Purity Law for Generalizable Neural TSP Solvers

TL;DR

This paper introduces the Purity Law, a structural principle for TSP solutions, and proposes Purity Policy Optimization to improve neural solver generalization across diverse instances.

Contribution

The paper uncovers the Purity Law for TSP solutions and develops PUPO, a training method that aligns neural solutions with this law to enhance generalization.

Findings

PULO reveals a bias toward local sparsity in optimal solutions.

PUPO improves neural solver generalization without extra inference cost.

Experimental results show significant performance gains across diverse TSP instances.

Abstract

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.