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 $\{r(u,v): u,v \in V\}$, find a minimum size set I of new edges (any edge is allowed) such that the graph J∪I contains r(u,v) internally-disjoint uv-paths, for all u,v∈V. In Rooted NCA there is s∈V such 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 an approximation ratio polylogarithmic in k. Rooted NCA is at least as hard to approximate as Hitting-Set. The previously best approximation ratios for the problem were O(kln n) for NCA and O(ln n) for Rooted NCA. In this paper we give an approximation algorithm with ratios O(kln 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 $\mathcal{D}$ 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 $\mathcal{D}$ is O(l 2), where l is the maximum connectivity in J of a pair in $\mathcal{D}$.