A Simple Combinatorial Proof of Duality of Multiroute Flows and Cuts

A classical flow is a nonnegative linear combination of unit flows along simple paths. A multiroute flow, first considered by Kishimoto and Takeuchi, generalizes this concept. The basic building blocks are not single paths with unit flows but rather tuples consisting of k edge disjoint paths, each path with a unit flow. A multiroute flow is a nonnegative linear combination of such tuples. We present a simple combinatorial proof of the duality theorem for multiroute flows and cuts and its corollary which characterizes multiroute flows in terms of classical flows. Specifically, we show that a (classical) flow of size F is a k-flow if and only if the flow through every edge is at most F/k. This duality then immediately yields an efficient algorithm. 1 On leave from Department of Applied Mathematics, Faculty of Mathematics and Physics, Charles University in Prague, Czech Republic Email addresses: {bagchi,amic}@ics.uci.edu (Amitabha Bagchi, Amitabh Chaudhary), [email protected] (Petr Kolman), [email protected] (Jǐŕı Sgall). Preprint submitted to Elsevier Science 24 March 2004