Approximating subset k-connectivity problems

A subset T⊆V of terminals is k-connected to a root s in a directed/undirected graph J if J has k internally-disjoint vs-paths for every v∈T; T is k-connected in J if T is k-connected to every s∈T. We consider the Subsetk-Connectivity Augmentation problem: given a graph G=(V,E) with edge/node-costs, a node subset T⊆V, and a subgraph J=(V,EJ) of G such that T is (k−1)-connected in J, find a minimum-cost augmenting edge-set F⊆E∖EJ such that T is k-connected in J∪F. The problem admits trivial ratio O(|T|2). We consider the case |T|>k and prove that for directed/undirected graphs and edge/node-costs, a ρ-approximation algorithm for Rooted Subsetk-Connectivity Augmentation implies the following approximation ratios for Subsetk-Connectivity Augmentation: (i) b(\rho+k) + {\left(\frac{|T|}{|T|-k}\right)}^2 O\left(\log \frac{|T|}{|T|-k}\right) and (ii) \rho \cdot O\left(\frac{|T|}{|T|-k} \log k \right), where b=1 for undirected graphs and b=2 for directed graphs. The best known values of ρ on undirected graphs are min {|T|,O(k)} for edge-costs and min {|T|,O(k log|T|)} for node-costs; for directed graphs ρ=|T| for both versions. Our results imply that unless k=|T|−o(|T|), Subsetk-Connectivity Augmentation admits the same ratios as the best known ones for the rooted version. This improves the ratios in [19,14].