An Achievable Rate Region for Two Groupcast Messages over the K-User Broadcast Channel and Capacity Results for the Combination Network

An Achievable Rate Region for Two Groupcast Messages over the K-User Broadcast Channel and Capacity Results for the Combination Network

Mohamed Salman and Mahesh K. Varanasi
This work was presented in part at the 2018 IEEE International Symposium on Information Theory, Vail, CO [1]. The authors are with the Electrical, Computer and Energy Engineering Department, University of Colorado, Boulder, CO, USA (emails: {mohamed.salman, varanasi}@colorado.edu).
Abstract

Using an order-theoretic framework, a novel achievable rate region is obtained for the general -receiver discrete memoryless broadcast channel over which two groupcast messages are to be transmitted, with each message required by an arbitrary group of receivers. The associated achievability scheme is an amalgamation of random coding techniques with novel features including up-set message-splitting, message set expansion including the generation of possibly multiple auxiliary codebooks for certain compositions of split messages using superposition coding with subset inclusion order, partial interference decoding at all receivers in general, joint unique decoding at receivers that desire both messages, and non-unique or indirect decoding at receivers that desire only one of the two messages. While the generality of such a scheme implies that its rate region coincides with all previously found capacity regions for special classes of broadcast channels with two private or two nested groupcast messages, wherein the group of receivers desiring one message is contained in that desiring the other, we show that, when specialized to the so-called combination network, our inner bound coincides with the capacity region for three different scenarios, namely, (a) the two messages are intended for two distinct sets of receivers each and (b) two nested messages in which one message is intended for one or (c) two (common) receivers and both messages are intended for all other (private) receivers. Moreover, we show that there is a trade-off between the complexity of the coding scheme and that of the distribution of the auxiliary random variables and the encoding function that must be chosen to achieve the capacity region in these scenarios.

I Introduction

The problem of sending two groupcast messages over the -receiver broadcast channel (BC) is studied. Each such message is intended for a distinct group of receivers, with the two groups of receivers assumed to be arbitrary in general. In spite of its apparent simplicity, this problem remains unsolved in general in the Shannon-theoretic sense. However, some partial capacity results, mainly in two- and three-receiver cases, have been obtained in the literature.

The most studied problem of sending two messages over the BC is the two-receiver discrete memoryless (DM) case with private messages. The capacity region is notoriously difficult in this case and remains unsolved in general to date. However, for the increasingly larger classes of degraded [2], less noisy [3, Definition 2] and more capable [3, Definition 3] channels, the capacity region was found in the series of papers [4, 2, 5, 6, 7] in the 1970s. In particular, the superposition coding scheme proposed in [2] was shown, using a clever identification of auxiliary random variable, to achieve the capacity region in [5] for the degraded BC. The same scheme was also shown to achieve the capacity region for the larger class of less noisy and more capable BCs in [3] and [7], where the images-of-a-set technique [6] and the Csiszar sum lemma [8, Lemma 7] were used to prove the converses, respectively.

The capacity region for the two-receiver DM BC with two nested (i.e., degraded) messages was found by Korner and Marton in 1977 [9]. Interestingly, with superposition coding as the achievability scheme and a converse based on the images-of-a-set technique [6], the authors therein established the capacity region without any restriction on the channel. However, the generalization of this result for three or more receivers has remained elusive for decades.

In the -receiver BC with two nested messages the receivers can be classified into common receivers that require only one (common) message and private receivers that require both messages (with ). The result of Korner and Marton in [9] might suggest that the nested structure of the messages might render a straightforward extension of their superposition coding scheme to be capacity-optimal even in this -receiver setting. However, the authors of [10] and [11] showed that superposition coding alone is not optimal for the three-receiver DM BC with one and two common receivers, respectively. In the latter case, they proposed a more general scheme that involves a simple form of rate-splitting along with superposition coding [11]. However, even this scheme was only shown to achieve capacity for the restricted class of DM BCs wherein the private receiver is less noisy than one of the two common receivers.

One of the challenges of obtaining capacity results for rate-splitting based schemes beyond the three-receiver case is the difficulty of obtaining a closed-form polyhedral description for the inner bound in terms of the message rates due to the large number of split rates possible. We make progress on this problem in [12] where an achievable rate region that generalizes in one direction the capacity result for the three-user, two-common receiver problem in [11] to arbitrary and arbitrary is obtained. In particular, the private message is split into sub-messages, and each common receiver decodes the common message uniquely, and certain sub-messages of the private message assigned to it, indirectly [11]. The inner bound is presented in terms of the two nested message rates only, by eliminating all split rates for any and any , in general. Also, this inner bound is shown to be capacity-optimal for classes of channels characterized by certain pair-wise relationships between and among the common and private receivers. For example, the scheme is optimal for the class of four-receiver DM BCs with and in which the private receiver is less noisy than two of the three common receivers.

Inspired by the general order-theoretic framework of Romero and Varanasi on rate-splitting and superposition coding for the -receiver DM BC with general message sets in [13], we propose a new inner bound for the DM BC with two groupcast messages that, when specialized to the case of nested messages, relates to our previous results in [12] in the following ways (a) it incorporates more general message splitting, wherein the private message is now split into parts, instead of parts, hence (b) subsuming the inner bound of [12], and thereby being a fortiori capacity-optimal for all classes of DM BCs for which capacity was characterized in our work in [12] and (c) testing its efficacy beyond those classes, by the criterion of whether or not, when specialized to the so-called combination network (cf. [14, 15]), it yields its capacity region. In a way, our aim is to have proposed an inner bound for the DM BC that is sufficiently strong, in spite of not incorporating binning, so as to be capacity-optimal when specialized to the combination network, for any and any two groupcast messages. While the establishment of such a result appears to be elusive, we provide instances of special message sets, but for general , for which the inner bound proposed in this work does in fact achieve our aim in those instances, namely, (a) two messages each intended by a distinct set of receivers and (b) two nested messages with one or (c) two common receivers.

Note that in [13], the authors consider general message sets with any number of messages, and propose a coding scheme which depends on a flexible form of message set expansion, up-set rate-splitting, superposition coding, and unique decoding. In this paper, we use a coding scheme similar to that in [13] but with non-unique decoding. In particular, each receiver decodes only the desired message uniquely and any other sub-messages assigned to the receiver due to message splitting, non-uniquely. The general rate-splitting strategy that we use here comes at the expense of the ability to eliminate the split rates for arbitrary and , in contrast to our work in [12]. The description of the achievable region uses the order-theoretic framework developed for the multiple-access channel with general message sets by Romero and Varanasi in [16] and its conference versions, which the same authors later applied to the DM BC in [13]. One of the features of this paper not seen in [13] however is an investigation of a possible trade-off between a choice of expanded message set and the choice of random coding distribution that suffices to attain capacity in the three instances of messaging in the combination network mentioned previously for which we find capacity.

The capacity regions of the combination network for the two- and three-receiver cases were established in [17] under the guise of fundamental constraints in multicast capacity regions and where the transmitter must transmit all possible independent groupcast messages. The achievability scheme depends mainly on the rate transfer argument. For example, in a two-receiver combination network, we have three possible independent messages; two private messages and one common message. If the transmitter is able to simultaneously send a rate of 1 bit per channel use for each of the three messages, then by sending the same information in each of the two private messages, it must be able to send a common message at rate of 2 bits per channel use for both receivers. Another possible rate transfer operation is when the transmitter merely uses the common bit to send private information to one of the receivers. Then, the channel can deliver 2 bits of private message per channel use to that receiver (and 1 bit to the other). In other words, the achievability of any -dimensional rate of messages implies the achievability of another -dimension rate vector regardless of the channel. The approach of [17] is to exhaustively determine all possibilities for rate transfer to characterize the inner bound for , whereas for , rate transfer and network coding are employed to establish the inner bound. On the other hand, the outer bound depends on cut-set bounds with some extensions. These proofs of the converse are specific to being two or three, and hard to extend to . In fact, the capacity region of the general -receiver combination network is an open problem to date for .

Because it is unclear how to generalize the approach of [17] to more that three receivers since the complexity of rate transfer increases exponentially with the number of users, Tian in [18], under the guise of latent capacity regions, effectively considers a restricted class of symmetric -receiver combination networks with the capacities of certain sets of finite capacity links being the same, and with symmetric message rates, wherein the messages required by the same number of receivers have the same rate. By simplifying the channel model and the message structure in this manner, Tian was able to establish the symmetric capacity region (where the rates of all messages of the same order are equal) of the symmetric -receiver combination network by extending the rate-transfer approach of [17] to this scenario.

Later, in [15], Salimi et al. proposed a general framework for the outer bound of broadcast networks in which they obtain a large family of outer bounds based on the sub-modularity of entropy they call generalized cut-set bounds. These bounds are used to reproduce the outer bounds of [17] for the two- and three-receiver combination networks and, along with an explicit polyhedral description, the symmetric capacity region of the -receiver symmetric combination network of [18].

Romero and Varanasi in [19] established the capacity region for the combination networks via a different approach from the one used in [17] for and that used in [18] for -user symmetric combination networks. They started with the general inner bound proposed in [13] for -user DM BC with general message sets and then specialized to the combination network to recover the results in [17] and [18]. This bolsters the case for considering superposition coding and rate splitting (i.e., without binning) for achieving the capacity region, not only the symmetric capacity region, of the general (asymmetric) combination network with . We adopt the approach in [19] to strengthen this case and achieve some success in this regard.

In this paper, instead of following the approach in [18, 15] by simplifying the channel model and the message structure, we consider the general, i.e., asymmetric, combination network but, for the sake of simplicity, restrict attention to two groupcast messages. We establish the capacity region for different message set scenarios mentioned previously using a similar approach to the one in [19] which is to start with an inner bound for the more general DM BC and specialize it to the combination network. In particular, we show that a single coding distribution of the auxiliary random variables is “extremal” in that the rate region corresponding to that distribution subsumes the rate regions associated with all other admissible coding distributions. A converse result is supplied to establish that this rate region is the capacity region of the combination network for the aforementioned special cases of message sets with two messages.

Note that the special case of the problem of transmitting two nested messages for the -receiver combination network was also addressed in [20]. It was shown that a linear network coding scheme, wherein the source transmits linear combinations of the information symbols, achieves the capacity region for the combination network with two common receivers. More precisely, the transmitted signal is obtained by the multiplication of a carefully designed matrix over a finite field with the information symbols vector over that field. The structure of this matrix follows the zero-structured matrices [20, Definition 2] while the rank of this matrix dominates the decoding feasibility analysis.

The main difference between that result and our result on the combination network is again that we establish the capacity region in a top-down manner, by starting with the DM BC and then specializing to the combination network. More importantly, the particular description of the capacity region for the combination network given in this paper is more structured and succinct. Our work here also provides what we believe to be the right framework in which the capacity of the combination network in the general cases of two groupcast messages (not just nested mesages) can be addressed. To prove this point, we establish the capacity region of the combination network for two messages, each intended for a distinct set of receivers. In contrast, the framework used in [20] does not appear to lend itself to an extension to two general messages.

The rest of this paper is organized as follows. In Section II, we state the system model and present the notation and definitions. We devote Section III to establish the new inner bound for two general messages. This inner bound is specified for the nested messages case in Section IV. Then, in Section V, we establish the capacity region for combination networks for three different message sets. In Section VI, a trade-off between complexity of coding scheme (via message set expansion) and choice of random coding distributions is studied. Finally, the paper is concluded in Section VII.

Ii System Model and Preliminaries

Ii-a System Model

Encoder

DM BC

Decoder

Decoder

Decoder

Fig. 1: The three-receiver DM BC where are the messages sent through the channel and are the messages decoded by receiver .

The DM BC consists of one transmitter , receivers , and the channel transition probability where the conditional probability of channel outputs (, conditioned on channel inputs () is given by

(1)

The message of rate is indexed by the subset of receivers it is intended for. Define as the set of all message indices (which are subsets of ) and let be the power set of excluding the empty set. In general, .

For any and , define as

(2)

Denote the set of all messages to be sent over a -user DM BC as . A code consists of (i) an encoder that assigns to each message tuple a codeword (ii) a decoder at each receiver, with the decoder mapping the received sequence for each into the respective decoded messages , denoted as . The three-receiver DM BC is illustrated in Fig. 1. The probability of error is the probability that not all receivers decode their intended messages correctly. The rate tuple is said to be achievable if there exists a sequence of codes with as . The closure of the union of achievable rates is the capacity region.

When describing examples, we find it convenient to make certain notational simplifications when no confusion arises. For example, consider the three-receiver DM-BC with the message index set , so that there are two messages and , the first one intended for the first receiver and the second for all three receivers. For simplicity, we will denote these messages as and . Similarly, we will write their rates and simply as and . Also, for convenience, we denote in this case. In other words, for simplicity, and when there is no confusion, we abbreviate the set for any positive number as (adopting the convention that ). Note that with this notational simplification, when , we have .

In some cases, especially when the set has many elements, we find it more convenient to denote it by its complement. For example, the common message intended for all receivers is denoted by . It is simpler to denote it as where is the empty set and for any . Similarly, we can represent the message index set of two messages each required by receivers as .

The combination network [15, 19], which is a special case of the general DM-BC, is described next. It consists, as described in [15], of three layers of nodes, as shown in Fig. 2 for the three-receiver case. The top layer consists of a single source node , and the bottom layer consists of receivers , . The middle layer consists of intermediate nodes, denoted for all . The source is connected to each of the intermediate nodes through a noiseless link of capacity (per channel use). Receiver is connected to the intermediate nodes for all via noiseless links of unlimited capacity. An equivalent representation for the combination networks is given in [19] wherein the combination network is considered to be a network of noiseless DM BCs with the channel input connected in different ways to the channel outputs () each through a noiseless BC. In particular, the channel input contains components , for all . For each , the component , where , is noiselessly received at each receiver for all and not received at the receivers with , i.e., .

Fig. 2: A combination network with intermediate nodes and three receivers. The dark lines represent finite capacity links while the dashed lines represent infinite capacity links. The capacity of the dark line connecting the node to the node is per channel use for each . For brevity, the source/destination nodes are denoted by their transmitted/received symbols and the intermediate nodes by their output symbols.

Ii-B Just Enough Order Theory

We introduce ideas from order theory following the notation in [21]. Any set equipped with an order is an ordered set. Let be such an ordered set and be a subset of . We say that is

  1. an up-set if , , and implies .

  2. a down-set if , , and implies .

Note that these two types of subsets are duals of each other, i.e., if is a down-set then is an up-set. Moreover, for any subset , we define the smallest down-set containing as and the smallest up-set containing as . Further, for any , denote the part of the smallest down-set containing that is also in , i.e., , as . Similarly, , the smallest up-set of that is in , is denoted as . Henceforth, for brevity, (or ) is referred to as the down-set (or up-set, respectively) of in .

Also, let denote the family of all down-sets of and denote the family of all up-sets of . Finally, let and denote the family of all up-sets and all down-sets of that contain , respectively.

In this paper, we will take the ground set to be a set of sets, such as the set of non-empty subsets of , the receiver index set. We will denote a set of sets in sans-serif font to distinguish it from sets. The order on the ground set considered in this paper is exclusively that of set inclusion, i.e., if and only if . Recall that, for simplicity, we write the index set as (adopting the convention that ). To illustrate such notation, consider the example of . The ground set in this case could be the set of all non-empty subsets of , denoted as . The down-set of, say, in is , and the up-set of in is . For the same , we have , whereas .

To illustrate families of up-sets and down-sets, consider the ground set . Then, , while . For the same , we have and .

Lemma 1.

The following relationships are true:

  1. For any set , we have

    (3)
    (4)
  2. For any set of sets ,

    (5)
    (6)
  3. For any set and

    (7)
    (8)
Proof.

The proofs of all the above equalities are straightforward given the order theoretic definitions except that of (7), which is given in Appendix A. ∎

Iii Two Messages

This paper is devoted to the problem of sending two groupcast messages over the -receiver DM-BC. Let the two general messages be and , so that the message index set is . Without loss of generality, we let and . The set of indices of receivers that decode both messages is denoted as , that decode only is denoted by , and that decode only is denoted by . The receivers with indices in can be thought of as private receivers, the receivers with indices in can be thought of the first group of common receivers that decode only , and the receivers with indices in can be thought of the second group of common receivers that decode only . Note that . Of special interest in this paper are two special cases (a) two nested messages so that either or and (b) two order- messages so that and .

Next, we obtain a new inner bound for the -user DM BC with two general messages. We use order theory to describe our result. In particular, let , the set of all non-empty subsets of receiver indices , be the ground set. As stated previously, we will think of as an ordered set with the order relation defined by set inclusion, i.e., if and only if . Evidently, the message index set .

Theorem 1.

Let be some message index superset so that . The rate pair is achievable if there exist non-negative up-set split rates () such that for each

(9)

and reconstruction rates

(10)

that satisfy the inequalities

(11)

and, for each , the inequalities

(12)

for some time sharing and auxiliary random variables and with a joint distribution that factors as and taken to be a deterministic function of .

Proof.

A detailed proof is given in Appendix B. We only provide an outline here. Each of the two messages is divided into a collection of sub-messages, , for each as per (9). This form of rate-splitting is called up-set rate splitting in [13] which considers general message sets. By reassembling the sub-messages, we obtain the reconstructed message for all with rate given by (10). We will refer to the expansion of the original message index set to that of reconstructed message index set (via message splitting and reconstruction) succinctly as message set expansion. The set of reconstructed messages with indices in are encoded using superposition coding as described in Appendix B. Private receiver (with ) jointly decodes the desired messages and via the unique joint decoding of the set of reconstructed messages that contain those two messages. As shown in Appendix B, the reconstructed messages can be reliably transmitted over the DM BC if the partial sums of the reconstructed message rates satisfy the inequalities given by (11). On the other hand, the common receiver (with , ) only needs to decode the message . Hence, non-unique decoding can be employed by these receivers. Note that for each , the reconstructed messages contain the desired message as well as partial interference via up-set message splitting and reconstruction. Thus, among these reconstructed messages, only the reconstructed messages with indices in are uniquely decoded because these messages contain the desired message , whereas the rest of the reconstructed messages do not, and these messages are hence decoded non-uniquely. This happens successfully with high probability if the partial sums of the reconstructed message rates satisfy the inequalities given by (12). ∎

Remark 1.

Note that a common receiver , which needs to decode only one message, is required to uniquely decode all reconstructed messages with indices in because when up-set rate splitting is used, a part of the receiver’s desired message becomes some part of all such reconstructed messages as per (9) and (10).

Remark 2.

In [13, Theorem 2], an inner bound for a general message set was proposed that used the same encoding scheme but with a different decoding strategy. In particular, each receiver () uniquely decodes all reconstructed messages with indices in that contain the desired messages as well as, for some receivers, partial interference assigned to it via message-splitting and reconstruction. This causes some receivers to decode uniquely undesired messages which in turn produces more inequalities on the reconstruction rates. In Theorem 1, we avoid this by employing non-unique decoding at the common receivers , i.e., for , instead.

Fig. 3: A Hasse diagram for the coding scheme for Example 1 where the message index superset . Each line represents superposition coding with codebooks generated top to bottom, i.e., we first generate the codebook (whose codewords are “cloud centers”) that represents . Then, using superposition coding, we conditionally independently generate the codebooks and (conditioned on the cloud centers) that represent and , respectively. The corresponding codewords form three primary satellite codebooks for each cloud center. Finally, for each and satellite codeword pair, we generate the secondary satellite codebook that represents the message . Moreover, for each and satellite codeword pair, we generate a single codeword dependent on that pair (since ). Similarly, for each and satellite codeword pair, we generate a single codeword dependent on that pair (since ). See Remarks 4 and 3 for further discussion of this point.
Example 1.

Consider the case and so that , , and and . Choose the message index superset . Up-set message splitting described in the proof of Theorem 1 yields and with split rates defined according to (9). The reconstructed messages and their rates as per (10) are given as

The resulting rate-splitting/superposition coding scheme described in the proof of Theorem 1 is illustrated in Fig. 3 with the specifics explained in its caption. From the conditions for reliable communication of the messages at their desired destinations given in (12) (note that (11) is vacuous since in this example) of Theorem 1, we get that the reconstructed message rates must satisfy the inequalities

for some are achievable.

Note that the inner bound for the same example using the result in [13, Theorem 2] has the two additional inequalities

because in that scheme receiver uniquely decodes the undesired sub-message and receiver uniquely decodes the undesired sub-message , whereas in the scheme of Theorem 1 those sub-messages are decoded non-uniquely at Receivers 2 and 3, respectively.

We will see later that the non-unique decoding employed in proving Theorem 1 is useful for simpler characterizations of the capacity region of the combination network for certain pairs of messages.

Remark 3.

In Theorem 1, for every possible message set expansion from to such that , we get a different achievable region which involves a different set of auxiliary random variables. Expanding leads to finer message splitting (and hence using more auxiliary random variables/codebooks) and it therefore cannot reduce the achievable region. Hence, the full power of the coding scheme of Theorem 1 is realized by setting . Nevertheless, we prefer to leave as a parameter to be chosen rather than replace it with in Theorem 1 since a smaller leads to a simpler coding scheme and sometimes a specific such choice suffices to achieve capacity (as we illustrate later). Interestingly, note that choosing any yields some zero reconstruction rates in (10) and this point is illustrated in the next remark.

Remark 4.

When we choose in Example 1, we get two zero reconstruction rates, namely, and , per (10). This is reflected in Fig. 3 which depicts the superposition coding scheme described in Appendix B for Example 1. In particular, the codewords and do not encode more messages than those already encoded in , , and , i.e., for every pair of codewords and , we generate a single codeword according to . Similarly, for every pair of codewords and , we generate a single codeword according to . However, since , we generate codewords for every pair of codewords and . Hence, in general, in the coding scheme of Theorem 1, superposition coding is not only used to encode a message over other messages (satellites over cloud centers), but also to encode some messages multiple times using different distributions. This novel feature of generating a single satellite per cloud center will be present in general as long as we choose such that .

Iii-a Explicit polyhedral representation for the inner bound with

For this case, we have , and . Hence, . We use Theorem 1 to get a polyhedral description of the inner bound by eliminating the split rates. Here, the message is split into two parts via (9), i.e., while the other message is split into and . The polyhedral representation is presented in the next corollary.

Corollary 1.

An inner bound of -user DM BC for the message index set is the set of rate pairs () satisfying

(13)
(14)
(15)
(16)
(17)
(18)

for some and as a deterministic function of .

Proof.

The proof begins with the result of Theorem 1 by setting . Since we have and , we have only three non-zero reconstruction rates from (10), namely, , such that

(19)
(20)
(21)

Also, from (9), we know that each message is split only into two parts so that

(22)
(23)

Hence, we can write (11) as follow