Multicommodity Flows in Polymatroidal Capacity Networks

A classical result in undirected edge-capaciated networks is the approximate optimality of routing (flow) for multiple-unicast: the min-cut upper bound is within a logarithmic factor of the number of sources of the max flow [2, 3]. In this paper we focus on extending this result to a more general network model, where there are joint polymatroidal constraints on the rates of the edges that meet at a node. For directed polymatroidal networks with symmetric demands we show that the maximum concurrent flow is within O(log k) of the sparsest edge cut. We also show that for bidirected polymatroidal networks (with general demands), the maximum concurrent flow is within O(log k) of the sparsest edge cut. We finally show that the rate region achievable by flows is within a factor of O(log k) of the rate region defined by the cut-set bounds.