Survivable network activation problems

In the Survivable Networks Activation problem we are given a graph G=(V,E), S⊂V, a family {f_uv(x_u,x_v):uv∈E} of monotone non-decreasing activating functions from R+2 to {0,1} each, and connectivity requirements {r(uv):uv∈R} over a set R of requirement edges on V. The goal is to find a weight assignment w={w_v:v∈V} of minimum total weight w(V)=Σ_v∈Vw_v, such that in the activated graph G_w=(V, E_w), where E_w={uv: f_uv(w_u,w_v)=1}, the following holds: for each uv∈R, the activated graph ^Gw contains r(uv) pairwise edge-disjoint uv-paths such that no two of them have a node in S\-{u,v} in common. This problem was suggested recently by Panigrahi (2011) [19], generalizing the Node-Weighted Survivable Network and the Minimum-Power Survivable Network problems, as well as several other problems with motivation in wireless networks. We give new approximation algorithms for this problem. For undirected/directed graphs, our ratios are O(klogn) for k-Out/In-connected Subgraph Activation and k-Connected Subgraph Activation. For directed graphs this solves a question from Panigrahi (2011) [19] for k=1, while for the min-power case and k arbitrary this solves a question from Nutov (2010) [16]. For other versions on undirected graphs, our ratios match the best known ones for the Node-Weighted Survivable Network problem (Nutov, 2009 [14]).