# An Overview and Experimental Study of Learning-based Optimization Algorithms for Vehicle Routing Problem

arXiv: https://arxiv.org/abs/2107.07076 · HTML: https://mlsanity.com/paper/2107.07076/

## Key facts

- Cited by 22 papers in the map (data through 14 Oct 2026).
- Builds on 56 of its 203 references that are in the map.
- First posted on arXiv 15 Jul 2021.

## Builds on (most cited first, 30 of 56)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Neural Machine Translation by Jointly Learning to Align and Translate](https://mlsanity.com/paper/1409.0473/) (2014)
- [Sequence to Sequence Learning with Neural Networks](https://mlsanity.com/paper/1409.3215/) (2014)
- [Show, Attend and Tell: Neural Image Caption Generation with Visual Attention](https://mlsanity.com/paper/1502.03044/) (2015)
- [Effective Approaches to Attention-based Neural Machine Translation](https://mlsanity.com/paper/1508.04025/) (2015)
- [Deep Visual-Semantic Alignments for Generating Image Descriptions](https://mlsanity.com/paper/1412.2306/) (2014)
- [Pointer Networks](https://mlsanity.com/paper/1506.03134/) (2015)
- [End-to-End Training of Deep Visuomotor Policies](https://mlsanity.com/paper/1504.00702/) (2015)
- [Deep Learning in Neural Networks: An Overview](https://mlsanity.com/paper/1404.7828/) (2014)
- [Learning Dexterous In-Hand Manipulation](https://mlsanity.com/paper/1808.00177/) (2018)
- [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)
- [Learning Combinatorial Optimization Algorithms over Graphs](https://mlsanity.com/paper/1704.01665/) (2017)
- [A Brief Survey of Deep Reinforcement Learning](https://mlsanity.com/paper/1708.05866/) (2017)
- [Reinforcement Learning for Solving the Vehicle Routing Problem](https://mlsanity.com/paper/1802.04240/) (2018)
- [An Efficient Graph Convolutional Network Technique for the Travelling Salesman Problem](https://mlsanity.com/paper/1906.01227/) (2019)
- [Combinatorial optimization and reasoning with graph neural networks](https://mlsanity.com/paper/2102.09544/) (2021)
- [A Review of Relational Machine Learning for Knowledge Graphs](https://mlsanity.com/paper/1503.00759/) (2015)
- [Differentiation of Blackbox Combinatorial Solvers](https://mlsanity.com/paper/1912.02175/) (2019)
- [A Critical Review of Recurrent Neural Networks for Sequence Learning](https://mlsanity.com/paper/1506.00019/) (2015)
- [Learning to Perform Local Rewriting for Combinatorial Optimization](https://mlsanity.com/paper/1810.00337/) (2018)
- [Learning Improvement Heuristics for Solving Routing Problems](https://mlsanity.com/paper/1912.05784/) (2019)
- [Generalize a Small Pre-trained Model to Arbitrarily Large TSP Instances](https://mlsanity.com/paper/2012.10658/) (2020)
- [Discriminative Embeddings of Latent Variable Models for Structured Data](https://mlsanity.com/paper/1603.05629/) (2016)
- [Neural Large Neighborhood Search for the Capacitated Vehicle Routing Problem](https://mlsanity.com/paper/1911.09539/) (2019)
- [Learning 2-opt Heuristics for the Traveling Salesman Problem via Deep Reinforcement Learning](https://mlsanity.com/paper/2004.01608/) (2020)
- [Multi-Decoder Attention Model with Embedding Glimpse for Solving Vehicle Routing Problems](https://mlsanity.com/paper/2012.10638/) (2020)
- [Deep Reinforcement Learning for Multi-objective Optimization](https://mlsanity.com/paper/1906.02386/) (2019)
- [On Using Very Large Target Vocabulary for Neural Machine Translation](https://mlsanity.com/paper/1412.2007/) (2014)
- [Combinatorial Optimization by Graph Pointer Networks and Hierarchical Reinforcement Learning](https://mlsanity.com/paper/1911.04936/) (2019)

## Built on by (most cited first, 22 of 22)

- [Neural Combinatorial Optimization with Heavy Decoder: Toward Large Scale Generalization](https://mlsanity.com/paper/2310.07985/) (2023)
- [Backpropagation through Combinatorial Algorithms: Identity with Projection Works](https://mlsanity.com/paper/2205.15213/) (2022)
- [Neural Combinatorial Optimization Algorithms for Solving Vehicle Routing Problems: A Comprehensive Survey with Perspectives](https://mlsanity.com/paper/2406.00415/) (2024)
- [Learning to Reduce Search Space for Generalizable Neural Routing Solver](https://mlsanity.com/paper/2503.03137/) (2025)
- [Instance-Conditioned Adaptation for Large-scale Generalization of Neural Routing Solver](https://mlsanity.com/paper/2405.01906/) (2024)
- [URS: A Unified Neural Routing Solver for Cross-Problem Zero-Shot Generalization](https://mlsanity.com/paper/2509.23413/) (2025)
- [A Survey on Learnable Evolutionary Algorithms for Scalable Multiobjective Optimization](https://mlsanity.com/paper/2206.11526/) (2022)
- [Learning for routing: A guided review of recent developments and future directions](https://mlsanity.com/paper/2507.00218/) (2025)
- [Prompt Learning for Generalized Vehicle Routing](https://mlsanity.com/paper/2405.12262/) (2024)
- [A Scalable Test Problem Generator for Sequential Transfer Optimization](https://mlsanity.com/paper/2304.08503/) (2023)
- [Survey on Neural Routing Solvers](https://mlsanity.com/paper/2602.21761/) (2026)
- [From Inverse Optimization to Feasibility to ERM](https://mlsanity.com/paper/2402.17890/) (2024)
- [Vision-Assisted Foundation Model for Solving Multi-Task Vehicle Routing Problems](https://mlsanity.com/paper/2606.10431/) (2026)
- [Multi-Agent Environments for Vehicle Routing Problems](https://mlsanity.com/paper/2411.14411/) (2024)
- [REMS: a unified solution representation, problem modeling and metaheuristic algorithm design for general combinatorial optimization problems](https://mlsanity.com/paper/2505.17108/) (2025)
- [Test-Time Search in Neural Graph Coarsening Procedures for the Capacitated Vehicle Routing Problem](https://mlsanity.com/paper/2510.00958/) (2025)
- [A Unified Deep Reinforcement Learning Approach for Close Enough Traveling Salesman Problem](https://mlsanity.com/paper/2510.03065/) (2025)
- [Hybrid Quantum Reinforcement Learning with QAOA for Improved Vehicle Routing Optimization](https://mlsanity.com/paper/2605.01574/) (2026)
- [Teacher-Aware Evolution of Heuristic Programs from Learned Optimization Policies](https://mlsanity.com/paper/2605.10634/) (2026)
- [RACL: Reasoning-Agent Control Layers for Continuous Metaheuristic Learning](https://mlsanity.com/paper/2606.20142/) (2026)
- [Adaptive Cluster-First Route-Second Decomposition for Industrial-Scale Vehicle Routing](https://mlsanity.com/paper/2606.31820/) (2026)
- [Robust Feasible Route Construction through Collaborative Partition Optimization](https://mlsanity.com/paper/2607.03694/) (2026)

## Abstract (arXiv)

Vehicle routing problem (VRP) is a typical discrete combinatorial optimization problem, and many models and algorithms have been proposed to solve the VRP and its variants. Although existing approaches have contributed a lot to the development of this field, these approaches either are limited in problem size or need manual intervening in choosing parameters. To solve these difficulties, many studies have considered the learning-based optimization (LBO) algorithms to solve the VRP. This paper reviews recent advances in this field and divides relevant approaches into end-to-end approaches and step-by-step approaches. We performed a statistical analysis of the reviewed articles from various aspects and designed three experiments to evaluate the performance of four representative LBO algorithms. Finally, we conclude the applicable types of problems for different LBO algorithms and suggest directions in which researchers can improve LBO algorithms.