Hardness of Approximation for Vertex-Connectivity Network Design Problems

In the survivable networkdesign problem (SNDP), the goal is to find a minimum-cost spanning 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 strong lower bound on the approximability of SNDP, showing that the problem admits no efficient 2^log1-ε n ratio approximation for any fixed ε\! >\! 0, unless ⊆ (n⁽ⁿ⁾). We show hardness of approximation results for some important special cases of SNDP, and we exhibit the first lower bound on the approximability of the related classical NP-hard problem of augmenting the connectivity of a graph using edges from a given set.