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

TL;DR

MMHCVRP is addressed by ECHO, which uses a dual-modality node encoder to capture local topology, a PFCA decoder to reduce myopic decisions, and a data augmentation strategy leveraging vehicle permutation invariance and node symmetry. The approach achieves state-of-the-art performance across multiple vehicle and node scales, demonstrates robust generalization to unseen distributions, and ablations validate the contribution of each component. It also delivers significant speedups over prior MMHCVRP solvers, making it more suitable for time-sensitive deployment. The results highlight the practical value of incorporating topology-aware encoding and symmetry-aware training in neural combinatorial optimization.

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.

Figures (7)

• Figure 1: Illustration on the decoding processes of 2D-Ptr 2dptr (a) and the proposed ECHO (b). 2D-Ptr myopically relies on cumulative travel time to select the vehicle-node pair at each time step, which may lead to suboptimal solutions. Whereas the proposed ECHO alleviates this problem by emphasizing the vehicle selected in the preceding time step.
• Figure 2: Illustration of solution process for an MMHCVRP instance with three vehicles and eight nodes (seven customer nodes and one depot node). At each time step, the policy network selects a vehicle and a node as the action, repeating until all customer nodes have been served and all vehicles have returned to the depot.
• Figure 3: Overview of the proposed ECHO solver. To capture local topological information, ECHO exploits the proposed dual-modality node encoder to fuse node and edge features. Subsequently, the vehicle encoder integrates the node embedding with vehicle-specific attributes to produce the vehicle embedding. Finally, to highlight the importance of the vehicle selected at the $t-1$th time step, ECHO employs a decoder incorporating the proposed PFCA mechanism to efficiently select vehicle-node pairs.
• Figure 4: Illustration of the proposed augment method.
• Figure 5: Generalization performance of NCO solvers across different vehicle scales.