We consider multicommodity flow and cut problems in polymatroidal networks where there are submodular capacity constraints on the edges incident to a node. Polymatroidal networks were introduced by Lawler and Martel [Math. Oper. Res., 7 (1982), pp. 334--347] and Hassin [On Network Flows, Ph.D. dissertation, Yale University, New Haven, CT, 1978] in the single-commodity setting and are closely related to the submodular flow model of Edmonds and Giles [Ann. Discrete Math., 1 (1977), pp. 185--204]; the well-known maxflow-mincut theorem holds in this more general setting. Polymatroidal networks for the multicommodity case have not, as far we are aware, been previously explored. Our work is primarily motivated by applications to information flow in wireless networks. We also consider the notion of undirected polymatroidal networks and observe that they provide a natural way to generalize flows and cuts in edge and node capacitated undirected networks. We establish flow-cut gap results in several scenarios that have been previously considered in the standard network flow models where capacities are on the edges or nodes. Our results are based on analyzing the dual of the flow relaxations via continuous extensions of submodular functions, in particular, the Lovász extension. For directed graphs we rely on a simple yet useful reduction from polymatroidal networks to standard networks. For undirected graphs we rely on the interplay between the Lovász extension of a submodular function and line embeddings with low average distortion introduced by Matousek and Rabinovich [Israel J. Math., 123 (2001), pp. 285--301]; this connection is inspired by, and generalizes, the work of Feige, Hajiaghayi, and Lee on node-capacitated multicommodity flows and cuts. Our results have found applications in wireless network information flow [S. Kannan and P. Viswanath, IEEE Trans. Inform. Theory, 60 (2014), pp. 6303--6328] and we anticipate others in the future.

In this paper, a compress-and-forward scheme with backward decoding is presented for the unicast wireless relay network. The encoding at the source and relay is a generalization of the noisy network coding (NNC) scheme. While it achieves the same reliable data rate as NNC scheme, the backward decoding allows for a better decoding complexity as compared with the joint decoding of the NNC scheme. Characterizing the layered decoding scheme is shown to be equivalent to characterizing an information flow for the wireless network. A node-flow for a graph with bisubmodular capacity constraints is presented and a max-flow min-cut theorem is presented. This generalizes many well-known results of flows over capacity constrained graphs studied in computer science literature. The results for the unicast relay network are generalized to the network with multiple sources with independent messages intended for a single destination.

We consider multicommodity flow and cut problems in polymatroidal networks where there are submodular capacity constraints on the edges incident to a node. Polymatroidal networks were introduced by Lawler and Martel [20] and Hassin [15] in the single-commodity setting and are closely related to the submodular flow model of Edmonds and Giles [10]; the well-known maxflow-mincut theorem holds in this more general setting. Polymatroidal networks for the multicommodity case have not, as far as the authors are aware, been previously explored. Our work is primarily motivated by applications to information flow in wireless networks.

We study a wireless broadcast network, where a single source reliably communicates independent messages to multiple destinations, with the potential aid of relays and cooperation between destinations. The wireless nature of the medium is captured by the broadcast nature of transmissions as well as the superposition of transmitted signals plus independent Gaussian noise at the received signal at any radio. We propose a scheme that can achieve rate tuples within a constant gap away from the cut-set bound, where the constant is independent of channel coefficients and power constraints. First, for a deterministic broadcast network, we propose a new coding scheme, constructed by adopting a “receiver-centric” viewpoint, that uses quantize-and-forward relaying as an inner code concatenated with an outer Marton code for the induced deterministic broadcast channel. This scheme is shown to achieve the cut-set bound evaluated with product form distributions. This result is then lifted to the Gaussian network by using a deterministic network called the discrete superposition network as a formal quantization interface. This two-stage construction circumvents the difficulty involved in working with a vector nonlinear non-Gaussian broadcast channel that arises if we construct a similar scheme directly for the Gaussian network.

A classical result in undirected wireline networks is the near optimality of routing (flow) for multiple-unicast: the min cut upper bound is within a logarithmic factor of the number of sources of the max flow. Wireless channels differ from wireline ones in two primary ways: the signal out of a transmitting node is broadcast and the signals at a receiving node superpose. In this paper we focus on “extending” the wireline result to the wireless context, by separately considering the broadcast and superposition constraints. Our main result is the approximate optimality of a simple layering principle: local physical-layer schemes combined with global routing. We show this in the context of both Gaussian networks and packet erasure networks. The key technical contribution is an approximation of min cut in a bidirected graph with submodular constraints on the edge capacities by max flow.

We study a wireless broadcast-cum-multicast network, where a single source reliably communicates independent messages to multiple destinations, with the aid of relays. In addition, we assume there are nodes that demand all the messages at the source. We propose a compress-and-forward scheme that can achieve rates within a constant gap away from the cut-set bound. The proposed scheme operates in two steps: the inner code induces a broadcast channel with sufficient mutual information between the source and the destinations, and the outer code is basically a Marton code for broadcast channels. The inner code is constructed by lifting a scheme designed for a corresponding discrete superposition network.

We study a wireless relay network, with a single source and a single destination. Our main result is to show that an appropriate compress-and-forward scheme supports essentially the same reliable data rate as the quantize-map-and-forward and noisy network coding schemes [1], [2]; thus, it is approximately optimal - in the sense the data rate is a universal constant away from the cut-set upper bound. We characterize the compress-and-forward scheme through an abstract flow formulation, a generalization of flow on linking systems. This characterization allows for efficient computation of the minimal amount of information that has to flow through each node in the network.

Diversity-multiplexing tradeoff (DMT) is a coarse high SNR approximation of the fundamental tradeoff between data rate and reliability in a slow fading channel. In this paper, we characterize the fundamental DMT of the two-user single antenna Gaussian interference channel. We show that the class of multilevel superposition coding schemes universally achieves (for all fading statistics) the DMT for the two-user interference channel. For the special case of symmetric DMT, when the two users have identical rate and diversity gain requirements, we characterize the DMT achieved by the Han-Kobayashi scheme, which corresponds to two level superposition coding.

Diversity-multiplexing tradeoff (DMT) is a coarse high SNR approximation of the fundamental tradeoff between data rate and reliability in a slow fading channel. In this paper, we characterize the fundamental DMT of the two-user single antenna Gaussian interference channel. We show that the class of multilevel superposition coding schemes universally achieves (for all fading statistics) the DMT for the two-user interference channel. For the special case of symmetric DMT, when the two users have identical rate and diversity gain requirements, we characterize the DMT achieved by the Han-Kobayashi scheme, which corresponds to two level superposition coding.

The linear deterministic model has been used recently to get a first order understanding of many wireless communication network problems. In many of these cases, it has been pointed out that the capacity regions of the network and its reciprocal (where the communication links are reversed and the roles of the sources and the destinations are swapped) are the same. In this paper, we consider a linear deterministic communication network with multiple unicast information flows. For this model and under the restriction to the class of linear coding, we show that the rate regions for a network and its reciprocal are the same. This can be viewed as a generalization of the linear reversibility of wireline networks, already known in the network coding literature.

We introduce the two-user finite state compound interference channel. The main contributions involve both novel inner and outer bounds. For the Gaussian case, we characterize its capacity region to within one bit. The inner bound is multilevel superposition coding but the decoding of the levels is opportunistic, depending on the channel state. The genie aided outer bound is motivated by the typical error events of the achievable scheme.

We introduce the two user finite state compound Gaussian interference channel and characterize its capacity region to within one bit. The main contributions involve both novel inner and outer bounds. The inner bound is multilevel superposition coding but the decoding of the levels is opportunistic, depending on the channel state. The genie aided outer bound is motivated by the typical error events of the achievable scheme.