Multicommodity flows and cuts in polymatroidal networks

We consider multicommodity flow and cut problems in polymatroidal networks where there are submodular capacity constraints on the edges incident to a node. Polymatroidal networks were introduced by Lawler and Martel [20] and Hassin [15] in the single-commodity setting and are closely related to the submodular flow model of Edmonds and Giles [10]; the well-known maxflow-mincut theorem holds in this more general setting. Polymatroidal networks for the multicommodity case have not, as far as the authors are aware, been previously explored. Our work is primarily motivated by applications to information flow in wireless networks. We also consider the notion of undirected polymatroidal networks and observe that they provide a natural way to generalize flows and cuts in edge and node capacitated undirected networks. We establish poly-logarithmic flow-cut gap results in several scenarios that have been previously considered in the standard network flow models where capacities are on the edges or nodes [21, 22, 13, 19, 12]. Our results from a preliminary version have already found applications in wireless network information flow [16, 7] and we anticipate more in the future. On the technical side our key tools are the formulation and analysis of the dual of the flow relaxations via continuous extensions of submodular functions, in particular the Lovász extension. For directed graphs we rely on a simple yet useful reduction from polymatroidal networks to standard networks. For undirected graphs we rely on the interplay between the Lovász extension of a submodular function and line embeddings with low average distortion introduced by Matoušek and Rabinovich [25]; this connection is inspired by, and generalizes, the work of Feige, Hajiaghayi and Lee [12] on node-capacitated multicommodity flows and cuts. The applicability of embeddings to flow-cut gaps in polymatroidal networks is of independent mathematical interest.