Approximating connectivity augmentation problems

Let G = (V, E) be a graph and let S ⊆ V. The S-connectivity λs(u,v;G) of u and v in G is the maximum number of uv-paths that no two of them have an edge or a node in S -{u,v} in common. The corresponding Connectivity Augmentation Problem (CAP) is: given a graph G = (V, E), a node subset S ⊆ V, and a nonnegative integer requirement function r(u, v) on the set of pairs of nodes, add a minimum size set F of new edges to G so that λ _S(u,v; G + F) ≥ r(u, v) holds for all u,v ∈ V. Three extensively studied particular cases are: the edge- (S = θ), the node- (S = V), and the element- (r(u,v) = 0 whenever u ∈ S or v ∈ S) CAP. A polynomial algorithm for edge-CAP was developed by A. Frank. In this paper we consider the element-CAP and the node-CAP, that are NP-hard even for r(u,v) ∈{0,2}. Our main result is a 7/4-approximation algorithm for the element-CAP, improving the previously best known 2-approximation. For the {0, k}-element-CAP (with r(u,v) ∈ {0, k}) and for the {0, 1, 2}-element-CAP we give a 3/2-approximation algorithm. The approximation ratios are based on a new lower bound on the number of edges needed to cover a skew-supermodular set function. For the node-CAP we establish the following approximation threshold: the {0, k}-node-CAP cannot be approximated within O(2 ^{log 1-ε n}) for any fixed ∈ > 0, unless NP ⊆DTIME(n ^polylog(n)); thus the node-CAP is unlikely to have a polylogarithmic approximation.