Hardness of Approximation for Vertex-Connectivity Network Design Problems

Abstract

In the survivable network design 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 $2^{\log^{1-\epsilon} n}$ ratio approximation for any fixed $\epsilon\! >\! 0$, unless $\NP\subseteq \DTIME(n^{\polylog(n)})$. 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.