Reinforced Lin-Kernighan-Helsgaun Algorithms for the Traveling Salesman Problems

The Traveling Salesman Problem (TSP) is a classical NP-hard combinatorial optimization problem with many practical variants. The Lin-Kernighan-Helsgaun (LKH) algorithm is one of the state-of-the-art local search algorithms for the TSP, and LKH-3 is a powerful extension of LKH that can solve many TSP variants. Both LKH and LKH-3 associate a candidate set to each city to improve the algorithm efficiency, and have two different methods, called $\alpha$-measure and POPMUSIC, to decide the candidate sets. In this work, we first propose a Variable Strategy Reinforced LKH (VSR-LKH) algorithm, which incorporates three reinforcement learning methods (Q-learning, Sarsa, and Monte Carlo) with the LKH algorithm, for solving the TSP. We further propose a new algorithm, denoted as VSR-LKH-3, that combines the variable strategy reinforcement learning method with LKH-3 for typical TSP variants, including the TSP with time windows (TSPTW) and Colored TSP (CTSP). The proposed algorithms replace the inflexible traversal operations in LKH and LKH-3 and let the algorithms learn to make a choice at each search step by reinforcement learning. Both LKH and LKH-3, with either the $\alpha$-measure or the POPMUSIC method, can be significantly improved by our methods. Specifically, empirical results on 236 public and widely-used TSP benchmarks with up to 85,900 cities demonstrate the excellent performance of VSR-LKH, and the extended VSR-LKH-3 also significantly outperforms the state-of-the-art heuristics for TSPTW and CTSP.