RELAGRAM: Recursive Latent Graph Model for the Capacitated Vehicle Routing Problem

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 $N=100$.

Over the past decade, neural network solvers powered by generative artificial intelligence have garnered significant attention in the domain of vehicle routing problems (VRPs), owing to their exceptional computational efficiency and superior reasoning capabilities. In particular, autoregressive solvers integrated with reinforcement learning have emerged as a prominent trend. However, much of the existing work emphasizes large-scale generalization of neural approaches while neglecting the limited robustness of attention-based methods across heterogeneous distributions of problem parameters. Their improvements over heuristic search remain largely restricted to hand-curated, fixed-distribution benchmarks. Furthermore, these architectures tend to degrade significantly when node representations are highly similar or when tasks involve long decision horizons. To address the aforementioned limitations, we propose a novel fusion neural network framework that employs a discrete noise graph diffusion model to learn the underlying constraints of vehicle routing problems and generate a constraint assignment matrix. This matrix is subsequently integrated adaptively into the feature representation learning and decision process of the autoregressive solver, serving as a graph structure mask that facilitates the formation of solutions characterized by both global vision and local feature integration. To the best of our knowledge, this work represents the first comprehensive experimental investigation of neural network model solvers across a 378-combinatorial space spanning four distinct dimensions within the CVRPlib public dataset. Extensive experimental evaluations demonstrate that our proposed fusion model effectively captures and leverages problem constraints, achieving state-of-the-art performance across multiple benchmark datasets.

Most neural methods for Vehicle Routing Problems (VRPs) are limited to Euclidean settings or simple graphs. In this work, we instead consider multigraphs, where parallel edges represent distinct travel options with varying trade-offs (e.g., distance vs. time). Multigraphs are highly relevant in practice, yet few neural methods are designed for them, and those that do exist face major scalability issues. We address these scalability issues with Node-Edge Policy Factorization (NEPF), which splits the routing policy into a node permutation stage and an edge selection stage. To enable the decomposition, we introduce a pre-encoding edge aggregation scheme and a non-autoregressive architecture for the edge stage, as well as a hierarchical reinforcement learning method to train the stages jointly. Our experiments across six VRP variants demonstrate that NEPF trains and runs up to orders of magnitude faster than prior neural multigraph methods and scales to considerably larger instances, while matching or improving on their solution quality.