Learning to Solve the Quadratic Assignment Problem with Warm-Started MCMC Finetuning

2026-04-21

AI Summary

This work addresses the challenge of simultaneously achieving high solution quality and robustness across diverse real-world instances of the Quadratic Assignment Problem (QAP). The authors propose the PLMA framework, which integrates neural networks with Markov Chain Monte Carlo (MCMC) fine-tuning. Their approach introduces a warm-start MCMC mechanism and an O(1) time-complexity 2-swap sampling strategy, alongside a cross-graph attention network to model interactions between facilities and locations. By combining an energy-based model with efficient Metropolis–Hastings sampling and short-chain MCMC optimization, the method achieves near-zero average optimality gaps on QAPLIB benchmarks and demonstrates superior performance on challenging instances such as Taixxeyy and bandwidth minimization tasks.

Technology Category

• Search and Optimization: Sampling/Simulation-based Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Quantum Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

The quadratic assignment problem (QAP) is a fundamental NP-hard task that poses significant challenges for both traditional heuristics and modern learning-based solvers. Existing QAP solvers still struggle to achieve consistently competitive performance across structurally diverse real-world instances. To bridge this performance gap, we propose PLMA, an innovative permutation learning framework. PLMA features an efficient warm-started MCMC finetuning procedure to enhance deployment-time performance, leveraging short Markov chains to anchor the adaptation to the promising regions previously explored. For rapid exploration via MCMC over the permutation space, we design an additive energy-based model (EBM) that enables an $O(1)$-time 2-swap Metropolis-Hastings sampling step. Moreover, the neural network used to parameterize the EBM incorporates a scalable and flexible cross-graph attention mechanism to model interactions between facilities and locations in the QAP. Extensive experiments demonstrate that PLMA consistently outperforms state-of-the-art baselines across various benchmarks. In particular, PLMA achieves a near-zero average optimality gap on QAPLIB, exhibits remarkably superior robustness on the notoriously difficult Taixxeyy instances, and also serves as an effective QAP solver in bandwidth minimization.

Problem

• Quadratic Assignment Problem
• NP-hard
• combinatorial optimization
• permutation learning
• real-world instances

Innovation

• warm-started MCMC
• energy-based model
• permutation learning
• cross-graph attention
• quadratic assignment problem