On multicommodity flows in planar graphs

Abstract

Abstract Okamura and Seymour recently proved two properties of multicommodity flows in undirected planar networks where all the sources and the sinks are on a common face of the underlying graph. One is that a feasible solution is guaranteed whenever each cut's capacity is at least as large as the cut's demand. The second is that if all demands and capacities are integers then the flow values may be chosen half‐integer‐valued. In this paper we use the first property to construct two computational procedures; one examines the existence of a feasible flow, and the other constructs such a flow if one exists. We also show that the construction procedure can be used as an alternative proof to the above properties. Finally we show, by presenting counterexamples, that the half‐integrality property does not necessarily hold when either the graph cannot be drawn in the plane with all sources and sinks on a common face, or the graph is directed.