# Pareto Set Learning for Neural Multi-objective Combinatorial Optimization

arXiv: https://arxiv.org/abs/2203.15386 · HTML: https://mlsanity.com/paper/2203.15386/

## Key facts

- Cited by 56 papers in the map (data through 14 Oct 2026).
- Builds on 41 of its 89 references that are in the map.
- First posted on arXiv 29 Mar 2022.

## Builds on (most cited first, 30 of 41)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Rethinking the Inception Architecture for Computer Vision](https://mlsanity.com/paper/1512.00567/) (2015)
- [HyperNetworks](https://mlsanity.com/paper/1609.09106/) (2016)
- [Pointer Networks](https://mlsanity.com/paper/1506.03134/) (2015)
- [Benchmarking Graph Neural Networks](https://mlsanity.com/paper/2003.00982/) (2020)
- [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)
- [Reinforcement Learning for Solving the Vehicle Routing Problem](https://mlsanity.com/paper/1802.04240/) (2018)
- [Exact Combinatorial Optimization with Graph Convolutional Neural Networks](https://mlsanity.com/paper/1906.01629/) (2019)
- [Continual learning with hypernetworks](https://mlsanity.com/paper/1906.00695/) (2019)
- [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)
- [Reinforcement Learning for Combinatorial Optimization: A Survey](https://mlsanity.com/paper/2003.03600/) (2020)
- [Learning to Perform Local Rewriting for Combinatorial Optimization](https://mlsanity.com/paper/1810.00337/) (2018)
- [Combinatorial Optimization with Graph Convolutional Networks and Guided Tree Search](https://mlsanity.com/paper/1810.10659/) (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)
- [Learning the Pareto Front with Hypernetworks](https://mlsanity.com/paper/2010.04104/) (2020)
- [Erdos Goes Neural: an Unsupervised Learning Framework for Combinatorial Optimization on Graphs](https://mlsanity.com/paper/2006.10643/) (2020)
- [Learning to Delegate for Large-scale Vehicle Routing](https://mlsanity.com/paper/2107.04139/) (2021)
- [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)
- [Deep Policy Dynamic Programming for Vehicle Routing Problems](https://mlsanity.com/paper/2102.11756/) (2021)
- [Neural Algorithmic Reasoning](https://mlsanity.com/paper/2105.02761/) (2021)
- [Deep Reinforcement Learning for Multi-objective Optimization](https://mlsanity.com/paper/1906.02386/) (2019)
- [Learning Combinatorial Optimization on Graphs: A Survey with Applications to Networking](https://mlsanity.com/paper/2005.11081/) (2020)
- [Graph Neural Network Guided Local Search for the Traveling Salesperson Problem](https://mlsanity.com/paper/2110.05291/) (2021)

## Built on by (most cited first, 30 of 56)

- [Multi-Objective GFlowNets](https://mlsanity.com/paper/2210.12765/) (2022)
- [Learning to Search Feasible and Infeasible Regions of Routing Problems with Flexible Neural k-Opt](https://mlsanity.com/paper/2310.18264/) (2023)
- [Efficient Evolutionary Search Over Chemical Space with Large Language Models](https://mlsanity.com/paper/2406.16976/) (2024)
- [Smooth Tchebycheff Scalarization for Multi-Objective Optimization](https://mlsanity.com/paper/2402.19078/) (2024)
- [Algorithm Evolution Using Large Language Model](https://mlsanity.com/paper/2311.15249/) (2023)
- [Pareto Set Learning for Expensive Multi-Objective Optimization](https://mlsanity.com/paper/2210.08495/) (2022)
- [Sample-efficient Multi-objective Molecular Optimization with GFlowNets](https://mlsanity.com/paper/2302.04040/) (2023)
- [Multi-Task Learning for Routing Problem with Cross-Problem Zero-Shot Generalization](https://mlsanity.com/paper/2402.16891/) (2024)
- [Three-Way Trade-Off in Multi-Objective Learning: Optimization, Generalization and Conflict-Avoidance](https://mlsanity.com/paper/2305.20057/) (2023)
- [Evolutionary Reinforcement Learning: A Survey](https://mlsanity.com/paper/2303.04150/) (2023)
- [Large Language Model for Multi-objective Evolutionary Optimization](https://mlsanity.com/paper/2310.12541/) (2023)
- [Neural Combinatorial Optimization Algorithms for Solving Vehicle Routing Problems: A Comprehensive Survey with Perspectives](https://mlsanity.com/paper/2406.00415/) (2024)
- [Efficient Meta Neural Heuristic for Multi-Objective Combinatorial Optimization](https://mlsanity.com/paper/2310.15196/) (2023)
- [Pareto Set Learning for Multi-Objective Reinforcement Learning](https://mlsanity.com/paper/2501.06773/) (2025)
- [Gradient-Based Multi-Objective Deep Learning: Algorithms, Theories, Applications, and Beyond](https://mlsanity.com/paper/2501.10945/) (2025)
- [Neural Multi-Objective Combinatorial Optimization with Diversity Enhancement](https://mlsanity.com/paper/2310.15195/) (2023)
- [Continuation Path Learning for Homotopy Optimization](https://mlsanity.com/paper/2307.12551/) (2023)
- [Learning Pareto Set for Multi-Objective Continuous Robot Control](https://mlsanity.com/paper/2406.18924/) (2024)
- [ExLLM: Experience-Enhanced LLM Optimization for Molecular Design and Beyond](https://mlsanity.com/paper/2502.12845/) (2025)
- [URS: A Unified Neural Routing Solver for Cross-Problem Zero-Shot Generalization](https://mlsanity.com/paper/2509.23413/) (2025)
- [Learning for routing: A guided review of recent developments and future directions](https://mlsanity.com/paper/2507.00218/) (2025)
- [EmoDM: A Diffusion Model for Evolutionary Multi-objective Optimization](https://mlsanity.com/paper/2401.15931/) (2024)
- [Preference-Driven Multi-Objective Combinatorial Optimization with Conditional Computation](https://mlsanity.com/paper/2506.08898/) (2025)
- [A Unified Framework for Combinatorial Optimization Based on Graph Neural Networks](https://mlsanity.com/paper/2406.13125/) (2024)
- [Neural Solver Selection for Combinatorial Optimization](https://mlsanity.com/paper/2410.09693/) (2024)
- [Evolutionary Preference Sampling for Pareto Set Learning](https://mlsanity.com/paper/2404.08414/) (2024)
- [Ranking over Regression for Bayesian Optimization and Molecule Selection](https://mlsanity.com/paper/2410.09290/) (2024)
- [Improving Pareto Set Learning for Expensive Multi-objective Optimization via Stein Variational Hypernetworks](https://mlsanity.com/paper/2412.17312/) (2024)
- [Pareto-Grid-Guided Large Language Models for Fast and High-Quality Heuristics Design in Multi-Objective Combinatorial Optimization](https://mlsanity.com/paper/2507.20923/) (2025)
- [Survey on Neural Routing Solvers](https://mlsanity.com/paper/2602.21761/) (2026)

## Abstract (arXiv)

Multiobjective combinatorial optimization (MOCO) problems can be found in many real-world applications. However, exactly solving these problems would be very challenging, particularly when they are NP-hard. Many handcrafted heuristic methods have been proposed to tackle different MOCO problems over the past decades. In this work, we generalize the idea of neural combinatorial optimization, and develop a learning-based approach to approximate the whole Pareto set for a given MOCO problem without further search procedure. We propose a single preference-conditioned model to directly generate approximate Pareto solutions for any trade-off preference, and design an efficient multiobjective reinforcement learning algorithm to train this model. Our proposed method can be treated as a learning-based extension for the widely-used decomposition-based multiobjective evolutionary algorithm (MOEA/D). It uses a single model to accommodate all the possible preferences, whereas other methods use a finite number of solutions to approximate the Pareto set. Experimental results show that our proposed method significantly outperforms some other methods on the multiobjective traveling salesman problem, multiobjective vehicle routing problem, and multiobjective knapsack problem in terms of solution quality, speed, and model efficiency.