Multi-armed Bandit and Backbone boost Lin-Kernighan-Helsgaun Algorithm for the Traveling Salesman Problems

TL;DR

This paper enhances the LKH heuristic for TSP by integrating dynamic backbone information and a multi-armed bandit model to adaptively select evaluation metrics, leading to significant performance improvements across various problem variants.

Contribution

It introduces a novel dynamic backbone extraction method and a multi-armed bandit framework to improve the guidance of local search algorithms for TSP and VRP variants.

Findings

Significantly improves LKH performance on TSP and VRP variants.

Demonstrates strong generalization across multiple problem types.

Outperforms existing methods in solution quality and efficiency.

Abstract

The Lin-Kernighan-Helsguan (LKH) heuristic is a classic local search algorithm for the Traveling Salesman Problem (TSP). LKH introduces an $\alpha$-value to replace the traditional distance metric for evaluating the edge quality, which leads to a significant improvement. However, we observe that the $\alpha$-value does not make full use of the historical information during the search, and single guiding information 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, which is dynamic and can be updated once a local optimal solution is found. We further propose to combine backbone information, $\alpha$-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 use an MAB model to help the algorithm select an appropriate evaluation metric dynamically. Both the backbone information and MAB can provide diverse guiding information and learn from the search history to suggest the best metric. We apply our methods to LKH and LKH-3, which is an extension version of LKH that can be used to solve about 40 variant problems of TSP and Vehicle Routing Problem (VRP). Extensive experiments show the excellent performance and generalization capability of our proposed method, significantly improving LKH for TSP and LKH-3 for two representative TSP and VRP variants, the Colored TSP (CTSP) and Capacitated VRP with Time Windows (CVRPTW).