Recently various optimization problems, such as Mixed Integer Linear Programming Problems (MILPs), have undergone comprehensive investigation, leveraging the capabilities of machine learning. This work focuses on learning-based solutions for efficiently solving the Quadratic Assignment Problem (QAPs), which stands as a formidable challenge in combinatorial optimization. While many instances of simpler problems admit fully polynomial-time approximate solution (FPTAS), QAP is shown to be strongly NP-hard. Even finding a FPTAS for QAP is difficult, in the sense that the existence of a FPTAS implies $P = NP$. Current research on QAPs suffer from limited scale and computational inefficiency. To attack the aforementioned issues, we here propose the first solution of its kind for QAP in the learn-to-improve category. This work encodes facility and location nodes separately, instead of forming computationally intensive association graphs prevalent in current approaches. This design choice enables scalability to larger problem sizes. Furthermore, a \textbf{S}olution \textbf{AW}are \textbf{T}ransformer (SAWT) architecture integrates the incumbent solution matrix with the attention score to effectively capture higher-order information of the QAPs. Our model's effectiveness is validated through extensive experiments on self-generated QAP instances of varying sizes and the QAPLIB benchmark.

翻译：近年来，各类优化问题（如混合整数线性规划问题）已借助机器学习能力得到全面研究。本文聚焦于基于学习的方法以高效求解二次分配问题——该问题是组合优化领域的一项严峻挑战。尽管许多较简单问题实例存在完全多项式时间近似方案，但QAP被证明是强NP难问题。甚至为QAP寻找FPTAS也十分困难，因为FPTAS的存在意味着$P = NP$。当前QAP研究受限于问题规模与计算效率。为攻克上述难题，我们首次提出"学习改进"范式的QAP求解方案。本工作分别对设施节点与位置节点进行编码，而非采用当前主流方法中计算密集的关联图构建方式。该设计使得模型能够扩展到更大规模问题。此外，\textbf{解感知Transformer}架构通过将当前解矩阵与注意力分数相融合，有效捕捉QAP的高阶信息。我们在自生成的不同规模QAP实例及QAPLIB基准测试上进行了广泛实验，验证了模型的有效性。