Network Design for Vertex Connectivity

We study the survivable network design problem (SNDP) for vertex connectivity. Given a graph G(V, E) with costs on edges, the goal of SNDP is to find a minimum cost subset of edges that ensures a given set of pairwise vertex connectivity requirements. When all connectivity requirements are between a special vertex, called the Source, and vertices in a subset T subset of V, called terminals, the problem is called the single-source SNDP. Our main result is a randomized k(O(k2)) log(4) n-approximation algorithm for single-source SNDP where k denotes the largest connectivity requirement for any source-terminal pair. In particular, we get a polylogarithmic approximation for my constant k. Prior to our work, no non-trivial approximation guarantees were known for this problem for any k >= 3. We also show that SNDP is k(Omega(1))-hard to approximate and provide art elementary construction that shows that the well-studied set-pair linear programming relaxation for this problem has an (Omega) over tilde (k(1/3)) integrality gap.