Multi-Task Learning (MTL) in Neural Combinatorial Optimization (NCO) is a promising approach to train a unified model capable of solving multiple Vehicle Routing Problem (VRP) variants. However, existing Reinforcement Learning (RL)-based multi-task methods can only train light decoder models on small-scale problems, exhibiting limited generalization ability when solving large-scale problems. To overcome this limitation, this work introduces a novel multi-task learning method driven by knowledge distillation (MTL-KD), which enables the efficient training of heavy decoder models with strong generalization ability. The proposed MTL-KD method transfers policy knowledge from multiple distinct RL-based single-task models to a single heavy decoder model, facilitating label-free training and effectively improving the model's generalization ability across diverse tasks. In addition, we introduce a flexible inference strategy termed Random Reordering Re-Construction (R3C), which is specifically adapted for diverse VRP tasks and further boosts the performance of the multi-task model. Experimental results on 6 seen and 10 unseen VRP variants with up to 1000 nodes indicate that our proposed method consistently achieves superior performance on both uniform and real-world benchmarks, demonstrating robust generalization abilities.

Existing neural methods for multi-task vehicle routing problems (VRPs) typically learn unified solvers to handle multiple constraints simultaneously. However, they often underutilize the compositional structure of VRP variants, each derivable from a common set of basis VRP variants. This critical oversight causes unified solvers to miss out the potential benefits of basis solvers, each specialized for a basis VRP variant. To overcome this limitation, we propose a framework that enables unified solvers to perceive the shared-component nature across VRP variants by proactively reusing basis solvers, while mitigating the exponential growth of trained neural solvers. Specifically, we introduce a State-Decomposable MDP (SDMDP) that reformulates VRPs by expressing the state space as the Cartesian product of basis state spaces associated with basis VRP variants. More crucially, this formulation inherently yields the optimal basis policy for each basis VRP variant. Furthermore, a Latent Space-based SDMDP extension is developed by incorporating both the optimal basis policies and a learnable mixture function to enable the policy reuse in the latent space. Under mild assumptions, this extension provably recovers the optimal unified policy of SDMDP through the mixture function that computes the state embedding as a mapping from the basis state embeddings generated by optimal basis policies. For practical implementation, we introduce the Mixture-of-Specialized-Experts Solver (MoSES), which realizes basis policies through specialized Low-Rank Adaptation (LoRA) experts, and implements the mixture function via an adaptive gating mechanism. Extensive experiments conducted across VRP variants showcase the superiority of MoSES over prior methods.

Multi-task neural routing solvers have emerged as a promising paradigm for their ability to solve multiple vehicle routing problems (VRPs) using a single model. However, existing neural solvers typically rely on predefined problem constraints or require per-problem fine-tuning, which substantially limits their zero-shot generalization ability to unseen VRP variants. To address this critical bottleneck, we propose URS, a unified neural routing solver capable of zero-shot generalization across a wide range of unseen VRPs using a single model without any fine-tuning. The key component of URS is the unified data representation (UDR), which replaces problem enumeration with data unification, thereby broadening the problem coverage and reducing reliance on domain expertise. In addition, we propose a Mixed Bias Module (MBM) to efficiently learn the geometric and relational biases inherent in various problems. On top of the proposed UDR, we further develop a parameter generator that adaptively adjusts the decoder and bias weights of MBM to enhance zero-shot generalization. Moreover, we propose an LLM-driven constraint satisfaction mechanism, which translates raw problem descriptions into executable stepwise masking functions to ensure solution feasibility. Extensive experiments demonstrate that URS can consistently produce high-quality solutions for more than 100 distinct VRP variants without any fine-tuning, which includes more than 90 unseen variants. To the best of our knowledge, URS is the first neural solver capable of handling over 100 VRP variants with a single model.

In real-world routing problems, users often propose conflicting or unreasonable requirements, which result in infeasible optimization models due to overly restrictive or contradictory constraints, leading to an empty feasible solution set. Existing Large Language Model (LLM)-based methods attempt to diagnose infeasible models, but modifying such models often involves multiple potential adjustments that these methods do not consider. To fill this gap, we introduce Multi-Objective Infeasibility Diagnosis (MOID), which combines LLM agents and multi-objective optimization within an automatic routing solver, to provide a set of representative actionable suggestions. Specifically, MOID employs multi-objective optimization to consider both path cost and constraint violation, generating a set of trade-off solutions, each encompassing varying degrees of model adjustments. To extract practical insights from these solutions, MOID utilizes LLM agents to generate a solution analysis function for the infeasible model. This function analyzes these distinct solutions to diagnose the original infeasible model, providing users with diverse diagnostic insights and suggestions. Finally, we compare MOID with several LLM-based methods on 50 types of infeasible routing problems. The results indicate that MOID automatically generates multiple diagnostic suggestions in a single run, providing more practical insights for restoring model feasibility and decision-making compared to existing methods.

Vehicle routing problems (VRPs) are central to combinatorial optimization with significant practical implications. Recent advancements in neural combinatorial optimization (NCO) have demonstrated promising results by leveraging neural networks to solve VRPs, yet the exploration of model scaling within this domain remains underexplored. Inspired by the success of model scaling in large language models (LLMs), this study introduces a Large Routing Model with 1 billion parameters (LRM-1B), designed to address diverse VRP scenarios. We present a comprehensive evaluation of LRM-1B across multiple problem variants, distributions, and sizes, establishing state-of-the-art results. Our findings reveal that LRM-1B not only adapts to different VRP challenges but also showcases superior performance, outperforming existing models. Additionally, we explore the scaling behavior of neural routing models from 1M to 1B parameters. Our analysis confirms power-law between multiple model factors and performance, offering critical insights into the optimal configurations for foundation neural routing solvers.

This paper introduces TuneNSearch, a hybrid transfer learning and local search approach for addressing different variants of vehicle routing problems (VRP). Recently, multi-task learning has gained much attention for solving VRP variants. However, this adaptability often compromises the performance of the models. To address this challenge, we first pre-train a reinforcement learning model on the multi-depot VRP, followed by a short fine-tuning phase to adapt it to different variants. By leveraging the complexity of the multi-depot VRP, the pre-trained model learns richer node representations and gains more transferable knowledge compared to models trained on simpler routing problems, such as the traveling salesman problem. TuneNSearch employs, in the first stage, a Transformer-based architecture, augmented with a residual edge-graph attention network to capture the impact of edge distances and residual connections between layers. This architecture allows for a more precise capture of graph-structured data, improving the encoding of VRP's features. After inference, our model is also coupled with a second stage composed of a local search algorithm, which yields substantial performance gains with minimal computational overhead added. Results show that TuneNSearch outperforms many existing state-of-the-art models trained for each VRP variant, requiring only one-fifth of the training epochs. Our approach demonstrates strong generalization, achieving high performance across different tasks, distributions and problem sizes, thus addressing a long-standing gap in the literature.

Vehicle Routing Problems (VRPs) are significant Combinatorial Optimization (CO) problems holding substantial practical importance. Recently, Neural Combinatorial Optimization (NCO), which involves training deep learning models on extensive data to learn vehicle routing heuristics, has emerged as a promising approach due to its efficiency and the reduced need for manual algorithm design. However, applying NCO across diverse real-world scenarios with various constraints necessitates cross-problem capabilities. Current NCO methods typically employ a unified model lacking a constraint-specific structure, thereby restricting their cross-problem performance. Current multi-task methods for VRPs typically employ a constraint-unaware model, limiting their cross-problem performance. Furthermore, they rely solely on global connectivity, which fails to focus on key nodes and leads to inefficient representation learning. This paper introduces a Constraint-Aware Dual-Attention Model (CaDA), designed to address these limitations. CaDA incorporates a constraint prompt that efficiently represents different problem variants. Additionally, it features a dual-attention mechanism with a global branch for capturing broader graph-wide information and a sparse branch that selectively focuses on the most relevant nodes. We comprehensively evaluate our model on 16 different VRPs and compare its performance against existing cross-problem VRP solvers. CaDA achieves state-of-the-art results across all the VRPs. Our ablation study further confirms that each component of CaDA contributes positively to its cross-problem learning performance.

Vehicle Routing Problems (VRPs) are optimization problems with significant real-world implications in logistics, transportation, and supply chain management. Despite the recent progress made in learning to solve individual VRP variants, there is a lack of a unified approach that can effectively tackle a wide range of tasks, which is crucial for real-world impact. This paper introduces RouteFinder, a framework for developing foundation models for VRPs. Our key idea is that a foundation model for VRPs should be able to model variants by treating each variant as a subset of a larger VRP problem, equipped with different attributes. We introduce a parallelized environment that can handle any combination of attributes at the same time in a batched manner, and an efficient sampling procedure to train on a mix of problems at each optimization step that can greatly improve convergence robustness. We also introduce novel Global Feature Embeddings that project instance-wise attributes efficiently onto the latent space and help the model understand different VRP variants. Finally, we introduce Efficient Adapter Layers, a simple yet effective technique to finetune pre-trained RouteFinder models to solve novel variants with previously unseen attributes outside of the original feature space.

Learning to solve vehicle routing problems (VRPs) has garnered much attention. However, most neural solvers are only structured and trained independently on a specific problem, making them less generic and practical. In this paper, we aim to develop a unified neural solver that can cope with a range of VRP variants simultaneously. Specifically, we propose a multi-task vehicle routing solver with mixture-of-experts (MVMoE), which greatly enhances the model capacity without a proportional increase in computation. We further develop a hierarchical gating mechanism for the MVMoE, delivering a good trade-off between empirical performance and computational complexity. Experimentally, our method significantly promotes zero-shot generalization performance on 10 unseen VRP variants, and showcases decent results on the few-shot setting and real-world benchmark instances. We further conduct extensive studies on the effect of MoE configurations in solving VRPs, and observe the superiority of hierarchical gating when facing out-of-distribution data. The source code is available at: https://github.com/RoyalSkye/Routing-MVMoE.

We introduce PyVRP, a Python package that implements hybrid genetic search in a state-of-the-art vehicle routing problem (VRP) solver. The package is designed for the VRP with time windows (VRPTW), but can be easily extended to support other VRP variants. PyVRP combines the flexibility of Python with the performance of C++, by implementing (only) performance critical parts of the algorithm in C++, while being fully customisable at the Python level. PyVRP is a polished implementation of the algorithm that ranked 1st in the 2021 DIMACS VRPTW challenge and, after improvements, ranked 1st on the static variant of the EURO meets NeurIPS 2022 vehicle routing competition. The code follows good software engineering practices, and is well-documented and unit tested. PyVRP is freely available under the liberal MIT license. Through numerical experiments we show that PyVRP achieves state-of-the-art results on the VRPTW and capacitated VRP. We hope that PyVRP enables researchers and practitioners to easily and quickly build on a state-of-the-art VRP solver.