通过构造连接：学习旅行商问题的可处理近巡回边际
Connected by Construction: Learning Tractable Near-Tour Marginals for Traveling Salesman Problems
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- Department of Civil and Environmental Engineering, Rice University（土木与环境工程系，莱斯大学）
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中文总结 AI 辅助
研究旅行商问题，提出端到端无监督学习管道C2TSP，基于构造连接的有根1-树吉布斯族，通过隐式微分学习残差边扰动，经平滑的Held-Karp层和证书引导锐化，在保持结构信息时提升解码性能，改善巡回成本和结构。
中文摘要 AI 辅助
基于学习的旅行商问题（TSP）方法通常通过解码或搜索后的巡回路线进行评估，但学习对象本身常处于替代空间，如热图、分配、构造策略或搜索指导分数。这掩盖了一个基本问题：解码前实际学到了什么哈密顿结构？本研究通过有结构意义的潜在对象学习TSP直接回答该问题。基于构造连接的有根1-树吉布斯族，提出端到端无监督学习管道C2TSP。通过隐式微分从无偏TSP成本学习残差边扰动，用平滑的Held-Karp层恢复期望度平衡，证书引导锐化推动连接分布向更类似巡回的结构发展。实验表明C2TSP在保持可解释结构信息的同时产生强大的解码性能。消融实验进一步验证边扰动和证书引导锐化共同改善巡回成本和类似巡回的结构。
英文摘要
Learning-based methods for the traveling salesman problem (TSP) are often evaluated through the tours produced after decoding or search, but the learned object itself frequently lives in a surrogate space such as heatmaps, assignments, construction policies, or search-guidance scores. This hides the fundamental question: what Hamiltonian structure has actually been learned before decoding? In this study, we directly answer this question by learning TSP through a structurally meaningful latent object, rather than leaving most of the Hamiltonian structure to the final decoding stage. Based on a connected-by-construction rooted $1$-tree Gibbs family, we propose an end-to-end unsupervised learning pipeline called \emph{C2TSP}. The pipeline learns residual edge perturbations from unbiased TSP cost through implicit differentiation. For structural correction, a smoothed Held--Karp layer restores expected degree balance, while certificate-guided sharpening further pushes the connected distribution toward more tour-like structures. Experiments show that C2TSP yields strong decoding performance while preserving interpretable structural information. Ablations further verify that edge perturbation and certificate-guided sharpening jointly improve both tour cost and tour-like structure.
