神经聚类先，路径后：通过可微最优传输实现的一次性有容量车辆路径问题
Neural Cluster First, Route Second: Capacitated Vehicle Routing via Differentiable Optimal Transport
AI总结：
本文提出神经CFRS框架，通过可微最优传输层实现一次性非自回归解决有容量车辆路径问题，理论保证了对空间对称性的抽象，并在实际应用中表现出鲁棒性和高效性。
摘要：
有容量车辆路径问题（CVRP）是现代最后一公里物流的基础。当前神经组合优化（NCO）方法通过自回归方式构建CVRP解决方案，继承了顺序解码瓶颈、对空间对称性的敏感性和对分布外行为的脆弱性。我们重新审视经典的聚类先、路径后（CFRS）范式——长期以来被认为在渐近上最优但被NCO largely忽视——并论证其结构与深度学习的核心优势一致：相似性和全局上下文中的分配，而不是长序列巡回的构建。我们引入神经CFRS，这是首个纯粹非自回归的一次性神经CFRS框架用于CVRP。它通过可微熵最优传输层端到端地强制全局车队容量约束，生成连续传输计划以稀疏化精确容量分配求解器。我们提供形式理论保证，证明我们的架构本质上抽象了E(2)空间、跨路径排列和内路径遍历对称性。通过为框架配备预训练的空间词汇，我们解锁了极高的参数效率和零样本扩展能力。主要针对现实世界空间分布下的恒定容量设置，神经CFRS能够稳健地扩展到分布外N=1000实例，差距小于4%——即使在该规模下，超轻量单层架构仍保持约5%的差距。此外，当在标准基准上部署时，我们实现了在大小为100的问题上的高度竞争性2.73%的最优性差距。
英文摘要：
The Capacitated Vehicle Routing Problem (CVRP) underpins modern last-mile logistics, where routing decisions recur over the same fixed service area, like a city. In this setting, routing problems share a fixed set of potential customer locations, while active customers and demands vary between instances. We study how this spatial support can be exploited through reusable learned representations and design our method around three symmetries of the symmetric Euclidean CVRP: $E(2)$ transformations, vehicle-route permutations, and tour reversal. We introduce Neural Cluster-First--Route-Second (CFRS), a neural extension of the Fisher--Jaikumar framework that predicts seed-selection scores and customer-to-cluster assignment costs non-autoregressively and respects the three symmetries. A differentiable entropic optimal transport layer provides capacity-aware supervision and guides discrete capacitated assignment, followed by independent traveling salesman subproblems for route recovery. Component ablations show consistent benefits from learned seed selection, while learned assignment costs perform best near the training size and classical FJ costs perform better at larger sizes under exact decoding. On the fixed-support distribution with constant capacity, a model trained on $N=100$ achieves a $3.77\%$ routing gap relative to HGS at $N=1000$ without retraining. A shallow variant with one attention layer in each transformer achieves a $5.08\%$ gap at this scale, with spatial embeddings consistently improving routing quality over raw coordinates. Embedding interpolation further accommodates entirely unseen customer locations without retraining. On standard CVRP benchmarks, a separately trained model achieves a $2.73\%$ routing gap relative to LKH-3 at $N=100$.
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