U = 1 , and D = 2 , that is, (1 | , , , (2 | , , , (3 | , , , (4 | , , , (5 | , , , (6 | , , , the symmetric rate of / is optimal. In fact, simplifyingTheorems 1 and 2 yields the capacity region that consists ofthe rate sextuples ( R , . . . , R ) such that R j + R j +2 ≤ , j ∈ [1 : 6] ,R j + R j +3 ≤ , j ∈ [1 : 6] ,R j + R j +1 + R j +2 + R j +3 + R j +4 ≤ , j ∈ [1 : 6] . In particular, this region is achievable by using compositeindices W , W , W , W , W , W , W , , W , , W , , W , , W , , and W , .VI. C ONCLUDING R EMARKS
Based on a first principle in Shannon’s random coding, thispaper has established the composite-coding inner bound onthe general index coding problem. This inner bound is simple,easy to compute, yet is tight for all index coding problems ofup to five messages as well as many existing examples. In asense, random coding is a “jackknife” rather than a “hammer.”The polymatroidal structure of the composite-coding innerbound and the submodularity of the outer bound suggest adeeper connection rooted in matroid theory [20], [28]. Inaddition to evaluating the inner and outer bounds for moreexamples (there are 1540944 nonisomorphic six-message in-dex coding problems), future studies will focus on analyzingthe algebraic structures of these bounds to investigate whatlies in the path to establishing the capacity region of a generalindex coding problem. R
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