Survivable Network Activation Problems

Abstract

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 $\mathbb{R}^2_+$ 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(k logn) for k -Out/In-connected Subgraph Activation and k -Connected Subgraph Activation. For directed graphs this solves a question from [12] for k = 1, while for the min-power case and k 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].