Cuts and Disjoint Paths in the Valley-Free Path Model of Internet BGP Routing

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 BGP routing policies of autonomous systems in the Internet. Robustness considerations lead to the problem of computing a maximum number of disjoint paths between two nodes, and the minimum size of a cut that separates them. We study these problems in the valley-free path model. For the problem of computing a maximum number of edgeor vertex-disjoint valid paths between two given vertices s and t, we give a 2-approximation algorithm and show that no better approximation ratio is possible unless P = NP . For the problem of computing a minimum vertex cut that separates s and t with respect to all valid paths, we give a 2-approximation algorithm 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.