The application of learning based methods to vehicle routing problems has emerged as a pivotal area of research in combinatorial optimization. These problems are characterized by vast solution spaces and intricate constraints, making traditional approaches such as exact mathematical models or heuristic methods prone to high computational overhead or reliant on the design of complex heuristic operators to achieve optimal or near optimal solutions. Meanwhile, although some recent learning-based methods can produce good performance for VRP with straightforward constraint scenarios, they often fail to effectively handle hard constraints that are common in practice. This study introduces a novel end-to-end framework that combines constraint-oriented hypergraphs with reinforcement learning to address vehicle routing problems. A central innovation of this work is the development of a constraint-oriented dynamic hyperedge reconstruction strategy within an encoder, which significantly enhances hypergraph representation learning. Additionally, the decoder leverages a double-pointer attention mechanism to iteratively generate solutions. The proposed model is trained by incorporating asynchronous parameter updates informed by hypergraph constraints and optimizing a dual loss function comprising constraint loss and policy gradient loss. The experiment results on benchmark datasets demonstrate that the proposed approach not only eliminates the need for sophisticated heuristic operators but also achieves substantial improvements in solution quality.
The Capacitated Vehicle Routing Problem (CVRP) is a fundamental NP-hard problem with broad applications in logistics and transportation. Real-world CVRPs often involve diverse objectives and complex constraints, such as time windows or backhaul requirements, motivating the development of a unified solution framework. Recent reinforcement learning (RL) approaches have shown promise in combinatorial optimization, yet they rely on end-to-end learning and lack explicit problem-solving knowledge, limiting solution quality. In this paper, we propose a knowledge-embedded framework inspired by the Route-First Cluster-Second heuristics. It incorporates knowledge at two levels: (1) decomposing CVRPs into the route-first and cluster-second subproblems, and (2) leveraging dynamic programming to solve the second subproblem, whose results guide the RL-based constructive solver to solve the first problem. To mitigate partial observability caused by problem decomposition, we introduce a unified history-enhanced context processing module. Extensive experiments show that this framework achieves superior solution quality compared with state-of-the-art learning-based methods, with a smaller gap to classical heuristics, demonstrating strong generalization across diverse CVRP variants.
Vehicle Routing Problems (VRPs) are fundamental combinatorial optimization problems with widespread applications in various scenarios. The advanced optimization solvers can effectively solve such problems. However, modeling complex VRP variants for solvers often requires substantial domain expertise, which limits the accessibility of advanced optimization technologies. In this paper, we propose Reinforcement Learning Enhanced LLMAgents(RLEA), a multi-agent framework designed to automate the modeling of complex VRPs. RLEA introduces a lightweight neural Planner trained with Soft Q-learning to efficiently orchestrate the actions of LLM-based agents. In addition, we equip the system with an evolutionary memory module and retrieval-augmented generation, enabling the agent to leverage both accumulated experience and external solver knowledge during program generation and refinement for solving VRPs. We evaluated 48 distinct VRP variants across various solvers. The experimental results demonstrate that RLEA outperforms the previous state-of-the-ar method, achieving a 16.67% higher success rate while significantly reducing runtime errors. These results validate that integrating reinforcement learning with LLM-based reasoning is highly effective for automated optimization modeling. The appendix is available at: https://doi.org/10.5281/zenodo.19134435.
Multi-task vehicle routing problems play a critical role in enhancing efficiency across various industries and service sectors. These problems consist of multiple variants that optimize routing costs while meeting diverse customer constraints. Existing multi-task VRP solvers solely utilize a graph-based modality, limiting their ability to address variants with multiple constraints. As a format to represent complex semantics, vision modality shows great potential for encoding diverse VRP constraints. This motivates us to learn patch-level semantics from the vision images, and then integrate them into a graph-based model to solve various VRP variants simultaneously. However, directly applying this approach to multi-task VRPs presents three challenges: 1) existing VRP images lack constraint representations, which are essential for multi-task VRPs, 2) the fixed receptive field of individual patches cannot effectively accommodate varying requirements across tasks, and 3) imbalanced pixel distribution among constraints may cause the model to overlook constraints with fewer pixels. In this paper, we propose a vision-assisted foundation model (VaFM) to address these challenges. In the vision modality, input images tailored to all constraints are encoded by a convolutional neural network. The obtained patch embeddings are fused with graph-based nodes to generate solutions, with an auxiliary task designed to address the pixel-imbalanced issue. The performance of VaFM is evaluated across 16 different VRP variants. The experimental results demonstrate the superiority of VaFM over state-of-the-art methods, especially for variants with complex constraints.