# NeuroLKH: Combining Deep Learning Model with Lin-Kernighan-Helsgaun Heuristic for Solving the Traveling Salesman Problem

arXiv: https://arxiv.org/abs/2110.07983 · HTML: https://mlsanity.com/paper/2110.07983/

## Key facts

- Cited by 103 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 37 references that are in the map.
- First posted on arXiv 15 Oct 2021.

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

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift](https://mlsanity.com/paper/1502.03167/) (2015)
- [Pointer Networks](https://mlsanity.com/paper/1506.03134/) (2015)
- [Attention, Learn to Solve Routing Problems!](https://mlsanity.com/paper/1803.08475/) (2018)
- [Machine Learning for Combinatorial Optimization: a Methodological Tour d'Horizon](https://mlsanity.com/paper/1811.06128/) (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)
- [POMO: Policy Optimization with Multiple Optima for Reinforcement Learning](https://mlsanity.com/paper/2010.16011/) (2020)
- [Residual Gated Graph ConvNets](https://mlsanity.com/paper/1711.07553/) (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)
- [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)
- [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 Policy Dynamic Programming for Vehicle Routing Problems](https://mlsanity.com/paper/2102.11756/) (2021)
- [Heterogeneous Attentions for Solving Pickup and Delivery Problem via Deep Reinforcement Learning](https://mlsanity.com/paper/2110.02634/) (2021)
- [Combining Reinforcement Learning with Lin-Kernighan-Helsgaun Algorithm for the Traveling Salesman Problem](https://mlsanity.com/paper/2012.04461/) (2020)
- [Learning to Solve Vehicle Routing Problems with Time Windows through Joint Attention](https://mlsanity.com/paper/2006.09100/) (2020)
- [Generalization of Machine Learning for Problem Reduction: A Case Study on Travelling Salesman Problems](https://mlsanity.com/paper/2005.05847/) (2020)

## Built on by (most cited first, 30 of 103)

- [DIFUSCO: Graph-based Diffusion Solvers for Combinatorial Optimization](https://mlsanity.com/paper/2302.08224/) (2023)
- [Neural Combinatorial Optimization with Heavy Decoder: Toward Large Scale Generalization](https://mlsanity.com/paper/2310.07985/) (2023)
- [DIMES: A Differentiable Meta Solver for Combinatorial Optimization Problems](https://mlsanity.com/paper/2210.04123/) (2022)
- [Simulation-guided Beam Search for Neural Combinatorial Optimization](https://mlsanity.com/paper/2207.06190/) (2022)
- [DeepACO: Neural-enhanced Ant Systems for Combinatorial Optimization](https://mlsanity.com/paper/2309.14032/) (2023)
- [RL4CO: an Extensive Reinforcement Learning for Combinatorial Optimization Benchmark](https://mlsanity.com/paper/2306.17100/) (2023)
- [MVMoE: Multi-Task Vehicle Routing Solver with Mixture-of-Experts](https://mlsanity.com/paper/2405.01029/) (2024)
- [Towards Omni-generalizable Neural Methods for Vehicle Routing Problems](https://mlsanity.com/paper/2305.19587/) (2023)
- [Learning Generalizable Models for Vehicle Routing Problems via Knowledge Distillation](https://mlsanity.com/paper/2210.07686/) (2022)
- [Pareto Set Learning for Neural Multi-objective Combinatorial Optimization](https://mlsanity.com/paper/2203.15386/) (2022)
- [Searching Large Neighborhoods for Integer Linear Programs with Contrastive Learning](https://mlsanity.com/paper/2302.01578/) (2023)
- [Rethinking Light Decoder-based Solvers for Vehicle Routing Problems](https://mlsanity.com/paper/2503.00753/) (2025)
- [Learning to Solve Routing Problems via Distributionally Robust Optimization](https://mlsanity.com/paper/2202.07241/) (2022)
- [Learning to Handle Complex Constraints for Vehicle Routing Problems](https://mlsanity.com/paper/2410.21066/) (2024)
- [Cross-Problem Learning for Solving Vehicle Routing Problems](https://mlsanity.com/paper/2404.11677/) (2024)
- [Meta-SAGE: Scale Meta-Learning Scheduled Adaptation with Guided Exploration for Mitigating Scale Shift on Combinatorial Optimization](https://mlsanity.com/paper/2306.02688/) (2023)
- [Neural Combinatorial Optimization Algorithms for Solving Vehicle Routing Problems: A Comprehensive Survey with Perspectives](https://mlsanity.com/paper/2406.00415/) (2024)
- [Deep Reinforcement Learning Guided Improvement Heuristic for Job Shop Scheduling](https://mlsanity.com/paper/2211.10936/) (2022)
- [Learning to Solve Vehicle Routing Problems: A Survey](https://mlsanity.com/paper/2205.02453/) (2022)
- [Imperative Learning: A Self-supervised Neuro-Symbolic Learning Framework for Robot Autonomy](https://mlsanity.com/paper/2406.16087/) (2024)
- [MTL-KD: Multi-Task Learning Via Knowledge Distillation for Generalizable Neural Vehicle Routing Solver](https://mlsanity.com/paper/2506.02935/) (2025)
- [Reinforced Lin-Kernighan-Helsgaun Algorithms for the Traveling Salesman Problems](https://mlsanity.com/paper/2207.03876/) (2022)
- [Neural Topological Ordering for Computation Graphs](https://mlsanity.com/paper/2207.05899/) (2022)
- [Improving Generalization of Neural Combinatorial Optimization for Vehicle Routing Problems via Test-Time Projection Learning](https://mlsanity.com/paper/2506.02392/) (2025)
- [Preference Optimization for Combinatorial Optimization Problems](https://mlsanity.com/paper/2505.08735/) (2025)
- [Adversarial Generative Flow Network for Solving Vehicle Routing Problems](https://mlsanity.com/paper/2503.01931/) (2025)
- [Learning to Insert for Constructive Neural Vehicle Routing Solver](https://mlsanity.com/paper/2505.13904/) (2025)
- [Collaboration! Towards Robust Neural Methods for Routing Problems](https://mlsanity.com/paper/2410.04968/) (2024)
- [Learning to Solve Multiple-TSP with Time Window and Rejections via Deep Reinforcement Learning](https://mlsanity.com/paper/2209.06094/) (2022)
- [GREAT: A Graph Edge Attention Network for the Electric Vehicle Routing Problem with Asymmetric Costs](https://mlsanity.com/paper/2408.16717/) (2024)

## Abstract (arXiv)

We present NeuroLKH, a novel algorithm that combines deep learning with the strong traditional heuristic Lin-Kernighan-Helsgaun (LKH) for solving Traveling Salesman Problem. Specifically, we train a Sparse Graph Network (SGN) with supervised learning for edge scores and unsupervised learning for node penalties, both of which are critical for improving the performance of LKH. Based on the output of SGN, NeuroLKH creates the edge candidate set and transforms edge distances to guide the searching process of LKH. Extensive experiments firmly demonstrate that, by training one model on a wide range of problem sizes, NeuroLKH significantly outperforms LKH and generalizes well to much larger sizes. Also, we show that NeuroLKH can be applied to other routing problems such as Capacitated Vehicle Routing Problem (CVRP), Pickup and Delivery Problem (PDP), and CVRP with Time Windows (CVRPTW).