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

arXiv: https://arxiv.org/abs/2405.17272 · HTML: https://mlsanity.com/paper/2405.17272/

## Key facts

- Cited by 18 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 63 references that are in the map.
- First posted on arXiv 27 May 2024.

## Builds on (most cited first, 23 of 23)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Attention, Learn to Solve Routing Problems!](https://mlsanity.com/paper/1803.08475/) (2018)
- [Neural Combinatorial Optimization with Reinforcement Learning](https://mlsanity.com/paper/1611.09940/) (2016)
- [POMO: Policy Optimization with Multiple Optima for Reinforcement Learning](https://mlsanity.com/paper/2010.16011/) (2020)
- [Reinforcement Learning for Solving the Vehicle Routing Problem](https://mlsanity.com/paper/1802.04240/) (2018)
- [ReZero is All You Need: Fast Convergence at Large Depth](https://mlsanity.com/paper/2003.04887/) (2020)
- [Multi-Agent Reinforcement Learning is a Sequence Modeling Problem](https://mlsanity.com/paper/2205.14953/) (2022)
- [Neural Combinatorial Optimization with Heavy Decoder: Toward Large Scale Generalization](https://mlsanity.com/paper/2310.07985/) (2023)
- [Sym-NCO: Leveraging Symmetricity for Neural Combinatorial Optimization](https://mlsanity.com/paper/2205.13209/) (2022)
- [Learning to Iteratively Solve Routing Problems with Dual-Aspect Collaborative Transformer](https://mlsanity.com/paper/2110.02544/) (2021)
- [Matrix Encoding Networks for Neural Combinatorial Optimization](https://mlsanity.com/paper/2106.11113/) (2021)
- [BQ-NCO: Bisimulation Quotienting for Efficient Neural Combinatorial Optimization](https://mlsanity.com/paper/2301.03313/) (2023)
- [Towards Omni-generalizable Neural Methods for Vehicle Routing Problems](https://mlsanity.com/paper/2305.19587/) (2023)
- [Towards Generalizable Neural Solvers for Vehicle Routing Problems via Ensemble with Transferrable Local Policy](https://mlsanity.com/paper/2308.14104/) (2023)
- [H-TSP: Hierarchically Solving the Large-Scale Travelling Salesman Problem](https://mlsanity.com/paper/2304.09395/) (2023)
- [ScheduleNet: Learn to solve multi-agent scheduling problems with reinforcement learning](https://mlsanity.com/paper/2106.03051/) (2021)
- [Learning the Multiple Traveling Salesmen Problem with Permutation Invariant Pooling Networks](https://mlsanity.com/paper/1803.09621/) (2018)
- [Equity-Transformer: Solving NP-hard Min-Max Routing Problems as Sequential Generation with Equity Context](https://mlsanity.com/paper/2306.02689/) (2023)
- [Distance-aware Attention Reshaping: Enhance Generalization of Neural Solver for Large-scale Vehicle Routing Problems](https://mlsanity.com/paper/2401.06979/) (2024)
- [DAN: Decentralized Attention-based Neural Network for the MinMax Multiple Traveling Salesman Problem](https://mlsanity.com/paper/2109.04205/) (2021)
- [Towards Global Optimality in Cooperative MARL with the Transformation And Distillation Framework](https://mlsanity.com/paper/2207.11143/) (2022)
- [The Multi-Agent Pickup and Delivery Problem: MAPF, MARL and Its Warehouse Applications](https://mlsanity.com/paper/2203.07092/) (2022)

## Built on by (most cited first, 18 of 18)

- [Monte Carlo Tree Search for Comprehensive Exploration in LLM-Based Automatic Heuristic Design](https://mlsanity.com/paper/2501.08603/) (2025)
- [Neural Combinatorial Optimization Algorithms for Solving Vehicle Routing Problems: A Comprehensive Survey with Perspectives](https://mlsanity.com/paper/2406.00415/) (2024)
- [MTL-KD: Multi-Task Learning Via Knowledge Distillation for Generalizable Neural Vehicle Routing Solver](https://mlsanity.com/paper/2506.02935/) (2025)
- [Improving Generalization of Neural Combinatorial Optimization for Vehicle Routing Problems via Test-Time Projection Learning](https://mlsanity.com/paper/2506.02392/) (2025)
- [Learning to Insert for Constructive Neural Vehicle Routing Solver](https://mlsanity.com/paper/2505.13904/) (2025)
- [Rethinking Neural Combinatorial Optimization for Vehicle Routing Problems with Different Constraint Tightness Degrees](https://mlsanity.com/paper/2505.24627/) (2025)
- [URS: A Unified Neural Routing Solver for Cross-Problem Zero-Shot Generalization](https://mlsanity.com/paper/2509.23413/) (2025)
- [PARCO: Parallel AutoRegressive Models for Multi-Agent Combinatorial Optimization](https://mlsanity.com/paper/2409.03811/) (2024)
- [Survey on Neural Routing Solvers](https://mlsanity.com/paper/2602.21761/) (2026)
- [Efficient Neural Combinatorial Optimization Solver for the Min-max Heterogeneous Capacitated Vehicle Routing Problem](https://mlsanity.com/paper/2507.21386/) (2025)
- [Learning to Solve the Min-Max Mixed-Shelves Picker-Routing Problem via Hierarchical and Parallel Decoding](https://mlsanity.com/paper/2502.10233/) (2025)
- [SPACE: Unifying Symmetric and Asymmetric Routing Problems for Generalist Neural Solver](https://mlsanity.com/paper/2605.24484/) (2026)
- [USPR: Learning a Unified Solver for Profiled Routing](https://mlsanity.com/paper/2505.05119/) (2025)
- [Efficient Decoder Scaling Strategy for Neural Routing Solvers](https://mlsanity.com/paper/2603.00430/) (2026)
- [Rethinking Constraint Awareness for Efficient State Embedding of Neural Routing Solver](https://mlsanity.com/paper/2605.10122/) (2026)
- [Onboard Mission Replanning for Adaptive Cooperative Multi-Robot Systems](https://mlsanity.com/paper/2506.06094/) (2025)
- [Multi-Action Self-Improvement for Neural Combinatorial Optimization](https://mlsanity.com/paper/2510.12273/) (2025)
- [A Unified Knowledge Embedded Reinforcement Learning-based Framework for Generalized Capacitated Vehicle Routing Problems](https://mlsanity.com/paper/2605.14416/) (2026)

## Abstract (arXiv)

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