Approximating connectivity augmentation problems

Let G = (V,E) be an undirected graph and let S V. The S-connectivity Λ^S_G(u,v) of a node pair (u,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 (CA) problem is: given a graph G = (V,E), a node subset S V, and a nonnegative integer requirement function r(u,v) on V × V, add a minimum size set F of new edges to G so that Λ^S_G+F(u,v) ≥ r(u,v) for all (u,v) ε V × V. Three extensively studied particular cases are: the Edge-CA (S = Ø), the Node-CA (S = V), and the Element-CA (r(u,v)= 0 whenever u S or v S). A polynomial-time algorithm for Edge-CA was developed by Frank. In this article we consider the Element-CA and the Node-CA, that are NP-hard even for r(u,v) ε {0,2}. The best known ratios for these problems were: 2 for Element-CA and O(r_max · ln n) for Node-CA, where r _max = max_u,vεV r(u,v) and n = V. Our main result is a 7/4-approximation algorithm for the Element-CA, improving the previously best known 2-approximation. For Element-CA with r(u,v) ε {0,1,2} we give a 3/2-approximation algorithm. These approximation ratios are based on a new splitting-off theorem, which implies an improved lower bound on the number of edges needed to cover a skew-supermodular set function. For Node-CA we establish the following approximation threshold: Node-CA with r(u,v) ε {0,k} cannot be approximated within O(2^log1-εn) for any fixed ε > 0, unless NP ⊆ DTIME(n^polylog(n)).