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, s(1) - t(1) and s(2) - t(2). 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 (S-i, T-j), where S-i subset of {s(1), s(2)} and T-j subset of {t(1), t(2)} for different subsets S-i and T-j. 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 t(i) is only guaranteed to decode information from the intended source s(i), 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.