# Neural Combinatorial Optimization with Reinforcement Learning

arXiv: https://arxiv.org/abs/1611.09940 · HTML: https://mlsanity.com/paper/1611.09940/

## Key facts

- Cited by 608 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Nov 2016.

## Built on by (most cited first, 30 of 608)

- [Gemma 2: Improving Open Language Models at a Practical Size](https://mlsanity.com/paper/2408.00118/) (2024)
- [What Can Transformers Learn In-Context? A Case Study of Simple Function Classes](https://mlsanity.com/paper/2208.01066/) (2022)
- [Combinatorial optimization and reasoning with graph neural networks](https://mlsanity.com/paper/2102.09544/) (2021)
- [Evolution of Heuristics: Towards Efficient Automatic Algorithm Design Using Large Language Model](https://mlsanity.com/paper/2401.02051/) (2024)
- [DIFUSCO: Graph-based Diffusion Solvers for Combinatorial Optimization](https://mlsanity.com/paper/2302.08224/) (2023)
- [PaliGemma 2: A Family of Versatile VLMs for Transfer](https://mlsanity.com/paper/2412.03555/) (2024)
- [End-to-End Constrained Optimization Learning: A Survey](https://mlsanity.com/paper/2103.16378/) (2021)
- [Neural Combinatorial Optimization with Heavy Decoder: Toward Large Scale Generalization](https://mlsanity.com/paper/2310.07985/) (2023)
- [Graph Learning: A Survey](https://mlsanity.com/paper/2105.00696/) (2021)
- [Training Agents Inside of Scalable World Models](https://mlsanity.com/paper/2509.24527/) (2025)
- [DIMES: A Differentiable Meta Solver for Combinatorial Optimization Problems](https://mlsanity.com/paper/2210.04123/) (2022)
- [Sym-NCO: Leveraging Symmetricity for Neural Combinatorial Optimization](https://mlsanity.com/paper/2205.13209/) (2022)
- [Monte Carlo Tree Search for Comprehensive Exploration in LLM-Based Automatic Heuristic Design](https://mlsanity.com/paper/2501.08603/) (2025)
- [Combinatorial Optimization with Physics-Inspired Graph Neural Networks](https://mlsanity.com/paper/2107.01188/) (2021)
- [Learning to Iteratively Solve Routing Problems with Dual-Aspect Collaborative Transformer](https://mlsanity.com/paper/2110.02544/) (2021)
- [LambdaNetworks: Modeling Long-Range Interactions Without Attention](https://mlsanity.com/paper/2102.08602/) (2021)
- [NeuroLKH: Combining Deep Learning Model with Lin-Kernighan-Helsgaun Heuristic for Solving the Traveling Salesman Problem](https://mlsanity.com/paper/2110.07983/) (2021)
- [Symbolic Regression via Neural-Guided Genetic Programming Population Seeding](https://mlsanity.com/paper/2111.00053/) (2021)
- [Efficient Active Search for Combinatorial Optimization Problems](https://mlsanity.com/paper/2106.05126/) (2021)
- [Deep Policy Dynamic Programming for Vehicle Routing Problems](https://mlsanity.com/paper/2102.11756/) (2021)
- [Matrix Encoding Networks for Neural Combinatorial Optimization](https://mlsanity.com/paper/2106.11113/) (2021)
- [Learning Collaborative Policies to Solve NP-hard Routing Problems](https://mlsanity.com/paper/2110.13987/) (2021)
- [GLOP: Learning Global Partition and Local Construction for Solving Large-scale Routing Problems in Real-time](https://mlsanity.com/paper/2312.08224/) (2023)
- [Size-Invariant Graph Representations for Graph Classification Extrapolations](https://mlsanity.com/paper/2103.05045/) (2021)
- [Simulation-guided Beam Search for Neural Combinatorial Optimization](https://mlsanity.com/paper/2207.06190/) (2022)
- [BQ-NCO: Bisimulation Quotienting for Efficient Neural Combinatorial Optimization](https://mlsanity.com/paper/2301.03313/) (2023)
- [Learning to Search Feasible and Infeasible Regions of Routing Problems with Flexible Neural k-Opt](https://mlsanity.com/paper/2310.18264/) (2023)
- [DeepACO: Neural-enhanced Ant Systems for Combinatorial Optimization](https://mlsanity.com/paper/2309.14032/) (2023)
- [Machine Learning for Electronic Design Automation: A Survey](https://mlsanity.com/paper/2102.03357/) (2021)
- [RL4CO: an Extensive Reinforcement Learning for Combinatorial Optimization Benchmark](https://mlsanity.com/paper/2306.17100/) (2023)

## Abstract (arXiv)

This paper presents a framework to tackle combinatorial optimization problems using neural networks and reinforcement learning. We focus on the traveling salesman problem (TSP) and train a recurrent network that, given a set of city coordinates, predicts a distribution over different city permutations. Using negative tour length as the reward signal, we optimize the parameters of the recurrent network using a policy gradient method. We compare learning the network parameters on a set of training graphs against learning them on individual test graphs. Despite the computational expense, without much engineering and heuristic designing, Neural Combinatorial Optimization achieves close to optimal results on 2D Euclidean graphs with up to 100 nodes. Applied to the KnapSack, another NP-hard problem, the same method obtains optimal solutions for instances with up to 200 items.