Integral symmetric 2-commodity flows

Abstract

We study integral 2-commodity flows in networks with a special characteristic, namely symmetry. We show that the Symmetric 2-Commodity Flow Problem is in P, by proving that the cut criterion is a necessary and sufficient condition for the existence of a solution. We also give a polynomial-time algorithm whose complexity is 6Cflow + O(|A|), where Cflow is the time complexity of your favorite flow algorithm (usually in O(|V| × |A|)). Our result closes an open question in a surprising way, since it is known that the Integral 2-Commodity Flow Problem is NP-complete for both directed and undirected graphs. This work finds application in optical telecommunication networks. α Springer-Verlag 2004.