Min-Power Strong Connectivity

Abstract

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 p _H (u) =  max _{uv ∈ 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) = ∑  _u ∈ V p _H (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 < B, we improve the best known approximation ratio from 1.8 to π ²/6 − 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.