Approximation Algorithms for the General Cluster Routing Problem

Abstract

Graph routing problem (GRP) and its generalizations have been extensively studied because of their broad applications in the real world. In this paper, we study a variant of GRP called the general cluster routing problem (GCRP). Let $G=\left( V,\,E\right)$ be a complete undirected graph with edge weight $c\left( e\right)$ satisfying the triangle inequality. The vertex set is partitioned into a family of clusters $C_{1},...,C_{m}$. We are given a required vertex subset $\overline{V} \subseteq V$ and a required edge subset $\overline{E} \subseteq E$. The aim is to compute a minimum cost tour that visits each vertex in $\overline{V}$ exactly once and traverses each edge in $\overline{E}$ at least once, while the vertices and edges of each cluster are visited consecutively. When the starting and ending vertices of each cluster are specified, we devise an approximation algorithm via incorporating Christofides’ algorithm with minimum weight bipartite matching, achieving an approximation ratio that equals the best approximation ratio of path TSP. Then for the case there are no specified starting and ending vertices for each cluster, we propose a more complicated algorithm with a compromised ratio of 2.67 by employing directed spanning tree against the designated auxiliary graph. Both approximation ratios improve the state-of-art approximation ratio that are respectively 2.4 and 3.25.