An O(k^3 log n)-Approximation Algorithm for Vertex-Connectivity Survivable Network Design

In the Survivable Network Design problem (SNDP), we are given an undirected graph G(V, E) with costs on edges, along with a connectivity requirement r(u, v) for each pair u, v of vertices. The goal is to find a minimum-cost subset E* of edges, that satisfies the given set of pairwise connectivity requirements. In the edge-connectivity version we need to ensure that there are r(u, v) edge-disjoint paths for every pair u, v of vertices, while in the vertex-connectivity version the paths are required to be vertex-disjoint. The edge-connectivity version of SNDP is known to have a 2-approximation. However, no non-trivial approximation algorithm has been known so far for the vertex version of SNDP, except for special cases of the problem. We present an extremely simple algorithm to achieve an O(k³ log |T|)-approximation for this problem, where k denotes the maximum connectivity requirement, and T is the set of vertices that participate in one or more pairs with non-zero connectivity requirements. We also give a simple proof of the recently discovered O(k³ log |T|)-approximation algorithm for the single-source version of vertex-connectivity SNDP. Our results establish a natural connection between vertex-connectivity and a well-understood generalization of edge-connectivity, namely, element-connectivity, in that, any instance of vertex-connectivity can be expressed by a small number of instances of the element-connectivity problem.