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(u,v) : u, v ∈ V}. The goal is to find a weight assignment w = {w _v : v ∈ V} of minimum total weight w(V) = _v∈V w _v, such that: for all u, v ∈ V, the activated graph G _w = (V,E _w ), where E _w = {uv : f ^uv (w _u ,w _v)=1}, contains r(u,v) 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 [12], 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(κ logn) for κ-Out/In-connected Subgraph Activation and κ-Connected Subgraph Activation. For directed graphs this solves a question from [12] for κ = 1, while for the min-power case and κ arbitrary this solves an open question from [9]. For other versions on undirected graphs, our ratios match the best known ones for the Node-Weighted Survivable Network problem [8].