[BEN07] The maximum integer multiterminal flow problem in directed graphs

Revue Internationale avec comité de lecture : Journal Operations Research Letters, vol. 35(2), pp. 195-200, 2007, (doi:10.1016/j.orl.2006.03.005)

Auteurs: C. Bentz

Résumé: This paper deals with the algorithmic aspects of one fundamental problem generalizing the famous maximum flow problem. This problem arises, in particular, as a requests routing problem in networks. Given an edge-capacitated graph and k terminal vertices, the maximum integer multiterminal flow problem (MaxIMTF) is to route the maximum number of flow units between the terminals. We study this problem in directed graphs and identify a key parameter, that was not considered previously: kL, the number of lonely terminals (a terminal is lonely if it lies on at least one directed cycle containing no other terminal). We prove that MaxIMTF is polynomial-time solvable when there is no lonely terminal and NP-hard to approximate within 2-ε (for any ε > 0), even when kL=1 and k=3 (and also when kL=k=2). We also give an (2 log (kL+2))-approximation algorithm for the general case, and show that the case kL=1 and k=2 is tractable. Moreover, some of our results for MaxIMTF extend to the well-known associated minimum multiterminal cut problem or match previous results known for this minimization problem.


@article {
title="{The maximum integer multiterminal flow problem in directed graphs}",
author="C. Bentz",
journal="Operations Research Letters",