# Reinforced Lin-Kernighan-Helsgaun Algorithms for the Traveling Salesman Problems

arXiv: https://arxiv.org/abs/2207.03876 · HTML: https://mlsanity.com/paper/2207.03876/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 57 references that are in the map.
- First posted on arXiv 8 Jul 2022.

## Builds on (most cited first, 14 of 14)

- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Pointer Networks](https://mlsanity.com/paper/1506.03134/) (2015)
- [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)
- [Reinforcement Learning for Solving the Vehicle Routing Problem](https://mlsanity.com/paper/1802.04240/) (2018)
- [Learning Improvement Heuristics for Solving Routing Problems](https://mlsanity.com/paper/1912.05784/) (2019)
- [Learning to Iteratively Solve Routing Problems with Dual-Aspect Collaborative Transformer](https://mlsanity.com/paper/2110.02544/) (2021)
- [NeuroLKH: Combining Deep Learning Model with Lin-Kernighan-Helsgaun Heuristic for Solving the Traveling Salesman Problem](https://mlsanity.com/paper/2110.07983/) (2021)
- [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)
- [Graph Neural Network Guided Local Search for the Traveling Salesperson Problem](https://mlsanity.com/paper/2110.05291/) (2021)
- [Combining Reinforcement Learning with Lin-Kernighan-Helsgaun Algorithm for the Traveling Salesman Problem](https://mlsanity.com/paper/2012.04461/) (2020)
- [Learning Permutations with Sinkhorn Policy Gradient](https://mlsanity.com/paper/1805.07010/) (2018)
- [Combining Reinforcement Learning and Optimal Transport for the Traveling Salesman Problem](https://mlsanity.com/paper/2203.00903/) (2022)

## Built on by (most cited first, 12 of 12)

- [Neural Combinatorial Optimization Algorithms for Solving Vehicle Routing Problems: A Comprehensive Survey with Perspectives](https://mlsanity.com/paper/2406.00415/) (2024)
- [Visual Reasoning and Multi-Agent Approach in Multimodal Large Language Models (MLLMs): Solving TSP and mTSP Combinatorial Challenges](https://mlsanity.com/paper/2407.00092/) (2024)
- [Looking Ahead to Avoid Being Late: Solving Hard-Constrained Traveling Salesman Problem](https://mlsanity.com/paper/2403.05318/) (2024)
- [Leader Reward for POMO-Based Neural Combinatorial Optimization](https://mlsanity.com/paper/2405.13947/) (2024)
- [LMask: Learn to Solve Constrained Routing Problems with Lazy Masking](https://mlsanity.com/paper/2505.17938/) (2025)
- [LocalEscaper: A Weakly-supervised Framework with Regional Reconstruction for Scalable Neural TSP Solvers](https://mlsanity.com/paper/2502.12484/) (2025)
- [SVRPBench: A Realistic Benchmark for Stochastic Vehicle Routing Problem](https://mlsanity.com/paper/2505.21887/) (2025)
- [Learn to Tour: Operator Design For Solution Feasibility Mapping in Pickup-and-delivery Traveling Salesman Problem](https://mlsanity.com/paper/2404.11458/) (2024)
- [ViTSP: A Vision Language Models Guided Framework for Solving Large-Scale Traveling Salesman Problems](https://mlsanity.com/paper/2509.23465/) (2025)
- [Adaptation and Fine-tuning with TabPFN for Travelling Salesman Problem](https://mlsanity.com/paper/2511.05872/) (2025)
- [The Traveling Thief Problem with Time Windows: Benchmarks and Heuristics](https://mlsanity.com/paper/2604.06724/) (2026)
- [Machine Learning-based Two-Stage Graph Sparsification for the Travelling Salesman Problem](https://mlsanity.com/paper/2604.20236/) (2026)

## Abstract (arXiv)

TSP is a classical NP-hard combinatorial optimization problem with many practical variants. LKH is one of the state-of-the-art local search algorithms for the TSP. LKH-3 is a powerful extension of LKH that can solve many TSP variants. Both LKH and LKH-3 associate a candidate set to each city to improve the efficiency, and have two different methods, α-measure and POPMUSIC, to decide the candidate sets. In this work, we first propose a Variable Strategy Reinforced LKH (VSR-LKH) algorithm, which incorporates three reinforcement learning methods (Q-learning, Sarsa, Monte Carlo) with LKH, for the TSP. We further propose a new algorithm called VSR-LKH-3 that combines the variable strategy reinforcement learning method with LKH-3 for typical TSP variants, including the TSP with time windows (TSPTW) and Colored TSP (CTSP). The proposed algorithms replace the inflexible traversal operations in LKH and LKH-3 and let the algorithms learn to make a choice at each search step by reinforcement learning. Both LKH and LKH-3, with either α-measure or POPMUSIC, can be significantly improved by our methods. Extensive experiments on 236 widely-used TSP benchmarks with up to 85,900 cities demonstrate the excellent performance of VSR-LKH. VSR-LKH-3 also significantly outperforms the state-of-the-art heuristics for TSPTW and CTSP.