The maximum integer multiterminal flow problem in directed graphs

Abstract : 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-? (for any ? 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.