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 paper, 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) >= 3, for all i = 1, ... , 3. In this paper, we consider connectivity-level vectors such that min(i=1, ... , 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.