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 motivated by routing policies of autonomous systems in the Internet. We give a $2$-approximation algorithm for the problem of computing a maximumn number of edge- or vertex-disjoint valid paths between two given vertices $s$ and $t$, and we show that no better approximation ratio is possible unless $P=NP$. Furthermore, we give a $2$-approximation algorithm 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. For the multiway variant of the cut problem, we give a $4$-approximation algorithm. We present additional results for acyclic graphs