Hardness of approximation for vertex-connectivity network-design problems

In the survivable network design problem SNDP, the goal is to find a minimum-cost subgraph satisfying certain connectivity requirements. We study the vertex-connectivity variant of SNDP in which the input specifies, for each pair. of vertices, a required number of vertex-disjoint paths connecting them. We give the first lower bound on the approximability of SNDP, showing that the problem admits no efficient 2(log1-epsilonn) ratio approximation for any fixed epsilon > 0 unless NP subset of or equal to DTIME(n(polylog(n))). We also show hardness of approximation results for several important special cases of SNDP, including constant factor hardness for the k-vertex connected spanning subgraph problem (k-VCSS) and for the vertex-connectivity augmentation problem, even when the edge costs are severely restricted.