Bandit based Dynamic Candidate Edge Selection in Solving Traveling Salesman Problems

Algorithms designed for routing problems typically rely on high-quality candidate edges to guide their search, aiming to reduce the search space and enhance the search efficiency. However, many existing algorithms, like the classical Lin-Kernighan-Helsgaun (LKH) algorithm for the Traveling Salesman Problem (TSP), often use predetermined candidate edges that remain static throughout local searches. This rigidity could cause the algorithm to get trapped in local optima, limiting its potential to find better solutions. To address this issue, we propose expanding the candidate sets to include other promising edges, providing them an opportunity for selection. Specifically, we incorporate multi-armed bandit models to dynamically select the most suitable candidate edges in each iteration, enabling LKH to make smarter choices and lead to improved solutions. Extensive experiments on multiple TSP benchmarks show the excellent performance of our method. Moreover, we employ this bandit-based method to LKH-3, an extension of LKH tailored for solving various TSP variant problems, and our method also significantly enhances LKH-3's performance across typical TSP variants.

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

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).