Achieving generalization in neural approaches across different scales and distributions remains a significant challenge for the Traveling Salesman Problem~(TSP). A key obstacle is that neural networks often fail to learn robust principles for identifying universal patterns and deriving optimal solutions from diverse instances. In this paper, we first uncover Purity Law (PuLa), a fundamental structural principle for optimal TSP solutions, defining that edge prevalence grows exponentially with the sparsity of surrounding vertices. Statistically validated across diverse instances, PuLa reveals a consistent bias toward local sparsity in global optima. Building on this insight, we propose Purity Policy Optimization~(PUPO), a novel training paradigm that explicitly aligns characteristics of neural solutions with PuLa during the solution construction process to enhance generalization. Extensive experiments demonstrate that PUPO can be seamlessly integrated with popular neural solvers, significantly enhancing their generalization performance without incurring additional computational overhead during inference.

本研究针对旅行商问题(TSP)在不同规模和分布中实现泛化的挑战，提出了纯度法则(PuLa)，揭示了优化TSP解的基本结构原则：边缘的普遍性随着周围顶点的稀疏性而指数增长。基于此，研究者提出了纯度政策优化(PUPO)这一新颖的训练范式，能够在解决方案构建过程中将神经网络解的特性与PuLa显式对齐，实验结果显示PUPO显著提高了流行神经求解器的泛化性能。 

通用神经旅行商问题求解器的纯度法则