Cuts and disjoint paths in the valley-free path model

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 maximum number of edgeor 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.

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