Approximating minimum power covers of intersecting families and directed connectivity problems

Given a (directed) graph with costs on the edges, the power of a node is the maximum cost of an edge leaving it, and the power of the graph is the sum of the powers of its nodes. Motivated by applications for wireless networks, we consider fundamental directed connectivity network design problems under the power minimization criteria: the k-outconnected and the k-connected spanning subgraph problems. For k = 1 these problems are at least as hard as the Set-Cover problem and thus have an Ω(ln |V|) approximation threshold, while for arbitrary k a polylogarithmic approximation algorithm is unlikely. We give an O(ln |V|)-approximation algorithm for any constant k. In fact, our results are based on a much more general O(ln |V|)-approximation algorithm for the problem of finding a min-power edge-cover of an intersecting set-family; a set-family F on a groundset V is intersecting if X ∩ Y, X ∪ Y ∈ F for any intersecting X, Y ∈ F, and an edge set I covers F if for every X ∈ F there is an edge in I entering X.