Cuts and disjoint paths in the valley-free path model

Abstract

In the valley-free path model, a path in a given directed graph is valid if it consists of a sequence of forward edges followed by a sequence of backward edges. This model is moti vated by routing policies of autonomous systems in the Internet. We give a 2-approximation algorithm for the problem of computing a maximum number of edge- or vertex-disjoint valid paths between two given vertices s and t, and we show that no better approximation ra tio is possible unless P = NP. Furthermore, we give a 2-approximation for the problem of computing a minimum vertex cut that separates s and t with respect to all valid paths and prove that the problem is APX-hard. The corresponding problem for edge cuts is shown to be polynomial-time solvable. We present additional results for acyclic graphs.