Approximating node-connectivity augmentation problems

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 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 D is O(ℓ ²), where ℓ is the maximum connectivity of a pair in D.