Purity Law for Generalizable Neural TSP Solvers

TL;DR

Purity Law (PuLa) reveals that edge usage in optimal TSP tours follows a negative exponential with respect to the edge purity order $K_p$, a property robust across scales and distributions. Building on PuLa, Purity Policy Optimization (PUPO) injects PuLa-informed signals into policy gradient training via purity availability φ and purity cost C, yielding learning signals that favor PuLa-consistent construction. Empirically, PUPO improves generalization of multiple neural TSP solvers on large-scale and real-world data (e.g., TSPLIB), with gains that grow when training on larger scales, while incurring negligible inference overhead. The work provides a principled, scalable approach to generalization in neural constructive solvers and suggests new directions for architectures that explicitly encode PuLa.

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.

Figures (6)

• Figure 1: Generalization performance of the neural TSP solver with PUPO is significantly improved across various instance scales and distributions compared to the vanilla solver without PUPO.
• Figure 2: Example of edge purity orders with the same length.
• Figure 3: Purity Law curves under varying scales and distributions.
• Figure 4: Mean purity order, proportion of $0$-order pure edges, and average order of non-$0$-order pure edges for varying instance.
• Figure 5: Heatmap of relatively improved accuracy after PUPO training of each model on the randomly generated dataset. From the figure, it can be observed that PUPO achieve outstanding improvement of generalizability on most instance types.

Theorems & Definitions (13)

• Definition 4.1: Purity Order
• Definition 5.1: Purity Availability
• Proposition 5.2: Supermodularity of Purity Availability
• Definition 5.3: Purity Cost
• Definition 5.4: Purity Weightings
• Proposition 2.1
• proof
• Proposition 2.2
• proof
• Proposition 2.3