DPN: Decoupling Partition and Navigation for Neural Solvers of Min-max Vehicle Routing Problems

TL;DR

This work tackles the min-max VRP objective of minimizing the longest route length by proposing Decoupling-Partition-Navigation (DPN), which separates partition and navigation through a bi-part Partition-and-Navigation encoder (P&N Encoder). It introduces an agent-permutation-symmetric (APS) loss to exploit permutation symmetry and a depot-aware Rotation-based positional encoding to improve partition representations. Empirical results on four min-max VRP variants, including single- and multi-depot settings, show that DPN consistently outperforms existing learning-based solvers and competes with traditional heuristics, with notable gains in mTSP and mPDP scenarios. The approach advances neural solvers for structured combinatorial problems by embedding problem-specific decoupling and symmetry into the learning process, and it opens avenues for extending decoupled representations to general VRPs in the future.

Abstract

The min-max vehicle routing problem (min-max VRP) traverses all given customers by assigning several routes and aims to minimize the length of the longest route. Recently, reinforcement learning (RL)-based sequential planning methods have exhibited advantages in solving efficiency and optimality. However, these methods fail to exploit the problem-specific properties in learning representations, resulting in less effective features for decoding optimal routes. This paper considers the sequential planning process of min-max VRPs as two coupled optimization tasks: customer partition for different routes and customer navigation in each route (i.e., partition and navigation). To effectively process min-max VRP instances, we present a novel attention-based Partition-and-Navigation encoder (P&N Encoder) that learns distinct embeddings for partition and navigation. Furthermore, we utilize an inherent symmetry in decoding routes and develop an effective agent-permutation-symmetric (APS) loss function. Experimental results demonstrate that the proposed Decoupling-Partition-Navigation (DPN) method significantly surpasses existing learning-based methods in both single-depot and multi-depot min-max VRPs. Our code is available at

Figures (20)

• Figure 1: (a) The instances and solutions of four involved min-max VRPs. (b) The sequential planning framework proposed in son2023solving. (c) The sequential planning framework of the proposed DPN. To decouple the partition and navigation features and improve the representation ability, we conduct modifications to the existing framework by proposing a novel P&N Encoder, utilizing agent-permutation-symmetries (APS) in loss calculation, and introducing a Roataion-based positional encoding (Rotation-based PE) for agent representations.
• Figure 2: Initial embeddings and the proposed P&N Encoder.
• Figure 3: Training curves for ablation study ($M$=5).
• Figure 4: An min-max mTSP example of constructive-based planning methods for neural solvers of min-max VRP.
• Figure 5: Attention score of min-max mTSP50 ($M$=10), each data is the average score of 8 heads and 3 attention layers.