Improved Approximation Algorithms for Minimum Power Covering Problems

Abstract

Given an undirected graph with edge costs, the power of a node is the maximum cost of an edge incident to it, and the power of a graph is the sum of the powers of its nodes. Motivated by applications in wireless networks, we consider two network design problems under the power minimization criteria. In both problems we are given a graph \(G=(V,E)\) with edge costs and a set \(T \subseteq V\) of terminals. The goal is to find a minimum power edge subset \(F \subseteq E\) such that the graph \(H=(V,F)\) satisfies some prescribed requirements. In the Min-Power Edge-Cover problem, H should contain an edge incident to every terminal. Using the Iterative Randomized Rounding (IRR) method, we give an algorithm with expected approximation ratio 1.41; the ratio is reduced to \(73/60<1.217\) when T is an independent set in G. In the case of unit costs we also achieve ratio 73 / 60, and in addition give a simple efficient combinatorial algorithm with ratio 5 / 4. For all these NP-hard problems the previous best known ratio was 3 / 2. In the related Min-Power Terminal Backup problem, H should contain a path from every \(t \in T\) to some node in \(T \setminus \{t\}\). We obtain ratio 3 / 2 for this NP-hard problem, improving the trivial ratio of 2.