Approximation algorithms for the min-max clustered k-traveling salesmen problems

Given a complete undirected graph G = (V, E), where V is the vertex set partitioned into K clusters V-1, V-2,...,V-K and E is the edge set with edge weights satisfying triangle inequality, and a positive integer k, the min-max clustered k-traveling salesmen problem(min-max Ck-TSP) asks to find a set of ktours to visit all vertices, such that each cluster is visited by exactly one tour and the vertices of each cluster are visited consecutively. The objective is to minimize the weight of the maximum weight tour. The problem is known to be NP-hard even when k = 1 and K = 1. In this paper, we consider two variants of the problem. The first one is all the ktours have a common predefined starting vertex, and the other one is no starting vertex of any tour is specified. For both the variants we propose the first constant-factor approximation algorithms with ratios 5.5 and 16, respectively. (c) 2022 Elsevier B.V. All rights reserved.