Approximation Algorithms and Hardness of the k -Route Cut Problem

We study the k -route cut problem: given an undirected edge-weighted graph G = ( V , E ), a collection {( s 1 , t 1 ), ( s 2 , t 2 ), …, ( s r , t r )} of source-sink pairs, and an integer connectivity requirement k , the goal is to find a minimum-weight subset E ′ of edges to remove, such that the connectivity of every pair ( s i , t i ) falls below k . Specifically, in the edge-connectivity version, EC-kRC, the requirement is that there are at most ( k − 1) edge-disjoint paths connecting s i to t i in G ∖ E ′, while in the vertex-connectivity version, VC-kRC, the same requirement is for vertex-disjoint paths. Prior to our work, poly-logarithmic approximation algorithms have been known for the special case where k ⩽ 3, but no non-trivial approximation algorithms were known for any value k > 3, except in the single-source setting. We show an O ( k log 3/2 r )-approximation algorithm for EC-kRC with uniform edge weights, and several polylogarithmic bi-criteria approximation algorithms for EC-kRC and VC-kRC, where the connectivity requirement k is violated by a constant factor. We complement these upper bounds by proving that VC-kRC is hard to approximate to within a factor of k ϵ for some fixed ϵ > 0. We then turn to study a simpler version of VC-kRC, where only one source-sink pair is present. We give a simple bi-criteria approximation algorithm for this case, and show evidence that even this restricted version of the problem may be hard to approximate. For example, we prove that the single source-sink pair version of VC-kRC has no constant-factor approximation, assuming Feige’s Random κ-AND assumption.