1.85 Approximation for Min-Power Strong Connectivity

Abstract: Given a directed simple graph G=(V,E) and a nonnegative-valued cost function the power of a vertex u in a directed spanning subgraph H is given by the maximum cost of an arcs of H exiting u. The power of H is the sum of the power of its vertices. Power Assignment seeks to minimize the power of H while H satisfies some connectivity constraint. In this paper, we assume E is bidirected (for every directed edge e in 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 introduced by Chen and Huang in 1989, who proved that bidirected minimum spanning tree has approximation ratio at most 2 (this is tight). In Approx 2010, we introduced a Greedy approximation algorithm and claimed a ratio of 1.992. Here we improve the analysis to 1.85. The proof also shows that a natural linear programming relaxation, introduced by us in 2012, has the same 1.85 integrality gap.