An Effective Iterated Two-stage Heuristic Algorithm for the Multiple Traveling Salesmen Problem

The multiple Traveling Salesmen Problem (mTSP) is a general extension of the famous NP-hard Traveling Salesmen Problem (TSP), that there are m (m > 1) salesmen to visit the cities. In this paper, we address the mTSP with both the minsum objective and minmax objective, which aims at minimizing the total length of the $m$ tours and the length of the longest tour among all the m tours, respectively. We propose an iterated two-stage heuristic algorithm called ITSHA for the mTSP. Each iteration of ITSHA consists of an initialization stage and an improvement stage. The initialization stage aims to generate high-quality and diverse initial solutions. The improvement stage mainly applies the variable neighborhood search (VNS) approach based on our proposed effective local search neighborhoods to optimize the initial solution. Moreover, some local optima escaping approaches are employed to enhance the search ability of the algorithm. Extensive experimental results on a wide range of public benchmark instances show that ITSHA significantly outperforms the state-of-the-art heuristic algorithms in solving the mTSP on both the objectives.