An algorithm for multicommodity flows in a class of planar networks

Abstract

AbstractThis paper presents a polynomial‐time algorithm for finding multicommodity flows in planar graphs. Suppose that G is an undirected planar graph and that some of the source‐sink pairs are located on the boundary of the outer face and all the other pairs share a common sink located on that boundary. The algorithm determines whether G has multicommodity flows each from a source to a sink and of a given demand, and if it does, actually finds them.Let n be the number of vertices in G and b be the number of edges on the boundary of the outer face. Then the time complexity of the algorithm is O(n(b2 + T+(n)), where T+(n) is the time required to find the shortest paths from a vertex to the others in a planar‐directed graph with n vertices and nonnegative real weighted edges. In the class of planar networks treated in this paper, it is known that the so‐called “Max Flow ‐ Min Cut theorem” holds.