Title: Learning the Quadratic Assignment Problem with Warm-Started MCMC Finetuning and Cross-Graph Attention
Primary area: optimization
Keywords: Quadratic Assignment Problem, Combinatorial Optimization, Permutation Learning, MCMC Sampling, Energy-Based Models, Deployment-time Adaptation

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. Current neural solvers for the QAP often struggle with poor scalability or limited flexibility when dealing with the coupled two-graph structure, and they underperform on real-world instances.  We propose PLMA, an innovative permutation learning framework to bridge this performance gap. 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, we design an additive energy-based model (EBM) over the permutation space, which enables an $O(1)$-time 2-swap Metropolis-Hastings sampling step.  Moreover, the network used to parameterize the EBM incorporates a cross-graph attention mechanism that directly models the coupled graph structure of the QAP, ensuring scalability and flexibility. Extensive experiments demonstrate the consistent superiority of PLMA over stat-of-the-art baseline methods across various benchmarks, highlighted by a near-zero average optimality gap on QAPLIB and remarkably superior robustness on the notoriously difficult Taixxeyy instances.