Efficient Neural Combinatorial Optimization Solver for the Min-max Heterogeneous Capacitated Vehicle Routing Problem

2025-07-28

AI Summary

This paper addresses the Min-Max Heterogeneous Capacitated Vehicle Routing Problem (MMHCVRP)—a practical NP-hard combinatorial optimization problem featuring heterogeneous fleets, capacity constraints, and the objective of minimizing the longest route length. To overcome limitations of existing neural solvers—such as myopic decoding, neglect of local topology, permutation invariance across vehicles, and node symmetry—we propose ECHO, an end-to-end neural combinatorial optimization solver. ECHO introduces a dual-modal node encoding to explicitly capture local structural information, employs a parameter-free cross-attention mechanism to mitigate myopia in sequential decision-making, and designs permutation- and symmetry-aware data augmentation to stabilize reinforcement learning training. Extensive experiments demonstrate that ECHO significantly outperforms state-of-the-art methods across diverse problem scales and demand distributions, exhibits strong generalization, and ablation studies confirm the effectiveness of each component.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Planning, Routing, and Scheduling: Planning with Language Models
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Numerous Neural Combinatorial Optimization (NCO) solvers have been proposed to address Vehicle Routing Problems (VRPs). However, most of these solvers focus exclusively on single-vehicle VRP variants, overlooking the more realistic min-max Heterogeneous Capacitated Vehicle Routing Problem (MMHCVRP), which involves multiple vehicles. Existing MMHCVRP solvers typically select a vehicle and its next node to visit at each decoding step, but often make myopic decoding decisions and overlook key properties of MMHCVRP, including local topological relationships, vehicle permutation invariance, and node symmetry, resulting in suboptimal performance. To better address these limitations, we propose ECHO, an efficient NCO solver. First, ECHO exploits the proposed dual-modality node encoder to capture local topological relationships among nodes. Subsequently, to mitigate myopic decisions, ECHO employs the proposed Parameter-Free Cross-Attention mechanism to prioritize the vehicle selected in the preceding decoding step. Finally, leveraging vehicle permutation invariance and node symmetry, we introduce a tailored data augment strategy for MMHCVRP to stabilize the Reinforcement Learning training process. To assess the performance of ECHO, we conduct extensive experiments. The experimental results demonstrate that ECHO outperforms state-of-the-art NCO solvers across varying numbers of vehicles and nodes, and exhibits well-performing generalization across both scales and distribution patterns. Finally, ablation studies validate the effectiveness of all proposed methods.

Problem

• Addresses min-max Heterogeneous Capacitated Vehicle Routing Problem (MMHCVRP)
• Overcomes myopic decisions and overlooked MMHCVRP key properties
• Improves performance and generalization in multi-vehicle routing scenarios

Innovation

• Dual-modality node encoder captures local topology
• Parameter-Free Cross-Attention reduces myopic decisions
• Tailored data augment stabilizes RL training