Approximation Algorithms for Min-Max Cycle Cover Problems

Abstract

As a fundamental optimization problem, the vehicle routing problem has wide application backgrounds and has been paid lots of attentions in past decades. In this paper we study its applications in data gathering and wireless energy charging for wireless sensor networks, by devising improved approximation algorithms for it and its variants. The key ingredients in the algorithm design include exploiting the combinatorial properties of the problems and making use of tree decomposition and minimum weighted maximum matching techniques. Specifically, given a metric complete graph $G$ and an integer $k>0$ , we consider rootless, uncapacitated rooted, and capacitated rooted min-max cycle cover problems in $G$ with an aim to find $k$ rootless (or rooted) edge-disjoint cycles covering the vertices in $V$ such that the maximum cycle weight among the $k$ cycles is minimized. For each of the mentioned problems, we develop an improved approximate solution. That is, for the rootless min-max cycle cover problem, we develop a $(5{ 1\over 3} +\epsilon)$ -approximation algorithm; for the uncapacitated rooted min-max cycle cover problem, we devise a $(6{ 1\over 3} +\epsilon)$ -approximation algorithm; and for the capacitated rooted min-max cycle cover problem, we propose a $(7+\epsilon)$ -approximation algorithm. These algorithms improve the best existing approximation ratios of the corresponding problems $6+\epsilon$ , $7+\epsilon$ , and $13+\epsilon$ , respectively, where $\epsilon$ is a constant with $0 . We finally evaluate the performance of the proposed algorithms through experimental simulations. Experimental results show that the actual approximation ratios delivered by the proposed algorithms are always no more than 2, much better than their analytical counterparts.