Topological Interference Management with
Decoded Message Passing
The topological interference management (TIM) problem studies partially-connected interference networks with no channel state information except for the network topology (i.e., connectivity graph) at the transmitters. In this paper, we consider a similar problem in the uplink cellular networks, while message passing is enabled at the receivers (e.g., base stations), so that the decoded messages can be routed to other receivers via backhaul links to help further improve network performance. For this TIM problem with decoded message passing (TIM-MP), we model the interference pattern by conflict digraphs, connect orthogonal access to the acyclic set coloring on conflict digraphs, and show that one-to-one interference alignment boils down to orthogonal access because of message passing. With the aid of polyhedral combinatorics, we identify the structural properties of certain classes of network topologies where orthogonal access achieves the optimal degrees-of-freedom (DoF) region in the information-theoretic sense. The relation to the conventional index coding with simultaneous decoding is also investigated by formulating a generalized index coding problem with successive decoding as a result of decoded message passing. The properties of reducibility and criticality are also studied, by which we are able to prove the linear optimality of orthogonal access in terms of symmetric DoF for the networks up to four users with all possible network topologies (218 instances). Practical issues of the tradeoff between the overhead of message passing and the achievable symmetric DoF are also discussed, in the hope of facilitating efficient backhaul utilization.
As the cellular network becomes larger, denser and more heterogeneous, interference management is increasingly crucial and challenging. The substantial gain promised by sophisticated interference management techniques (e.g., interference alignment ) requires usually that (almost) perfect and instantaneous channel state information at the transmitters (CSIT) is accessible. Nevertheless, to obtain CSIT perfectly and instantaneously is challenging, if not impossible. Especially when the number of users/antennas is large or the channel changes rapidly, it will be expensive to obtain CSIT timely with reasonable accuracy. Relaxations of the perfect and instantaneous CSIT requirements have been investigated in various networks (e.g., instantaneous CSIT with limited accuracy , perfect but delayed CSIT ). However, if only finite-precision CSIT is available, the system degrees of freedom (DoF) value, i.e., roughly speaking the number of non-interfering Gaussian channels that the system is able to support simultaneously, collapses to the situation as if no CSIT was available at all [4, 5]. Indeed, with no CSIT, the transmitters cannot distinguish different receivers, and are totally blind.
The no-CSIT assumption is somewhat too pessimistic. In fact, certain coarse channel information (e.g., channel fading statistics, strength, and users’ locations) is easily obtained even in today’s practical systems. For instance, if the fading channels of different users follow some structured patterns, then blind interference alignment could improve DoF beyond the absolutely no CSIT case . In addition, the DoF collapse was observed under the assumption that the wireless network is fully connected, so that interference is everywhere no matter whether it is strong or weak enough to be negligible. Intuitively, it makes no sense for a system designer to take into account the interference from very far away base stations. As the interference power rapidly decays with distance for distances beyond some critical threshold due to shadowing, blocking, and Earth curvature, interference from some sources is inevitably weaker than others, which suggests the use of a partially-connected bipartite graph to model, at least approximately, the network topology.
Interference networks with no channel state information (CSI) except for the knowledge of the connectivity graph at the transmitters have been considered under the name of the “topological interference management (TIM)” problem . It has been shown that substantial gains in terms of DoF can be obtained with only this topological information for partially-connected interference networks. Surprisingly, one half DoF per user, which is optimal for an interference channel with perfect and global CSIT, can be attained for some partially-connected interference channels with only topological information. Its substantial reduction of CSIT requirement has attracted a lot of followup works aiming at various aspects, such as the consideration of fast fading channels [8, 9], alternating connectivity [10, 11], multiple antennas , and cellular networks [13, 14, 15]. The TIM problem was also nicely bridged to the “index coding” problem , where the former offers well-developed interference management techniques (such as interference alignment) to attack the latter, and also serves as an intriguing application of great practical interest in wireless networks for the latter.
Recently, the TIM problem under a broadcast setting with distributed transmitter cooperation in the downlink cellular network was considered in . It has been shown that, if message sharing is enabled at the base stations, higher rate transmission can be created by allocating messages to the transmitters in a way such that the interference can be perfectly avoided or aligned. As a dual problem, a natural question then to ask is, whether receiver cooperation (or cooperative decoding at the base stations) in the uplink cellular networks also offers us some gains under the TIM setting.
Cooperative decoding at the base stations in the uplink cellular networks was widely studied (e.g., [18, 19]), where the received signals are shared among base stations via backhaul links so that joint signal processing is enabled. Nevertheless, joint signal processing and decoding results in huge amount of backhaul overhead, even if the quantized received signal samples are shared locally in a clustered decoding fashion . Most recently, a new type of local base station cooperation framework in uplink cellular networks was studied in  to boost the overall network performance. Differently from the strategy of sharing quantized received signals, the authors in  considered a successive decoding policy, in which the message at each receiver is decoded based on the locally received signal as well as the decoded messages passed from neighboring base stations that have already decoded their messages at an earlier stage. It has been shown that the local and single-round (non-iterative) message passing enables interference alignment without requiring symbol extensions or lattice alignment. For these results, it is crucial to exploit the partial connectivity of the interference graph while, as usual, the local interference alignment scheme requires perfect instantaneous CSIT. A natural question is whether the CSIT requirement can be relaxed in the decoded message passing setting. More specifically, with decoded message passing, is it possible to attain performance gain in partially-connected cellular networks with only topological information?
In this work, we formally formulate the TIM problem with decoded message passing at the receivers, referred to as the “TIM-MP” problem. As soon as a receiver decodes its own message, it can pass its message to any other receivers who are interested. Building on this decoded message passing setting, we model the interference pattern by conflict digraphs, and connect orthogonal access to the acyclic set coloring on conflict digraphs. With the aid of polyhedral combinatorics, we identify certain classes of network topologies for which orthogonal access achieves the optimal DoF region. The relation to index coding is also studied by formulating a generalized index coding problem with successive decoding. Reducibility and criticality are also discussed in the hope of reducing large-size problems to smaller ones. By reducibility and criticality, the linear optimality of orthogonal access in terms of symmetric DoF is also shown for the small-size networks up to four users with all possible 218 non-isomorphic topologies. Practical issues for TIM-MP problems such as the tradeoff between the overhead of message passing and the achievable symmetric DoF are also discussed in the hope of facilitating the most efficient backhaul utilization.
More specifically, our contributions are organized as follows.
In Section III, we model the interference pattern by a conflict directed graph, turning the interference between different transmitter-receiver pairs (i.e., the respectively desired messages) to the directed connectivity between nodes (representing the corresponding messages) in a directed graph. By this graphic modeling, we connect orthogonal access of TIM-MP problems to acyclic set coloring on conflict digraphs, where the latter is well-studied in the graph theory literature. The achievable symmetric DoF due to single-round and multiple-round message passing are connected to two graph theoretic parameters, dichromatic number and fractional dichromatic number. Thanks to the equivalence between local coloring and one-to-one alignment, we also show that one-to-one alignment boils down to orthogonal access as a result of decoded message passing, by proving that local acyclic set coloring is not better than acyclic set coloring.
We establish in Section IV the outer bound of the achievable DoF region by cycle and clique inequalities, and further connect it to set packing and covering polytopes. With the aid of polyhedral combinatorics, we identify sufficient conditions for which orthogonal access (i.e., fractional acyclic set coloring) achieves the optimal DoF region. Such conditions ensure the integrality of the outer bound of DoF region polytopes, where the integral extreme points of the polytopes can be achieved by acyclic set coloring. Time sharing among the integral extreme points yields the whole DoF region.
The relation to index coding is also studied in Section V, showing that TIM-MP corresponds to a generalized index coding problem, referred to as successive index coding (SIC). Generalizing conventional index coding, SIC allows successive decoding at the receivers, where as soon as a receiver decodes its desired message, it can declare it and pass it to other receivers as additional side information. The decoding and message passing orders play an crucial role, which makes SIC a more complex combinatorial problem. The analogous coding schemes to (partial) clique covering are also given. The vertex-reducibility and arc-criticality of SIC are also investigated, by which SIC problems with large vertex/arc size can be reduced to ones with smaller size.
The linear optimality of orthogonal access is considered in VI. We first consider some special network topologies that do not satisfy the sufficient conditions in Section IV, and then prove the linear optimality of orthogonal access with respect to symmetric DoF, if restricted to linear schemes. Thanks to the vertex-reducibility and arc-criticality investigated in Section V, the linear optimality of symmetric DoF or broadcast rate for small networks up to 4 users with all possible network topologies are fully characterized by orthogonal access.
The practical issue on the tradeoff between achievable symmetric DoF and the number of passed messages is also discussed in Section VII. We identify a sufficient condition under which only one message passing is helpful to improve the DoF region. The tradeoff between achievable symmetric DoF and the overhead of message passing is formulated as a matrix completion problem, which can be solved algorithmically although closed-form solution remains a challenge.
Notations: Throughout this paper, we define , and for any integer . Let , , and represent a variable, a set, and a matrix, respectively. In addition, is the complementary set of , and is the cardinality of the set . The set or represents a set or tuple indexed by . represents the -th entry of the matrix . Define and . and represent respectively the all-zero and all-one vectors.
Ii System Model
Ii-A Channel Model
We consider the uplink of a cellular network with user terminals (i.e., transmitters) that want to send messages to base stations (i.e., receivers), respectively. The base stations are connected with backhaul links, through which one base station could pass its own decoded message to its neighboring ones. Both user terminals and base stations are equipped with a single antenna each. It is assumed that, due to the scarce channel state feedback resource, the users have no access to channel realizations but only know the network connectivity graph, i.e., which user is connected to which base station. The received signals in this partially-connected network are modeled, for the base station at time instant , by
where is the transmitted signal subject to the average power constraint , is the Gaussian noise with zero-mean and unit-variance at the base stations, and is the channel coefficient between user and base station , and is not known by the users. Here represents the transmit set containing the indices of users that are connected to base station , for . We point out that channel coefficients are not available at the users, yet the network topology (i.e., ) is known by both users and base stations. The network topology is assumed to be fixed throughout the duration of communication. Such a setup is referred to as the “Topological Interference Management (TIM)” setting.
Ii-B Problem Statement
Similarly to the definition in , a decoding order is a partial order such that indicates the message should be decoded before . We assume that the base station only decodes its own desired message , and then passes it to the base station , even though sometimes the messages desired by other base stations are also decodable. As anticipated in Section I, we refer to this setting combining TIM and decoded message passing as “TIM-MP”. In the TIM-MP problem, given a decoding order , once is decoded, it can be passed to receiver to help decoding . Throughout this paper, we consider unconstrained message passing, that is, a message can be passed to any other receivers who are interested 111In fact, because of the locality of interference in physically motivated network topologies, only the neighboring receivers suffer from interference from message , and therefore messages are passed in a neighbor-propagation fashion..
Formally, the achievable rate of the TIM-MP problem can be defined as follows. For a network with topology represented by the bipartite graph , a rate tuple is said to be achievable under a specified decoding order if there exists a coding scheme consists of the following elements:
message sets , from which the message is uniformly chosen, ;
encoding functions , :
with power constraint , where each transmitter has only access to its own message and the network topology graph ;
decoding functions , :
where , and is a set of decoded messages passed from other receivers through backhaul links, defined as
such that the decoding error tends to zero when the code block length tends to infinity.
We consider the following message passing policies:
Single-round message passing: For a decoding order , for all such that , it is allowed to pass the messages only from receiver to receiver .
Multiple-round message passing: It consists of a sequence of multiple single-round message passing. Different rounds can have distinct decoding orders. For instance, it may happen that in one round and in another round with .
The achievable rate region with respect to the decoding order , denoted as , is the set of all achievable rate tuples , corresponding to the single-round message passing. The capacity region with multiple-round message passing over all possible decoding orders is given by
which can be obtained by time sharing among multiple single-round message passing with different decoding orders allowed in different rounds.
We follow the TIM setting and use symmetric DoF and DoF region as our main figures of merit.
Definition 1 (Symmetric DoF and DoF Region).
Ii-C Interference Modeling
We model the mutual interference in the network as a directed message conflict graph. A directed graph (digraph) consists of a set of vertices and a set of arcs between two vertices. We denote by the arc (i.e., directed edge) from vertex to vertex . More graph theoretic definitions are presented in Appendix -A.
Definition 2 (Conflict Digraph).
For a network topology, its directed conflict graph (briefly referred to as“conflict digraph”) is a digraph such that represent the message from transmitter to receiver and represents the interfering link from transmitter to receiver in the interference network.
The conflict digraphs indicate not only the message conflict due to mutual interference, but also the source and the sink of the interference. The conflict digraph captures exactly every instance of network topology. We refer to the TIM-MP problem with a specific conflict digraph as a TIM-MP instance.
Iii Orthogonal Access
Orthogonal access is the simplest transmission scheme of practical interest. For the TIM problem, orthogonal access is to schedule independent sets of the conflict graph across time or frequency , because simultaneous transmission of the messages in an independent set and orthogonal transmission of different independent sets across time or frequency avoid mutual interference. By contrast, message passing offers the possibility of interference cancelation for the messages that are not in an independent set. This complicates orthogonal access under the TIM-MP setting, as both interference avoidance and cancelation should be taken into account.
In what follows, we first introduce the concept of orthogonal access in the TIM-MP problem, and propose an inner bound of symmetric DoF for the single-round message passing setting, followed by the extension to the multiple-round message passing. The uselessness of one-to-one interference alignment is also shown from a graph theoretic perspective.
Iii-A What is Orthogonal Access?
Instead of scheduling independent sets in the TIM problem, we schedule acyclic set in the TIM-MP problem, where the acyclic set is an induced sub-digraph that contains no directed cycles (dicycles). More properties of the acyclic set can be found in Appendix -A. Thus, we have the following definition.
Definition 3 (Orthogonal Access).
Orthogonal access in the TIM-MP setting consists of scheduling orthogonally acyclic sets of conflict digraphs across time or frequency.
The messages in an acyclic set can be decoded successively via decoded message passing in one time slot. 222Note that the propagation of the messages over the backbone is much faster than the transmission over the wireless interface. Therefore, we may treat it, for conceptual simplicity, as one time slot counting the time to transmit (simultaneously) the codewords, and neglecting the message passing propagation time. In practice, a sequence of codewords can be multiplexed in time for the same acyclic set and decoding order, such that the propagation can be done in a pipelined way, such that effectively the time needed for end to end propagation of the decoded messages is much less than the duration of transmission in the state defined by the acyclic set. In an acyclic digraph , there always exists a topological ordering of such that a vertex comes before a vertex if there is an arc . Such a topological order gives us the decoding and message passing order. In particular, there exists at least one vertex with no incoming arcs in an acyclic set. We start with the decoding of these messages, which are free of interference. After decoding these messages, they are passed to the next ones which are only interfered by them such that the interference can be fully canceled out, and thus these next messages are also decodable. Keep doing this until all messages in this acyclic set are decoded in such a successive way. As such, simultaneous transmission of the messages in an acyclic set does not have residual interference left after the interference cancelation with passed messages.
Let us look at orthogonal access from a graph coloring perspective. If each acyclic set is assigned with one color, orthogonal access is equivalent to acyclic sets coloring of conflict digraphs. The messages assigned to the same color are simultaneously transmitted and successively decoded, whereas different colors are multiplexed over different time slots.
The dichromatic number of a digraph , denoted by , is the minimum number of colors required to color the vertices of in such a way that every set of vertices with the same color induces an acyclic sub-digraph in .
By this definition, we can immediately obtain an inner bound of symmetric DoF.
For the TIM-MP instance with conflict digraph , we have an inner bound
which is achieved by orthogonal access with single-round message passing.
Orthogonal access with single-round message passing can be seen as assigning a standard basis vector to each acyclic set. For instance, the assigning of the -th column of an identity matrix to an acyclic set is equivalent to the scheduling of this acyclic set in -th time slot.
For TIM-MP instances, the optimal symmetric DoF is if and only if the conflict digraphs are acyclic.
For TIM-MP instances, if only single-round message passing is allowed, the optimal symmetric DoF is if the conflict digraph contains either only directed odd cycles or only directed even cycles.
For the conflict digraphs in Fig. 1, there are only directed odd cycles in Fig. 1(a), only directed even cycles in Fig. 1(b), and both odd and even dicycles in Fig. 1(c) and (d). So, according to Corollary 2, the optimal symmetric DoF value with single-round message passing for both (a) and (b) is . We cannot expect that DoF is achievable in (c). In fact is optimal, which will be shown later. Nevertheless, the optimal symmetric DoF value of Fig. 1(d) is also , although the conflict digraph contains both even and odd cycles. This shows that the condition in Corollary 2 is only sufficient but not necessary.
The dichromatic number of a digraph can be represented as the solution to the following linear program:
where is the collection of all possible acyclic sets, and is the collection of all possible acyclic sets that involve the vertex . By relaxing to , as the fractionalized versions of other graph theoretic parameters, the linear program (9) yields the fractional dichromatic number , which can also serve as an inner bound of the symmetric DoF.
For the TIM-MP instance with conflict digraph , we have
which is achieved by orthogonal access with multiple-round message passing.
Fractional coloring, no matter whether independent or acyclic set coloring, can be treated as time sharing among a set of proper non-fractional coloring, where e.g., is the portion of the shared time of the acyclic set . Thus, fractional acyclic set coloring of conflict digraphs is equivalent to orthogonal access with multiple-round message passing. If there only exists the symmetric part in the conflict digraph (i.e., without uni-directed arcs, see Appendix -A), both dichromatic number and its fractional version reduce to their counterparts in the underlying undirected graph, as acyclic sets in the digraph reduce to independent sets in the underlying undirected graph.
The regular network  has a connectivity pattern where each receiver is connected to its paired transmitter and the next successive ones. By Lemma 2, we have the following corollary for the symmetric DoF inner bound for the regular network, whose proof is relegated to Appendix -F.
For the regular network with , we have
which is achieved by orthogonal access with multiple-round message passing.
Unless otherwise specified, orthogonal access in the rest of this paper is referred to multiple-round message passing.
Iii-B Can Interference Alignment Help?
Beyond the achievability through acyclic set coloring, one interesting question to ask is, if the more sophisticated achievability schemes, such as interference alignment, can outperform orthogonal access.
Roughly speaking, (subspace) interference alignment is to associate each interference with a subspace such that the superposition of interferences occupies a reduced dimensional subspace. One-to-one interference alignment is a special case of subspace alignment. It consists of aligning the interferences in a one-to-one manner, that is, given a one-dimensional subspace, two interferences are either completely aligned or disjoint.
Under the TIM setting, orthogonal access is equivalent to fractional vertex coloring on the undirected conflict graph [7, 15], and one-to-one interference alignment is a generalized version of orthogonal access . Let us associate each transmitter-receiver pair (i.e., each vertex in conflict graph) with a transmission scheduling vector of length , where is the number of scheduling intervals to properly serve all transmitter-receiver pairs without causing mutual interference (i.e., the number of colors for a proper vertex coloring on the conflict graph). Orthogonal access corresponds to assigning the basis vector (i.e., the -th column of ) to the transmitter-receiver pairs associated to the color , while one-to-one alignment consists of assigning general linearly independent vectors such that each receiver can recover its desired message by solving a linear system of equations (in the absence of noise). In general, this allows for a vector dimension , such that interference alignment may improve over orthogonal access. At a given receiver, only the transmitters that cause interference do appear in the linear system. Accordingly, the required vector dimension depends merely on the number of different colors in the in-neighborhood of the directed conflict graph. Thus, a feasible one-to-one interference alignment scheme under the TIM setting is equivalent to a proper local coloring on the directed conflict graph .
This also analogously applies to the TIM-MP setting. Given a proper acyclic set coloring for a conflict digraph, for an acyclic set, the maximum number of acyclic sets with different colors in the in-neighborhood (i.e., causing interference) does matter. In other words, the spanned subspace by the assigned vectors of these acyclic sets in the in-neighborhood should be minimized to make interference as aligned as possible.
Analogously to , we introduce a local version of fractional acyclic set coloring.
Definition 5 (Local Dichromatic Number).
The local dichromatic number of a digraph is defined as
where , is the set of vertices in the closed in-neighborhood of , and the minimum is over all possible acyclic set coloring . The closed in-neighborhood is defined as
The local dichromatic number of a digraph can be also represented as the solution to the following linear program:
Its fractional version can be similarly defined by replacing with . It is clear that fractional local coloring is built upon the feasible fractional coloring of acyclic sets, and the difference is that the local coloring only counts colors in the closed in-neighborhood. So, we always have , because an additional condition is imposed on the local version.
The linear program formulation of fractional local acyclic set coloring in (14) is a straightforward extension of fractional local independent set coloring, where the acyclic sets in (14) replace the independent sets. Similarly to the equivalence between interference alignment and local coloring shown in , it follows immediately that one-to-one interference alignment with message passing is equivalent to fractional local acyclic set coloring. Thus, we have a new inner bound for the symmetric DoF due to interference alignment.
For the TIM-MP instance with conflict digraph , we have
which is achieved by one-to-one interference alignment.
With message passing, one-to-one interference alignment boils down to orthogonal access due to
Let us consider an instance of the TIM-MP problem with . The conflict digraph is shown in Fig. 2(a), where a proper fractional acyclic set coloring is given for acyclic sets . It requires in total 5 colors for these acyclic sets, each of which receives 2 colors, such that the sub-digraph induced by the vertices with the same color is acyclic. Fig. 2 shows the in-neighborhood of the acyclic set , which includes all other vertices. The fractional dichromatic number is , which agrees with the fractional local dichromatic number.
Although one-to-one interference alignment does not outperform orthogonal access, it remains as an interesting and challenging open problem to exploit the potential benefit of subspace interference alignment, which has been shown to provide further gains in problems such as multiple groupcast TIM .
Iv The Optimality of Orthogonal Access via Polyhedral Combinatorics
Having assessed that one-one-one interference alignment under message passing decoding boils down to orthogonal access, a natural question is how powerful orthogonal access is, and under what condition orthogonal access is DoF-optimal in the information-theoretic sense. Before proceeding further, we introduce some outer bounds. The preliminaries related to polyhedral combinatorics can be found in Appendix -B.
Iv-A Outer Bounds via Polyhedral Combinatorics
By the nature of message passing, we conclude that cliques and dicycles are main obstacles, and thus have the following outer bounds.
Theorem 1 (Clique-Cycle Outer Bounds).
The DoF region of the TIM-MP problem is outer-bounded by
where is the collection of all minimal dicycles (i.e., dicycles without chord), and is the collection of all maximal cliques (i.e., cliques not a sub-digraph of other cliques).
See Appendix -H. ∎
We refer hereafter to the inequalities in (17) as individual inequalities, cycle inequalities, and clique inequalities, respectively. The cliques with size 1 are vertices, so clique inequalities imply the individual ones. The clique with size 2 is also a dicycle, such that clique and cycle inequalities have some inequalities in common. As all sub-digraphs of a clique are still cliques and the clique inequality of the maximal one implies all other ones, we only count the clique inequality with the maximal size. Moreover, if a dicycle has a chord, whatever its direction is, there exists a subset of vertices that form a shorter dicycle, rendering the constraint associated with the larger one redundant. As such, we only count the cycle inequalities corresponding to the dicycles without chord.
The outer bound with only individual and clique inequalities can be formed, by replacing by , as a set packing polytope (see Appendix -B)
The cycle inequalities, with a replacement of variables , can be equivalently rewritten as , and thus the outer bound with only individual and cycle inequalities can be formed, by replacing by , as a set covering polytope (see Appendix -B)
Taking all individual, clique, and cycle inequalities into account, we have the outer bound formed, by replacing by and removing redundant inequalities, as the mixed set covering and packing polytope 
The conditions and are to ensure that the redundancy between clique and cycle inequalities is removed. For some and , if , then the condition is redundant, so we add to avoid redundancy.
To rewrite the set packing and covering polytope into compact forms, we introduce two incidence matrices (see definitions in Appendix -B).
Definition 6 (Clique-Vertex Incidence Matrix).
Let be the collection of all induced maximal cliques of a digraph . The corresponding clique-vertex incidence matrix is a binary matrix, where
where , and is the -th clique in .
Definition 7 (Dicycle-Vertex Incidence Matrix).
Let be the collection of all induced minimal dicycles of a digraph . The corresponding dicycle-vertex incidence matrix is a binary matrix, where
where , and is the -th dicycle in .
As all clique inequalities correspond only to the maximal cliques, there are no dominating rows in the clique vertex incidence matrix . As all cycle inequalities corresponds only to the dicycles without chord, there are no dominating rows in the cycle-vertex incidence matrix . Nevertheless, there might be dominating rows in the concatenation of and .
By the above two incidence matrices, the compact representation of set packing and covering polytopes can respectively represented as
According to polyhedral combinatorics (see Appendix -B), the matrix is perfect if and only if has only integral extreme points, and the matrix is ideal if and only if has only integral extreme points. The widely-studied balanced and totally unimodular matrices (TUM) are special cases of perfect and ideal matrices. Fig. 3 presents their relations.
Iv-B The Optimality of Orthogonal Access
By the above outer bounds, we identify three families of network topologies for which orthogonal access achieves the optimal DoF region of TIM-MP problems. The conditions of the optimality of orthogonal access are summarized in Theorem 2, and will be detailed case by case in the ensuing theorems.
Theorem 2 (Optimality of Orthogonal Access).
For the TIM-MP problem with the conflict digraph , the clique-vertex incidence matrix and the dicycle-vertex incidence matrix , orthogonal access via fractional acyclic set coloring achieves the optimal DoF region, if it falls in any one of the following cases.
Case I: The conflict digraph contains no dicycles with , and is a perfect matrix;
Case II: The conflict digraph contains no cliques with , and is an ideal matrix;
Case III: The symmetric part is a perfect graph, and the dicycle-vertex incidence matrix of the remaining conflict digraph after removing cliques contains no minimally non-ideal submatrices.
The converse proof relies on the integrality of set packing and covering polytopes, which has been established in polyhedral combinatorics. The achievability is due to acyclic set coloring. The sub-digraphs in the conflict digraph induced by the coordinates of the extreme points of these polytopes are acyclic sets. The detailed proofs will be shown case by case in the ensuing theorems.
For the TIM setting, it has been shown in [15, Theorem 1] that orthogonal access achieves the all-unicast DoF region of the TIM problem if and only if the network topology is chordal bipartite. Analogously, we have the following theorem for the TIM-MP problem when message passing is enabled.
Theorem 3 (Case I).
For the family of networks in which conflict digraphs contain no dicycles with , and is a perfect matrix, the optimal DoF region achieved by orthogonal access can be characterized by the set packing polytope
where the undirected graph is the symmetric part of the conflict digraphs , and is the set of all maximal cliques in .
See Appendix -I. ∎
The condition that conflict digraph contains no dicycles with , and is a perfect matrix, indicates that is a perfect digraph . According to the definition of perfect digraphs in Appendix -A, Theorem 3 identifies the optimality of orthogonal access for the conflict digraph excluding the following cases:
contains dicycles with length as induced sub-digraph;
contains filled odd holes or filled odd antiholes, i.e., its symmetric part contains odd holes or odd antiholes.
Similarly to , the characterization of the optimal DoF region automatically yields the optimality of the traditional metrics such as sum or symmetric DoF.
For a perfect digraph , acyclic set coloring of reduces to vertex coloring of its symmetric part . Thus, any feasible coloring of the symmetric graph is also feasible for . If the conflict digraph only has symmetric part, then . As such, the orthogonal access of our problem is reduced to that of the TIM problem without message passing, because in this case interference is mutual, and message passing does not help.
Consider a 6-cell network topology shown in Fig. 4(a). In the conflict digraph in Fig. 4(b), the symmetric part in Fig. 4(c) is perfect, and there do not exist dicycles with as induced sub-digraph, although there exist dicycles for instance . As there is an arc , the sub-digraph induced by is not a dicycle. Thus, according to Theorem 3, we have the optimal DoF region
It immediately follows that the symmetric and sum DoF are and , respectively. To achieve the symmetric DoF of , we can simply schedule , , and in three time slots respectively. In each time slot, are free of interference, and can be subsequently decoded after passing the decoded messages at receivers 1, 3, 5 to receivers 2, 4, 6 respectively.
Note however that, the condition that the conflict digraph is perfect in Theorem 3 is only sufficient but not necessary. A counterexample is the dicycle , where the fractional acyclic set coloring achieves the DoF region
while the conflict digraph is not perfect.
Realizing that Case I focuses only on the integrality of the set packing polytopes, we identify another family of networks focusing on the integrality of the set covering polytopes in the following theorem, where orthogonal access is still DoF-optimal albeit the conflict digraph is not perfect.
Theorem 4 (Case II).
For the family of networks in which contains no cliques with , and is an ideal matrix, the optimal DoF region achieved by orthogonal access can be given by the set covering polytope
where is the collection of all dicycles without chord.
See Appendix -J. ∎
As a matter of fact, the condition that is ideal already excludes the existence of the cliques of size 3 or more in the conflict digraph. It is because otherwise a clique of size 3 or more will lead to the existence of a non-ideal circulant submatrix of which results in a contradiction to that any submatrix (minor) of an ideal matrix is also ideal. The definition of the circulant matrix can be found in Appendix -B. The circulant matrices that are ideal consists only of for even , , , and .
Differently from the perfect matrices, it is still an open problem to fully characterize all the ideal matrices. It has been shown in  that a matrix is ideal if and only if it does not contain a minimally non-ideal (MNI) submatrix minor (see definitions in Appendix -B). The MNI matrices are the “smallest” possible matrices that are not ideal [28, 29]. A submatrix of is a minor of if it can be obtained from by successively deleting a column and the rows with a ‘1’ in column . If a matrix is ideal then so are all its minors. A matrix is MNI, if it is not ideal but all its proper minors are. For instance, is a MNI matrix, so any matrix that contains as a minor is not ideal, e.g., Fig. 5(c).
As special cases of ideal matrices, balanced and totally unimodular (TU) matrices are completely characterizable. 333Note here that, although TU and balanced matrices are special cases of perfect matrices, the conflict digraphs with incidence matrices being TU or balanced are not the subclass of those in Theorem 3, because two different incidence matrices are considered. The characterization of balanced or totally unimodular matrices is well understood. A matrix is balanced if and only if it contains no odd hole matrices (i.e., with odd where ) as submatrices. A polynomial time recognition algorithm for balanced matrices was also given with the aid of decomposition . The full characterization of all TU matrices was also given in , where a matrix is TU if and only if it is a certain natural combination of some matrices and some copies of a particular 5-by-5 TU matrix.
Firstly, consider a 4-user TIM-MP instance, as shown in Fig. 5(a). There are two dicycles and without cliques, such that the outer bund is given by two cycle inequalities and as well as the individual ones , . It is not hard to verify that the dicycle-vertex incidence matrix
is totally unimodular, and thus ideal. The extreme points of the polytope consist of all possible 4-tuples excluding , and . It can be checked that all these extreme points can be achieved by acyclic set coloring. As such, the DoF region can be achieved by time sharing among these extreme points.
For the instance in Fig. 5(b), the corresponding dicycle-vertex incidence matrix is ideal, while in Fig. 5(c) with an arc added, the resulting dicycle-vertex incidence matrix is not ideal any more. In Fig. 5(c), the corresponding dicycle-vertex incidence matrix is
which contains an MNI matrix as a submatrix. So, it is not a balanced matrix, nor an ideal matrix.
For the TIM-MP problems, orthogonal access achieves the optimal DoF region given by (30), if any one of the following conditions is satisfied.
All the induced dicycles in conflict digraph are disjoint (i.e., none of two induced dicycles share vertices).
There exist at most two chordless dicycles in conflict digraph , including regular network whose conflict digraph is a single dicycle.
By Theorems 3 and 4, we show the optimality of orthogonal access for the three-user network with all possible topologies (16 non-isomorphic instances in total), whose proof is relegated to Appendix -L.
For the three-user TIM-MP problem, orthogonal access achieves the optimal DoF region.
Theorem 3 handles the perfect conflict digraphs with neither dicycles of length 3 or more, nor odd hole or antihole in the symmetric part, while Theorem 4 considers the case with dicycles but without clique of size 3 or more. The former only considers the tightness of clique inequalities, whereas the latter focuses only on the cycle inequalities. Beyond Theorems 3 and 4, we come up with anther sufficient condition on the optimality of orthogonal access, where both dicycles and cliques are contained in the conflict digraphs.
Theorem 5 (Case III).
For the family of networks in which is a perfect graph, and the dicycle-vertex incidence matrix of the remaining conflict digraph after removing cliques contains no MNI submatrices, the optimal DoF region achieved by orthogonal access can be given by the mixed set packing and covering polytope
where is the collection of all minimal dicycles and is the collection of all maximal cliques with size no less than 3, and
See Appendix -M. ∎
A special type of networks in case III, is that the concatenation of clique-/dicycle-incidence matrices is balanced, so that the polytope is integral for any integer .
It is still an open problem to fully sort out all the MNI matrices, while it has been proven that the MNI matrices do have some properties. It has been characterized by Lehman [28, 29] that if is an MNI matrix, then is isomorphic (up to a permutation of rows followed by a permutation of columns) to either (1) the degenerate projective plane with , or (2) where is a square nonsingular matrix with ‘1’s per row and per column, and each row of has at least ‘1’s. The known MNI matrices include the circulant matrices for odd , , , , , , , , , , and , the degenerate projective planes with , and the Fano plane . Fig. 6 presents some conflict digraphs whose dicycle-vertex incidence matrices are MNI matrices.
For the instance in Fig. 7(a), the optimal DoF region is given by
where the cycle inequalities for are replaced by a single clique inequality. The extreme points of are 5-tuple binary vectors apart from , , and , where denotes either 0 or 1. It can be easily checked that all extreme points are achievable by acyclic set coloring.
For the instance in Fig. 7(b), the optimal DoF region is given by
where the cycle inequalities for are replaced by a single clique inequality. After such a replacement, the resulting polytope is integral, and the extreme points are achievable by acyclic set coloring.
V A Generalized Index Coding Problem
Building upon the relation between index coding and TIM , we also establish an analogous relation to TIM-MP. As in Appendix -C, the goal of index coding is to minimize the number of transmissions such that all receivers are able to decode their own messages simultaneously. As TIM-MP is a generalization of TIM, we introduce a generalization of index coding, referred to as “successive index coding (SIC)”, where the message decoding is not necessarily simultaneous.
V-A Successive Index Coding (SIC)
In the SIC problem, the receivers are allowed to declare their own messages once they decode them. Such a declaration offers other receivers additional side information, by which the minimal number of transmissions can be further reduced.
The multiple-unicast SIC problem considers a noiseless broadcast channel, where a transmitter wants to send the message to the receiver who has access to prior knowledge of initial side information () as well as the additional side information due to the successive decoding and message passing. The goal is to find out the minimum number of transmissions (i.e., broadcast rate) over all possible successive decoding and message passing orders such that each receiver can successively decode its desired message.
The additional side information depends on the decoding order. For the single-round message passing, given a decoding order , the partial order indicates the message passing from receivers to , which is equivalent to enhancing the side information set . As such, the enhanced side information set can be written by
for a specific decoding order .
In what follows, we formally define the -receiver SIC problem. A successive index coding scheme with side information index sets and a given decoding order consists of the following:
an encoding function, at the transmitter that encodes -tuple of messages (codewords) to a length- index code.
a decoding function at the receiver , that decodes the received index code back to with initial side information held at receiver as well as the passed messages that decoded earlier .
The initial side information digraph or conflict digraph of a successive index coding instance is identical to that of the corresponding index coding instance.
Thus, for a given decoding order , a rate tuple is said to be achievable if there exists a successive index code with
such that any rate tuple in the rate region is achievable with
Similarly to the TIM-MP problem, the capacity region of the SIC problem is the set of all achievable rate tuples where time sharing among multiple single-round message passing is allowed. More specifically, it is the convex hull of the union of the achievable rate regions for all possible decoding orders, i.e., .
By the channel enhancement approach in , it is readily shown that the DoF region of every TIM-MP instance is outer bounded by the capacity region of the corresponding SIC instance, and both problems are equivalent for linear coding schemes (i.e., with linear encoding/decoding functions). In particular, for each single-round message passing, given a decoding order , the resulting TIM-MP (SIC) problem can be treated as a modified TIM (index coding) problem with updated side information set in (36).
As the linear coding schemes are considered in the previous sections, the results obtained for TIM-MP are applicable to SIC. Specifically, the achievable symmetric DoF or DoF region of TIM-MP with conflict digraph are also the achievable symmetric rate or rate region of SIC with initial side information digraph . Similarly, the sufficient or necessary conditions seen before for orthogonal access in TIM-MP are also applicable to the corresponding SIC setup.
In the rest of this section, we will focus on the broadcast rate, defined as
which is the minimum number of transmissions (i.e., the number of transmitted symbols over the shared link normalized by the total message length) for the SIC problem, also known as reciprocal symmetric capacity.
|Achievability||Index Coding||Successive Index Coding|
|Fractional Clique Covering|
|Partial Clique Covering|
V-B Analogy to Index Coding
Analogously to the index coding problem, we define some achievability schemes for the SIC problem. As partial clique covering is a generalized version of clique and cycle covering, we only present a definition of partial cliques in conflict digraphs of SIC instances, named weakly degenerate set.
Definition 8 (Weakly Degenerate Set).
A conflict sub-digraph is a weakly -degenerate set if any of its induced sub-digraph has a vertex of out-degree or in-degree no more than , i.e., for all , , .
The weakly degenerate set is a generalized version of partial clique to the SIC problem. The acyclic set is weakly 0-degenerate, the dicycle is weakly 1-degenerate, and the clique is weakly -degenerate.
Lemma 5 (Weakly Degenerate Set Covering).
The broadcast rate of the SIC problem with conflict digraph is upper bounded by
with weakly degenerate set covering, where the minimum is over all partitions of , and for all , is a weakly -degenerate set.
The analogy of achievability between index coding and successive index coding problems is summarized in Fig. 8. Note here that, for index coding, (fractional) clique covering on the side information digraph is equivalent to (fractional) vertex coloring on the underlying undirected conflict graph.
As a generalized version of acyclic set, however, weakly generate set covering does not offer any improvement over the acyclic set coloring, unlike that in the index coding problem where partial clique covering indeed outperforms clique covering (i.e., independent set coloring on conflict graphs). It is because if is a weakly -degenerate set, meaning that weakly degenerate set covering does not offer gains over acyclic set coloring. In other words, message passing renders the (weakly degenerate) set partition of conflict digraphs useless for SIC problems in terms of broadcast rate. Nevertheless, the weakly degenerate set partition has the potential to restrict message passing locally, shortens the decoding latency of the entire network, and facilitates the tradeoff between broadcast rate and message passing overhead.
V-C Reducibility and Criticality
V-C1 Reducibility: When is a Vertex/Message Reducible?
The objective of studying vertex-reducibility is to remove the vertices in the conflict graph without changing the broadcast rate, so as to reduce the large-size SIC instance to a smaller one with less vertices.
Definition 9 (Reducibility).
A vertex in the conflict digraph is reducible, if its removal does not decrease the broadcast rate of the corresponding SIC problem, i.e., .
By strong component decomposition (see Appendix -A), we have the following theorem.
Given the unique strong component decomposition , let be the strong component with the maximal fractional dichromatic number. If falls into the digraph classes in Theorem 2, then the vertices in are reducible.
See Appendix -N. ∎
The vertices that are not involved in any dicycles (e.g., with either only incoming or outgoing arcs) are reducible. Any vertex in a directed acyclic graph is reducible. This agrees with the fact that the broadcast rate of the SIC instances with conflict digraphs being directed acyclic is 1.
The strong component with the maximal fractional chromatic number is not necessarily the one with the largest size. For the strong component with the largest size, denoted by , the index coding problem corresponding to serves an upper bound of the original successive index coding problem, i.e., . That is, if a broadcast rate of the index coding instance is achievable, it is also achievable for the original SIC instance. It is because choosing one single vertex from each strong component forms an acyclic set, and thus the broadcast rate of for the index coding setting without message passing dominates.
Let us take two conflict digraphs shown in Fig. 9 as illustrative examples. The induced sub-digraphs in the shadow are the strong components with the maximal fractional dichromatic numbers. The strong components in shadow are a clique (on the left) and a dicycle , both of which are perfect digraphs and fall into the cases in Theorem 2. Thus, both SIC instances can be reduced without loss of broadcast rate to the ones with only the strong components in the shadow, so that for Fig. 9(a) and for Fig. 9(b). Note also in Fig. 9(b) that the dicycle is the strong component with the largest size, but its dichromatic number is not maximal, because the strong component in the shadow has . Thus, we have an upper bound . Together with the cycle bound , we also have the optimal broadcast rate .
V-C2 Criticality: When is an Arc/Interference Critical?
The objective of studying arc-criticality is to remove the arcs in the conflict graph without changing the broadcast rate, so as to reduce the SIC instance with a large arc set to a smaller one with less arcs.
Definition 10 (Criticality).
An arc in the conflict digraph is critical, if its removal strictly decreases the broadcast rate of the corresponding SIC problem, i.e., .
The removal of an arc (i.e., interference link) does not increase the interference in the network, so the broadcast rate should not be increased. A conflict digraph is said to be critical, if every arc in is critical.
In a conflict digraph , an arc is critical, if it belongs to the following:
the unique minimal dicycle when the dicycle-vertex incidence matrix of is ideal;
the unique maximal clique when is a perfect digraph.
See Appendix -O. ∎
In a conflict digraph , if an arc is critical, then it must belong to an induced dicycle. Otherwise, it can be removed without affecting the capacity region. Thus, if a conflict digraph is critical, then it must be strongly connected.
Vi Linear Optimality
In view of the equivalence between the TIM-MP and SIC problems with linear coding schemes, in this section, we restrict ourselves to linear coding schemes, and consider the optimality of orthogonal access for some instances in terms of linear symmetric DoF for TIM-MP (or linear symmetric rate for SIC).
Vi-A Linear Optimality of Some MNI Matrices
In what follows, we show that, for two network topologies that are not included in Theorem 2, orthogonal access is linearly optimal for both TIM-MP and SIC problems.
Theorem 8 (Linear Optimality for Some Special Structures).
For the TIM-MP and SIC instances with dicycle-vertex incidence matrix and , orthogonal access achieve the optimal linear symmetric DoF/rate, where
See Appendix -P. ∎
Vi-B Small Networks with Reduction
Together with reducibility and criticality of the TIM-MP/SIC problems, we show the linear optimality of orthogonal access for all 4-user network topologies (in total 218 non-isomorphic conflict graphs).
Theorem 9 (Linear Optimality for Small Networks).
For TIM-MP/SIC problems up to 4 users, orthogonal access achieves the linearly optimal symmetric DoF/rate for all topologies.
See Appendix -Q. ∎
Thanks to the reducibility and criticality, the network topologies that can be reduced to 3-user case are already done in Corollary 5, and we only need to focus on the case when each vertex is irreducible and each arc is critical. This substantially reduces the number of non-isomorphic instances that need to be considered from 218 to 6. It can also be checked that, for the TIM-MP/SIC instances up to 5 users, orthogonal access achieves the optimal linear symmetric DoF/rate for almost all topologies, i.e., except two out of in total 9608 non-isomorphic ones. Those two instances are and shown in Fig. 6(a) and Fig. 6(b), respectively. Such a reduction approach based on reducibility and criticality has great potential to further identify the symmetric DoF/rate for larger networks, although the number of non-isomorphic topologies increases dramatically as the number of users increases.
Vii Discussion: Message Passing Overhead and Achievable Rate Tradeoff
From the previous section, we have seen that decoded message passing is so powerful that it leads to orthogonal access with almost optimal DoF, if no constraints are imposed on message passing. In practice, however, message passing may incur some cost. For example, in the uplink of a cellular system there may be limitations of the usage of the wired network connecting the base station receivers. Then, it is meaningful to study the case where a limited number of messages can be passed along from each receiver. In this section, we first consider the case when only one passing message in the entire network can help improve DoF region of TIM problem, followed by the generalization to arbitrary number of passing messages and the formulation of a matrix completion problem for the tradeoff between achievable symmetric DoF and the passing messages overhead.
Vii-A When Does One Message Passing Help?
Given a specific decoding order, the decoded message passing is also determined. After interference cancelation with passed messages, the TIM-MP problem becomes a modified TIM problem with some interfering links removed. According to the equivalence between TIM and index coding problems , a decoding order corresponds to the arc removal in the conflict digraph of the TIM problem, and equivalently the arc adding in the side information digraphs of the index coding problem. A natural question is whether message passing helps in the sense that the corresponding arc removal from increases the DoF region of the TIM problem. The following theorem offers a sufficient condition to this question.
A message passing is helpful, if the addition of the corresponding arc in the side information digraph forms new dicycles as induced sub-digraphs.
See Appendix -R. ∎
The newly formed dicycle is not necessarily unique. It may form multiple dicycles. As message passing will not be harmful, as long as a new dicycle is formed, the DoF region will be enlarged.
While the above condition is only sufficient in general, it is also necessary for chordal bipartite networks. We have the following corollary, whose proof is presented in Appendix -S.
For chordal bipartite networks, a message passing is helpful if and only if the addition of the corresponding arc in the side information digraph forms new dicycles as induced sub-digraphs.
Let us consider a simple network topology shown in Fig. 11(a), which is a chordal bipartite network . Its conflict digraph and the side information digraph of the corresponding index coding problem are shown respectively in (b) and (c). The DoF region for this network topology is and the symmetric DoF are . In Fig. 11(d), the addition of the arc in forms a new dicycle , which increases the symmetric DoF to . By removing the uni-directed arc in in Fig. 11(e), the DoF region is enhanced to . The DoF region of Fig. 11(f) after removing the arc in is still , because it is not uni-directed in nor forming a new dicycle by its addition in .