On the Information Capacity of Layered Undirected Unicast Networks

An important unsolved conjecture in network coding theory states that network coding has no rate benefit over routing in undirected unicast networks. Recently, a first non-trivial information-theoretic bound, called partition bound, was characterized for symmetric rate in undirected unicast networks, and it was shown that the bound is achievable for two classes of networks by a routing scheme. In this letter, we focus on the problem of characterization of networks for which the partition bound is tight. In particular, we consider layered undirected unicast networks. We show a routing scheme achieving the partition bound for a class of 3-layer networks and thus establish explicit information capacity expression and prove the conjecture for the class of networks. We also show that there exists a 4-layer network for which the partition bound cannot be achieved by an optimal routing scheme.