# Preference Optimization for Combinatorial Optimization Problems

arXiv: https://arxiv.org/abs/2505.08735 · HTML: https://mlsanity.com/paper/2505.08735/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Placed in Reward modeling and RLHF, one of 10 directions in LLM alignment / RLHF / preference optimisation.
- Builds on 36 of its 59 references that are in the map.
- First posted on arXiv 13 May 2025.

## Placed in

- LLM alignment / RLHF / preference optimisation / [Reward modeling and RLHF](https://mlsanity.com/alignment/reward-modeling-and-rlhf/)
- Deep reinforcement learning methods / [RL for games and combinatorial optimization](https://mlsanity.com/deeprl/rl-for-games-and-combinatorial-optimization/)

## Builds on (most cited first, 30 of 36)

- [Direct Preference Optimization: Your Language Model is Secretly a Reward Model](https://mlsanity.com/paper/2305.18290/) (2023)
- [Deep reinforcement learning from human preferences](https://mlsanity.com/paper/1706.03741/) (2017)
- [SimPO: Simple Preference Optimization with a Reference-Free Reward](https://mlsanity.com/paper/2405.14734/) (2024)
- [A General Theoretical Paradigm to Understand Learning from Human Preferences](https://mlsanity.com/paper/2310.12036/) (2023)
- [Pointer Networks](https://mlsanity.com/paper/1506.03134/) (2015)
- [Reinforcement Learning with Deep Energy-Based Policies](https://mlsanity.com/paper/1702.08165/) (2017)
- [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)
- [ORPO: Monolithic Preference Optimization without Reference Model](https://mlsanity.com/paper/2403.07691/) (2024)
- [POMO: Policy Optimization with Multiple Optima for Reinforcement Learning](https://mlsanity.com/paper/2010.16011/) (2020)
- [DIFUSCO: Graph-based Diffusion Solvers for Combinatorial Optimization](https://mlsanity.com/paper/2302.08224/) (2023)
- [Disentangling Length from Quality in Direct Preference Optimization](https://mlsanity.com/paper/2403.19159/) (2024)
- [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 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)
- [Hybrid Genetic Search for the CVRP: Open-Source Implementation and SWAP* Neighborhood](https://mlsanity.com/paper/2012.10384/) (2020)
- [Sym-NCO: Leveraging Symmetricity for Neural Combinatorial Optimization](https://mlsanity.com/paper/2205.13209/) (2022)
- [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)
- [Matrix Encoding Networks for Neural Combinatorial Optimization](https://mlsanity.com/paper/2106.11113/) (2021)
- [BQ-NCO: Bisimulation Quotienting for Efficient Neural Combinatorial Optimization](https://mlsanity.com/paper/2301.03313/) (2023)
- [DeepACO: Neural-enhanced Ant Systems for Combinatorial Optimization](https://mlsanity.com/paper/2309.14032/) (2023)
- [Towards Omni-generalizable Neural Methods for Vehicle Routing Problems](https://mlsanity.com/paper/2305.19587/) (2023)
- [Inverse Preference Learning: Preference-based RL without a Reward Function](https://mlsanity.com/paper/2305.15363/) (2023)
- [Learning Generalizable Models for Vehicle Routing Problems via Knowledge Distillation](https://mlsanity.com/paper/2210.07686/) (2022)
- [Towards Generalizable Neural Solvers for Vehicle Routing Problems via Ensemble with Transferrable Local Policy](https://mlsanity.com/paper/2308.14104/) (2023)

## Built on by (most cited first, 11 of 11)

- [ADPO: Anchored Direct Preference Optimization](https://mlsanity.com/paper/2510.18913/) (2025)
- [VAGPO: Vision-augmented Asymmetric Group Preference Optimization for Graph Routing Problems](https://mlsanity.com/paper/2508.01774/) (2025)
- [Baseline-Free Policy Optimization for Neural Combinatorial Optimization](https://mlsanity.com/paper/2606.10321/) (2026)
- [HyCO: A Hybrid Neural Solver for Combinatorial Optimization](https://mlsanity.com/paper/2609.07990/) (2026)
- [Rethinking Efficiency in Neural Combinatorial Optimization: Batched Preference Optimization with Mamba](https://mlsanity.com/paper/2602.20730/) (2026)
- [FiLMMeD: Feature-wise Linear Modulation for Cross-Problem Multi-Depot Vehicle Routing](https://mlsanity.com/paper/2604.28102/) (2026)
- [WeCon: An Efficient Weight-Conditioned Neural Solver for Multi-Objective Combinatorial Optimization Problems](https://mlsanity.com/paper/2605.22876/) (2026)
- [MViewRouter: Internalizing Geometric Equivariance via Multi-view Alternating Attention for Combinatorial Routing](https://mlsanity.com/paper/2606.01084/) (2026)
- [Learning to Place Guards by Reinforcement: A Geo-Free Neural Policy for the Vertex-Guard Art Gallery Problem](https://mlsanity.com/paper/2606.21604/) (2026)
- [TSP with Predictions: Heatmap to Tour with Provable Guarantees](https://mlsanity.com/paper/2607.03791/) (2026)
- [Improving Cross-Problem Vehicle Routing with Locally Augmented Preferences and Representation Disentanglement](https://mlsanity.com/paper/2608.24859/) (2026)

## Abstract (arXiv)

Reinforcement Learning (RL) has emerged as a powerful tool for neural combinatorial optimization, enabling models to learn heuristics that solve complex problems without requiring expert knowledge. Despite significant progress, existing RL approaches face challenges such as diminishing reward signals and inefficient exploration in vast combinatorial action spaces, leading to inefficiency. In this paper, we propose Preference Optimization, a novel method that transforms quantitative reward signals into qualitative preference signals via statistical comparison modeling, emphasizing the superiority among sampled solutions. Methodologically, by reparameterizing the reward function in terms of policy and utilizing preference models, we formulate an entropy-regularized RL objective that aligns the policy directly with preferences while avoiding intractable computations. Furthermore, we integrate local search techniques into the fine-tuning rather than post-processing to generate high-quality preference pairs, helping the policy escape local optima. Empirical results on various benchmarks, such as the Traveling Salesman Problem (TSP), the Capacitated Vehicle Routing Problem (CVRP) and the Flexible Flow Shop Problem (FFSP), demonstrate that our method significantly outperforms existing RL algorithms, achieving superior convergence efficiency and solution quality.