Approximating minimum power covers of intersecting families and directed edge-connectivity problems

Abstract

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. Let G = ( V , E ) be a graph with edge costs { c ( e ) : e ∈ E } and let k be an integer. We consider problems that seek to find a min-power spanning subgraph G of G that satisfies a prescribed edge-connectivity property. In the Min-Power k -Edge-Outconnected Subgraph problem we are given a root r ∈ V , and require that G contains k pairwise edge-disjoint r v -paths for all v ∈ V − r . In the Min-Power k -Edge-Connected Subgraph problem G is required to be k -edge-connected. For k = 1 , these problems are at least as hard as the Set-Cover problem and thus have an Ω ( ln | V | ) approximation threshold. For k = Ω ( n ε ) , they are unlikely to admit a polylogarithmic approximation ratio [15]. We give approximation algorithms with ratio O ( k ln | V | ) . Our algorithms are based on a more general O ( ln | V | ) -approximation algorithm for the problem of finding a min-power directed edge-cover of an intersecting set-family; a set-family F 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 .