# Efficient Meta Neural Heuristic for Multi-Objective Combinatorial Optimization

arXiv: https://arxiv.org/abs/2310.15196 · HTML: https://mlsanity.com/paper/2310.15196/

## Key facts

- Cited by 21 papers in the map (data through 14 Oct 2026).
- Builds on 26 of its 52 references that are in the map.
- First posted on arXiv 22 Oct 2023.

## Builds on (most cited first, 26 of 26)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Model-Agnostic Meta-Learning for Fast Adaptation of Deep Networks](https://mlsanity.com/paper/1703.03400/) (2017)
- [On First-Order Meta-Learning Algorithms](https://mlsanity.com/paper/1803.02999/) (2018)
- [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)
- [Rapid Learning or Feature Reuse? Towards Understanding the Effectiveness of MAML](https://mlsanity.com/paper/1909.09157/) (2019)
- [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)
- [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)
- [Learning Improvement Heuristics for Solving Routing Problems](https://mlsanity.com/paper/1912.05784/) (2019)
- [Neural Large Neighborhood Search for the Capacitated Vehicle Routing Problem](https://mlsanity.com/paper/1911.09539/) (2019)
- [Learning to Iteratively Solve Routing Problems with Dual-Aspect Collaborative Transformer](https://mlsanity.com/paper/2110.02544/) (2021)
- [Multi-Decoder Attention Model with Embedding Glimpse for Solving Vehicle Routing Problems](https://mlsanity.com/paper/2012.10638/) (2020)
- [Matrix Encoding Networks for Neural Combinatorial Optimization](https://mlsanity.com/paper/2106.11113/) (2021)
- [Deep Reinforcement Learning for Multi-objective Optimization](https://mlsanity.com/paper/1906.02386/) (2019)
- [Learning Collaborative Policies to Solve NP-hard Routing Problems](https://mlsanity.com/paper/2110.13987/) (2021)
- [Pareto Set Learning for Neural Multi-objective Combinatorial Optimization](https://mlsanity.com/paper/2203.15386/) (2022)
- [A Deep Reinforcement Learning Algorithm Using Dynamic Attention Model for Vehicle Routing Problems](https://mlsanity.com/paper/2002.03282/) (2020)
- [Meta-Learning-Based Deep Reinforcement Learning for Multiobjective Optimization Problems](https://mlsanity.com/paper/2105.02741/) (2021)
- [MODRL/D-AM: Multiobjective Deep Reinforcement Learning Algorithm Using Decomposition and Attention Model for Multiobjective Optimization](https://mlsanity.com/paper/2002.05484/) (2020)
- [MODRL/D-EL: Multiobjective Deep Reinforcement Learning with Evolutionary Learning for Multiobjective Optimization](https://mlsanity.com/paper/2107.07961/) (2021)
- [PPLS/D: Parallel Pareto Local Search based on Decomposition](https://mlsanity.com/paper/1709.09785/) (2017)

## Built on by (most cited first, 21 of 21)

- [ReEvo: Large Language Models as Hyper-Heuristics with Reflective Evolution](https://mlsanity.com/paper/2402.01145/) (2024)
- [Neural Combinatorial Optimization Algorithms for Solving Vehicle Routing Problems: A Comprehensive Survey with Perspectives](https://mlsanity.com/paper/2406.00415/) (2024)
- [Gradient-Based Multi-Objective Deep Learning: Algorithms, Theories, Applications, and Beyond](https://mlsanity.com/paper/2501.10945/) (2025)
- [BOPO: Neural Combinatorial Optimization via Best-anchored and Objective-guided Preference Optimization](https://mlsanity.com/paper/2503.07580/) (2025)
- [Benchmarking PtO and PnO Methods in the Predictive Combinatorial Optimization Regime](https://mlsanity.com/paper/2311.07633/) (2023)
- [PARCO: Parallel AutoRegressive Models for Multi-Agent Combinatorial Optimization](https://mlsanity.com/paper/2409.03811/) (2024)
- [Preference-Driven Multi-Objective Combinatorial Optimization with Conditional Computation](https://mlsanity.com/paper/2506.08898/) (2025)
- [Moco: A Learnable Meta Optimizer for Combinatorial Optimization](https://mlsanity.com/paper/2402.04915/) (2024)
- [RedAHD: Reduction-Based End-to-End Automatic Heuristic Design with Large Language Models](https://mlsanity.com/paper/2505.20242/) (2025)
- [Experience-Guided Reflective Co-Evolution of Prompts and Heuristics for Automatic Algorithm Design](https://mlsanity.com/paper/2509.24509/) (2025)
- [Context-aware Diversity Enhancement for Neural Multi-Objective Combinatorial Optimization](https://mlsanity.com/paper/2405.08604/) (2024)
- [Beyond Simple Graphs: Neural Multi-Objective Routing on Multigraphs](https://mlsanity.com/paper/2506.22095/) (2025)
- [Game-Theoretic Co-Evolution for LLM-Based Heuristic Discovery](https://mlsanity.com/paper/2601.22896/) (2026)
- [Chain-of-Context Learning: Dynamic Constraint Understanding for Multi-Task VRPs](https://mlsanity.com/paper/2603.01667/) (2026)
- [On Size and Hardness Generalization in Unsupervised Learning for the Travelling Salesman Problem](https://mlsanity.com/paper/2403.20212/) (2024)
- [Multi-Objective Reinforcement Learning for Large Language Model Optimization: Visionary Perspective](https://mlsanity.com/paper/2509.21613/) (2025)
- [End-to-end Deep Reinforcement Learning for Stochastic Multi-objective Optimization in C-VRPTW](https://mlsanity.com/paper/2512.01518/) (2025)
- [Adversarial Instance Generation and Robust Training for Neural Combinatorial Optimization with Multiple Objectives](https://mlsanity.com/paper/2601.01665/) (2026)
- [Two-Stage Learned Decomposition for Scalable Routing on Multigraphs](https://mlsanity.com/paper/2605.05389/) (2026)
- [WeCon: An Efficient Weight-Conditioned Neural Solver for Multi-Objective Combinatorial Optimization Problems](https://mlsanity.com/paper/2605.22876/) (2026)
- [AGDN: Learning to Solve Traveling Salesman Problem with Anisotropic Graph Diffusion Network](https://mlsanity.com/paper/2606.19185/) (2026)

## Abstract (arXiv)

Recently, neural heuristics based on deep reinforcement learning have exhibited promise in solving multi-objective combinatorial optimization problems (MOCOPs). However, they are still struggling to achieve high learning efficiency and solution quality. To tackle this issue, we propose an efficient meta neural heuristic (EMNH), in which a meta-model is first trained and then fine-tuned with a few steps to solve corresponding single-objective subproblems. Specifically, for the training process, a (partial) architecture-shared multi-task model is leveraged to achieve parallel learning for the meta-model, so as to speed up the training; meanwhile, a scaled symmetric sampling method with respect to the weight vectors is designed to stabilize the training. For the fine-tuning process, an efficient hierarchical method is proposed to systematically tackle all the subproblems. Experimental results on the multi-objective traveling salesman problem (MOTSP), multi-objective capacitated vehicle routing problem (MOCVRP), and multi-objective knapsack problem (MOKP) show that, EMNH is able to outperform the state-of-the-art neural heuristics in terms of solution quality and learning efficiency, and yield competitive solutions to the strong traditional heuristics while consuming much shorter time.