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

An Achievable Region for the Double Unicast Problem Based on a Minimum Cut Analysis

We consider the multiple unicast problem under network coding over directed acyclic networks when there are two source-terminal pairs, s1 - t1 and s2 - t2. Current characterizations of the multiple unicast capacity region in this setting have a large number of inequalities, which makes them hard to explicitly evaluate. In this work we consider a slightly different problem. We assume that we only know certain minimum cut values for the network, e.g., mincut(Si, Tj), where Si ⊆ {si, s2} and Tj ⊆ {t1, t2} for different subsets Si and Tj. Based on these values, we propose an achievable rate region for this problem based on linear codes. Towards this end, we begin by defining a base region where both sources are multicast to both the terminals. Following this we enlarge the region by appropriately encoding the information at the source nodes, such that terminal ti is only guaranteed to decode information from the intended source si, while decoding a linear function of the other source. The rate region takes different forms depending upon the relationship of the different cut values in the network

On the multiple unicast capacity of 3-source, 3-terminal directed acyclic networks

We consider the multiple unicast problem with three source-terminal pairs over directed acyclic networks with unit-capacity edges. The three $s_i-t_i$ pairs wish to communicate at unit-rate via network coding. The connectivity between the $s_i - t_i$ pairs is quantified by means of a connectivity level vector, $[k_1 k_2 k_3]$ such that there exist $k_i$ edge-disjoint paths between $s_i$ and $t_i$. In this work we attempt to classify networks based on the connectivity level. It can be observed that unit-rate transmission can be supported by routing if $k_i \geq 3$, for all $i = 1, \dots, 3$. In this work, we consider, connectivity level vectors such that $\min_{i = 1, \dots, 3} k_i < 3$. We present either a constructive linear network coding scheme or an instance of a network that cannot support the desired unit-rate requirement, for all such connectivity level vectors except the vector $[1~2~4]$ (and its permutations). The benefits of our schemes extend to networks with higher and potentially different edge capacities. Specifically, our experimental results indicate that for networks where the different source-terminal paths have a significant overlap, our constructive unit-rate schemes can be packed along with routing to provide higher throughput as compared to a pure routing approach.Comment: To appear in the IEEE/ACM Transactions on Networkin

Linear Network Coding for Two-Unicast-ZZ Networks: A Commutative Algebraic Perspective and Fundamental Limits

We consider a two-unicast-$Z$ network over a directed acyclic graph of unit capacitated edges; the two-unicast-$Z$ network is a special case of two-unicast networks where one of the destinations has apriori side information of the unwanted (interfering) message. In this paper, we settle open questions on the limits of network coding for two-unicast-$Z$ networks by showing that the generalized network sharing bound is not tight, vector linear codes outperform scalar linear codes, and non-linear codes outperform linear codes in general. We also develop a commutative algebraic approach to deriving linear network coding achievability results, and demonstrate our approach by providing an alternate proof to the previous results of C. Wang et. al., I. Wang et. al. and Shenvi et. al. regarding feasibility of rate $(1,1)$ in the network.Comment: A short version of this paper is published in the Proceedings of The IEEE International Symposium on Information Theory (ISIT), June 201

A simple necessary and sufficient condition for the double unicast problem

Abstract—We consider a directed acyclic network where there are two source-terminal pairs and the terminals need to receive the symbols generated at the respective sources. Each source independently generates an i.i.d. random process over the same alphabet. Each edge in the network is error-free, delay-free, and can carry one symbol from the alphabet per use. We give a simple necessary and sufficient condition for being able to simultaneously satisfy the unicast requirements of the two source-terminal pairs at rate-pair (1, 1) using vector network coding. The condition is also sufficient for doing this using only “XOR ” network coding and is much simpler compared to the necessary and sufficient conditions known from previous work. Our condition also yields a simple characterization of the capacity region of a double-unicast network which does not support the rate-pair (1, 1). I