On Integer Multiflow Maximization

Generalizing the two-commodity ow theorem of Rothschild and Whinston and the multiiow theorem of Lovv asz and Cherkasskij, Karzanov and Lomonosov proved a min-max theorem on maximum multiiows in the seventies. Their original proof is quite long and technical and relies on earlier investigations on metrics. The main purpose of the present paper is to provide a relatively simple proof of this theorem. Our proof relies on the locking theorem, another result of Karzanov and Lomonosov, and the polymatroid intersection theorem of Edmonds. For completeness, we also provide a simpliied proof of the locking theorem. Finally, we introduce the notion of a node-demand problem and, as another application of the locking theorem, we derive a feasibility theorem concerning it. The presented approach gives rise to (combinatorial) polynomial time algorithms.