Approximating Node-Connectivity Augmentation Problems

Abstract

We consider the (undirected) Node Connectivity Augmentation (NCA) problem: given a graph J = (V ,E J ) and connectivity requirements {r (u ,v ):u ,v *** V }, find a minimum size set I of new edges (any edge is allowed) so that J + I contains r (u ,v ) internally disjoint uv -paths, for all u ,v *** V . In the Rooted NCA there is s *** V so that r (u ,v ) > 0 implies u = s or v = s . For large values of k = max u ,v *** V r (u ,v ), NCA is at least as hard to approximate as Label-Cover and thus it is unlikely to admit a polylogarithmic approximation. Rooted NCA is at least as hard to approximate as Hitting-Set. The previously best approximation ratios for the problem were O (k ln n ) for NCA and O (ln n ) for Rooted NCA. In [Approximating connectivity augmentation problems, SODA 2005] the author posed the following open question: Does there exist a function ρ (k ) so that NCA admits a ρ (k )-approximation algorithm? In this paper we answer this question, by giving an approximation algorithm with ratios O (k ln 2 k ) for NCA and O (ln 2 k ) for Rooted NCA. This is the first approximation algorithm with ratio independent of n , and thus is a constant for any fixed k . Our algorithm is based on the following new structural result which is of independent interest. If ${\cal D}$ is a set of node pairs in a graph J , then the maximum degree in the hypergraph formed by the inclusion minimal tight sets separating at least one pair in ${\cal D}$ is O (***2), where *** is the maximum connectivity of a pair in ${\cal D}$.