Approximating Steiner Network Activation Problems

In the Steiner Networks Activation problem we are given a graph G = (V,E), S ⊆ V , a family {fuv(xu, xv) : uv ∈ E} of monotone non-decreasing activating functions from R2+ to {0, 1} each, and connectivity requirements {r(u, v) : u, v ∈ V }. The goal is to find a weight assignment w = {wv : v ∈ V } of minimum total weight w(V ) = ∑ v∈V wv, such that: for all u, v ∈ V , the activated graph Gw = (V,Ew), where Ew = {uv : f uv(wu, wv) = 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 [14], generalizing the Node-Weighted Steiner Network and the Minimum-Power Steiner 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(k log n) for k-Out/Inconnected Subgraph Activation and k-Connected Subgraph Activation. For directed graphs this solves a question from [14] for k = 1, while for the min-power case and k arbitrary this solves an open question from [11]. For other versions on undirected graphs, our ratios match the best known ones for the Node-Weighted Steiner Network problem [10].