Are Generalized Cut-Set Bounds Tight for the Deterministic Interference Channel?

We propose the idea of extended networks, which is constructed by replicating the users in the two-user deterministic interference channel (DIC) and designing the interference structure among them, such that any rate that can be achieved by each user in the original network can also be achieved simultaneously by all replicas of that user in the extended network. We demonstrate that by carefully designing extended networks and applying the generalized cut-set (GCS) bound to them, we can derive a tight converse for the two-user DIC. Furthermore, we generalize our techniques to the three-user DIC, and demonstrate that the proposed approach also results in deriving a tight converse for the three-user DIC in the symmetric case.Comment: Part of this work has been presented in the 53rd Annual Allerton Conference on Communication, Control, and Computing, 201

A Generalized Cut-Set Bound for Deterministic Multi-Flow Networks and its Applications

We present a new outer bound for the sum capacity of general multi-unicast deterministic networks. Intuitively, this bound can be understood as applying the cut-set bound to concatenated copies of the original network with a special restriction on the allowed transmit signal distributions. We first study applications to finite-field networks, where we obtain a general outer-bound expression in terms of ranks of the transfer matrices. We then show that, even though our outer bound is for deterministic networks, a recent result relating the capacity of AWGN KxKxK networks and the capacity of a deterministic counterpart allows us to establish an outer bound to the DoF of KxKxK wireless networks with general connectivity. This bound is tight in the case of the "adjacent-cell interference" topology, and yields graph-theoretic necessary and sufficient conditions for K DoF to be achievable in general topologies.Comment: A shorter version of this paper will appear in the Proceedings of ISIT 201

The two-unicast problem

We consider the communication capacity of wireline networks for a two-unicast traffic pattern. The network has two sources and two destinations with each source communicating a message to its own destination, subject to the capacity constraints on the directed edges of the network. We propose a simple outer bound for the problem that we call the Generalized Network Sharing (GNS) bound. We show this bound is the tightest edge-cut bound for two-unicast networks and is tight in several bottleneck cases, though it is not tight in general. We also show that the problem of computing the GNS bound is NP-complete. Finally, we show that despite its seeming simplicity, the two-unicast problem is as hard as the most general network coding problem. As a consequence, linear coding is insufficient to achieve capacity for general two-unicast networks, and non-Shannon inequalities are necessary for characterizing capacity of general two-unicast networks.Comment: 23 pages, 22 figure

Rank Matching for Multihop Multiflow

We study the degrees of freedom (DoF) of the layered 2 X 2 X 2 MIMO interference channel where each node is equipped with arbitrary number of antennas, the channels between the nodes have arbitrary rank constraints, and subject to the rank-constraints the channel coefficients can take arbitrary values. The DoF outer bounds reveal a fundamental rank-matching phenomenon, reminiscent of impedance matching in circuit theory. It is well known that the maximum power transfer in a circuit is achieved not for the maximum or minimum load impedance but for the load impedance that matches the source impedance. Similarly, the maximum DoF in the rank- constrained 2 X 2 X 2 MIMO interference network is achieved not for the maximum or minimum ranks of the destination hop, but when the ranks of the destination hop match the ranks of the source hop. In fact, for mismatched settings of interest, the outer bounds identify a DoF loss penalty that is precisely equal to the rank-mismatch between the two hops. For symmetric settings, we also provide achievability results to show that along with the min-cut max-flow bounds, the rank-mismatch bounds are the best possible, i.e., they hold for all channels that satisfy the rank-constraints and are tight for almost all channels that satisfy the rank-constraints. Limited extensions - from sum-DoF to DoF region, from 2 unicasts to X message sets, from 2 hops to more than 2 hops and from 2 nodes per layer to more than 2 nodes per layer - are considered to illustrate how the insights generalize beyond the elemental 2 X 2 X 2 channel model

DMT of Multi-hop Cooperative Networks - Part I: Basic Results

In this two-part paper, the DMT of cooperative multi-hop networks is examined. The focus is on single-source single-sink (ss-ss) multi-hop relay networks having slow-fading links and relays that potentially possess multiple antennas. The present paper examines the two end-points of the DMT of full-duplex networks. In particular, the maximum achievable diversity of arbitrary multi-terminal wireless networks is shown to be equal to the min-cut. The maximum multiplexing gain of arbitrary full-duplex ss-ss networks is shown to be equal to the min-cut rank, using a new connection to a deterministic network. We also prove some basic results including a proof that the colored noise encountered in AF protocols for cooperative networks can be treated as white noise for DMT computations. The DMT of a parallel channel with independent MIMO links is also computed here. As an application of these basic results, we prove that a linear tradeoff between maximum diversity and maximum multiplexing gain is achievable for full-duplex networks with single antenna nodes. All protocols in this paper are explicit and rely only upon amplify-and-forward (AF) relaying. Half duplex networks are studied, and explicit codes for all protocols proposed in both parts, are provided in the companion paper.Comment: This submission is Part-I of a two-part paper, which is a detailed version of the previous submission arXiv:0802.188

On the Capacity of the Half-Duplex Diamond Channel

In this paper, a dual-hop communication system composed of a source S and a destination D connected through two non-interfering half-duplex relays, R1 and R2, is considered. In the literature of Information Theory, this configuration is known as the diamond channel. In this setup, four transmission modes are present, namely: 1) S transmits, and R1 and R2 listen (broadcast mode), 2) S transmits, R1 listens, and simultaneously, R2 transmits and D listens. 3) S transmits, R2 listens, and simultaneously, R1 transmits and D listens. 4) R1, R2 transmit, and D listens (multiple-access mode). Assuming a constant power constraint for all transmitters, a parameter $\Delta$ is defined, which captures some important features of the channel. It is proven that for $\Delta$=0 the capacity of the channel can be attained by successive relaying, i.e, using modes 2 and 3 defined above in a successive manner. This strategy may have an infinite gap from the capacity of the channel when $\Delta\neq$0. To achieve rates as close as 0.71 bits to the capacity, it is shown that the cases of $\Delta$>0 and $\Delta$<0 should be treated differently. Using new upper bounds based on the dual problem of the linear program associated with the cut-set bounds, it is proven that the successive relaying strategy needs to be enhanced by an additional broadcast mode (mode 1), or multiple access mode (mode 4), for the cases of $\Delta$0, respectively. Furthermore, it is established that under average power constraints the aforementioned strategies achieve rates as close as 3.6 bits to the capacity of the channel.Comment: 25 pages, 2 figures, submitted to IEEE Transactions on Information Theor

Cooperation for interference management: A GDoF perspective

The impact of cooperation on interference management is investigated by studying an elemental wireless network, the so called symmetric interference relay channel (IRC), from a generalized degrees of freedom (GDoF) perspective. This is motivated by the fact that the deployment of relays is considered as a remedy to overcome the bottleneck of current systems in terms of achievable rates. The focus of this work is on the regime in which the interference link is weaker than the source-relay link in the IRC. Our approach towards studying the GDoF goes through the capacity analysis of the linear deterministic IRC (LD-IRC). New upper bounds on the sum-capacity of the LD-IRC based on genie-aided approaches are established. These upper bounds together with some existing upper bounds are achieved by using four novel transmission schemes. Extending the upper bounds and the transmission schemes to the Gaussian case, the GDoF of the Gaussian IRC is characterized for the aforementioned regime. This completes the GDoF results available in the literature for the symmetric GDoF. It is shown that in the strong interference regime, in contrast to the IC, the GDoF is not a monotonically increasing function of the interference level

Informational Bottlenecks in Two-Unicast Wireless Networks with Delayed CSIT

We study the impact of delayed channel state information at the transmitters (CSIT) in two-unicast wireless networks with a layered topology and arbitrary connectivity. We introduce a technique to obtain outer bounds to the degrees-of-freedom (DoF) region through the new graph-theoretic notion of bottleneck nodes. Such nodes act as informational bottlenecks only under the assumption of delayed CSIT, and imply asymmetric DoF bounds of the form $mD_1 + D_2 \leq m$. Combining this outer-bound technique with new achievability schemes, we characterize the sum DoF of a class of two-unicast wireless networks, which shows that, unlike in the case of instantaneous CSIT, the DoF of two-unicast networks with delayed CSIT can take an infinite set of values.Comment: In proceedings of the 53rd Annual Allerton Conference on Communication, Control, and Computin

Cooperative Strategies for Simultaneous and Broadcast Relay Channels

Consider the \emph{simultaneous relay channel} (SRC) which consists of a set of relay channels where the source wishes to transmit common and private information to each of the destinations. This problem is recognized as being equivalent to that of sending common and private information to several destinations in presence of helper relays where each channel outcome becomes a branch of the \emph{broadcast relay channel} (BRC). Cooperative schemes and capacity region for a set with two memoryless relay channels are investigated. The proposed coding schemes, based on \emph{Decode-and-Forward} (DF) and \emph{Compress-and-Forward} (CF) must be capable of transmitting information simultaneously to all destinations in such set. Depending on the quality of source-to-relay and relay-to-destination channels, inner bounds on the capacity of the general BRC are derived. Three cases of particular interest are considered: cooperation is based on DF strategy for both users --referred to as DF-DF region--, cooperation is based on CF strategy for both users --referred to as CF-CF region--, and cooperation is based on DF strategy for one destination and CF for the other --referred to as DF-CF region--. These results can be seen as a generalization and hence unification of previous works. An outer-bound on the capacity of the general BRC is also derived. Capacity results are obtained for the specific cases of semi-degraded and degraded Gaussian simultaneous relay channels. Rates are evaluated for Gaussian models where the source must guarantee a minimum amount of information to both users while additional information is sent to each of them.Comment: 32 pages, 7 figures, To appear in IEEE Trans. on Information Theor

Lattice Coding and the Generalized Degrees of Freedom of the Interference Channel with Relay

The generalized degrees of freedom (GDoF) of the symmetric two-user Gaussian interference relay channel (IRC) is studied. While it is known that the relay does not increase the DoF of the IC, this is not known for the more general GDoF. For the characterization of the GDoF, new sum-capacity upper bounds and lower bounds are derived. The lower bounds are obtained by a new scheme, which is based on functional decode-and-forward (FDF). The GDoF is characterized for the regime in which the source-relay link is weaker than the interference link, which constitutes half the overall space of channel parameters. It is shown that the relay can indeed increase the GDoF of the IRC and that it is achieved by FDF.Comment: 5 pages, 3 figures, ISIT 201