Solving the Min-Max Multiple Traveling Salesmen Problem via Learning-Based Path Generation and Optimal Splitting

This study addresses the Min-Max Multiple Traveling Salesmen Problem ($m^3$-TSP), which aims to coordinate tours for multiple salesmen such that the length of the longest tour is minimized. Due to its NP-hard nature, exact solvers become impractical under the assumption that $P \ne NP$. As a result, learning-based approaches have gained traction for their ability to rapidly generate high-quality approximate solutions. Among these, two-stage methods combine learning-based components with classical solvers, simplifying the learning objective. However, this decoupling often disrupts consistent optimization, potentially degrading solution quality. To address this issue, we propose a novel two-stage framework named \textbf{Generate-and-Split} (GaS), which integrates reinforcement learning (RL) with an optimal splitting algorithm in a joint training process. The splitting algorithm offers near-linear scalability with respect to the number of cities and guarantees optimal splitting in Euclidean space for any given path. To facilitate the joint optimization of the RL component with the algorithm, we adopt an LSTM-enhanced model architecture to address partial observability. Extensive experiments show that the proposed GaS framework significantly outperforms existing learning-based approaches in both solution quality and transferability.