Construct, Merge, Solve & Adapt with Reinforcement Learning for the min-max Multiple Traveling Salesman Problem

Abstract

The Multiple Traveling Salesman Problem (mTSP) extends the Traveling Salesman Problem to m tours that start and end at a common depot and jointly visit all customers exactly once. In the min-max variant, the objective is to minimize the longest tour, reflecting workload balance. We propose a hybrid approach, Construct, Merge, Solve & Adapt with Reinforcement Learning (RL-CMSA), for the symmetric single-depot min-max mTSP. The method iteratively constructs diverse solutions using probabilistic clustering guided by learned pairwise q-values, merges routes into a compact pool, solves a restricted set-covering MILP, and refines solutions via inter-route remove, shift, and swap moves. The q-values are updated by reinforcing city-pair co-occurrences in high-quality solutions, while the pool is adapted through ageing and pruning. This combination of exact optimization and reinforcement-guided construction balances exploration and exploitation. Computational results on random and TSPLIB instances show that RL-CMSA consistently finds (near-)best solutions and outperforms a state-of-the-art hybrid genetic algorithm under comparable time limits, especially as instance size and the number of salesmen increase.