Multi-Armed Bandit and Backbone Boost Lin-Kernighan-Helsgaun Algorithm for the Traveling Salesman Problems

Abstract

The Lin-Kernighan-Helsguan (LKH) heuristic is a classic local search algorithm for the TravelingSalesman Problem (TSP). LKH introduces an 𝛼-value to replace the traditional distance metric forevaluating the edge quality, which leads to a significant improvement. However, we observe that the𝛼-value does not make full use of the historical information during the search, and single guidinginformation often makes LKH hard to escape from some local optima. To address the above issues,we propose a novel way to extract backbone information during the TSP local search process, whichis dynamic and can be updated once a local optimal solution is found. We further propose to combinebackbone information, 𝛼-value, and distance to evaluate the edge quality so as to guide the search.Moreover, we abstract their different combinations to arms in a multi-armed bandit (MAB) and usean MAB model to help the algorithm select an appropriate evaluation metric dynamically. Both thebackbone information and MAB can provide diverse guiding information and learn from the searchhistory to suggest the best metric. We apply our methods to LKH and LKH-3, which is an extensionversion of LKH that can be used to solve about 40 variant problems of TSP and Vehicle RoutingProblem (VRP). Extensive experiments show the excellent performance and generalization capabilityof our proposed method, significantly improving LKH for TSP and LKH-3 for two representative TSPand VRP variants, the Colored TSP (CTSP) and Capacitated VRP with Time Windows (CVRPTW).

Keywords: Traveling salesman problems, Multi-armed bandit, Backbone, Lin-Kernighan-Helsgaun algorithm, Local search