Hierarchical Learning-based Graph Partition for Large-scale Vehicle Routing Problems

TL;DR

The paper tackles scaling VRP/CVRP by introducing HLGP, a hierarchical partitioning framework that couples a global partition policy with level-specific local partition policies to progressively alleviate compounded misclusterings. It formalizes HLGP with both RL and SL training paradigms, providing a multi-level MDP formulation for RL and a self-imitation learning approach with curriculum-based labeling for SL. Empirical results across diverse CVRP benchmarks show HLGP improves generalization under distribution and scale shifts, achieving competitive or superior costs to state-of-the-art methods and scalability to CVRP10K within minutes per instance. The framework’s ability to treat partition subproblems as independent training instances, along with its theoretical underpinnings, suggests strong practical impact for large-scale COPs and potential extensions to other VRP variants.

Abstract

Neural solvers based on the divide-and-conquer approach for Vehicle Routing Problems (VRPs) in general, and capacitated VRP (CVRP) in particular, integrates the global partition of an instance with local constructions for each subproblem to enhance generalization. However, during the global partition phase, misclusterings within subgraphs have a tendency to progressively compound throughout the multi-step decoding process of the learning-based partition policy. This suboptimal behavior in the global partition phase, in turn, may lead to a dramatic deterioration in the performance of the overall decomposition-based system, despite using optimal local constructions. To address these challenges, we propose a versatile Hierarchical Learning-based Graph Partition (HLGP) framework, which is tailored to benefit the partition of CVRP instances by synergistically integrating global and local partition policies. Specifically, the global partition policy is tasked with creating the coarse multi-way partition to generate the sequence of simpler two-way partition subtasks. These subtasks mark the initiation of the subsequent K local partition levels. At each local partition level, subtasks exclusive for this level are assigned to the local partition policy which benefits from the insensitive local topological features to incrementally alleviate the compounded errors. This framework is versatile in the sense that it optimizes the involved partition policies towards a unified objective harmoniously compatible with both reinforcement learning (RL) and supervised learning (SL). (*Due to the notification of arXiv "The Abstract field cannot be longer than 1,920 characters", the appeared Abstract is shortened. For the full Abstract, please download the Article.)

Figures (8)

• Figure 1: The proposed HLGP framework. $I_{j\geq1}^{k}$ represents a sequence of subproblems. Following the HLGP framework, the sequence of subproblems $I_{j\geq1}^{K}$ are fed to a permutation policy to derive the respective subtours.
• Figure 2: RL-driven HLGP replaces the initially generated partial partition solution with the complete partition solution of subproblems within $\mathcal{C}^{(0)}$ at level 0. SL-driven HLGP requires labeled instances for training $\pi_{\theta_{G}}$ and $\pi_{\theta_{L}}$.
• Figure 3: The training curve (a) and the validation curve (b) of the global partition policy in RL-driven HLGP.
• Figure 4: The toy example of the overall HLGP framework.
• Figure 5: The toy example of the RL-driven HLGP training framework.

Theorems & Definitions (9)

• Theorem 1
• Definition 1
• Theorem 2
• Proposition 1
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6
• Proposition 2