Approximating Minimum-Power Network Design Problems

Given a 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 in wireless networks, we consider several network design problems under the power minimization criteria. Given a graph G = (V, E) with costs on the edges and requirements r(v) for each v ∈ V , the Min-Power Edge-Multi-Cover problem (MPEMC) is to find a min-power subgraph so that the degree of every node v is at least r(v). We give an O(log n)-approximation algorithms for MPEMC (improving the previously best known O(log n)-approximation [17]); this implies an O(log n+α)-approximation algorithm for the undirectedMin-Power k-Connected Subgraph (MPk-CS) problem, where α is the best known approximation for the min-cost variant of the problem. (Currently, α = O(ln k) for n ≥ 2k and α = O(ln k·min{ n n−k , √ k ln n}) otherwise.) We also consider the case of small requirements. Specifically, some of our approximation ratios are: 3/2 for MPEMC with r(v) ∈ {0, 1} (improving the ratio 2 by [17]) and 3 2 3 (improving the ratio 4 by [6]) for the min-power 2-connected and 2-edge-connected spanning subgraph problems. Finally, we give a 4rmax-approximation algorithm for the undirected min-power Steiner Network problem: find a min-power subgraph that contains r(u, v) pairwise edge-disjoint paths for every pair u, v of nodes. ∗Rutgers University, Camden, [email protected] †Microsoft Research, [email protected] ‡The Open University of Israel, Raanana [email protected] §The Open University of Israel, [email protected]