Min-power strong connectivity

• Authors: Gruia Calinescu
• Venue: APPROX/RANDOM'10 Proceedings of the 13th international conference on Approximation, and 14 the International conference on Randomization, and combinatorial optimization: algorithms and techniques
• Year: 2010

Given a directed simple graph G = (V,E) and a cost function c: E → R+, the power of a vertex u in a directed spanning subgraph H is given by pH(u) = maxuv∈E(H) c(uv), and corresponds to the energy consumption required for wireless node u to transmit to all nodes v with uv∈E(H). The power of H is given by p(H) = Σu∈V pH(u). Power Assignment seeks to minimize p(H) while H satisfies some connectivity constraint. In this paper, we assume E is bidirected (for every directed edge e ∈ E, the opposite edge exists and has the same cost), while H is required to be strongly connected. This is the original power assignment problem introduce in 1989 and since then the best known approximation ratio is 2 and is achieved by a bidirected minimum spanning tree. We improve this to 2 - ε for a small ε 0. We do this by combining techniques from Robins-Zelikovsky (2000) for Steiner Tree, Christofides (1976) for Metric Travelling Salesman, and Caragiannis, Flammini, and Moscardelli (2007) for the broadcast version of Power Assignment, together with a novel property on T-joins in certain two edge-connected hypergraphs. With the restriction that c: E → {A,B}, where 0 ≤ A , we improve the best known approximation ratio from 1.8 to π2/6-1/36+ε ≤ 1.61 using an adaptation of the algorithm developed by Khuller, Raghavachari, and Young (1995,1996) for (unweighted) Minimum Strongly Connected Subgraph.