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arXiv 2406.09899cs.LGcs.AI
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用于高效求解二次分配问题的解感知Transformer学习方法

Learning Solution-Aware Transformers for Efficiently Solving Quadratic Assignment Problem

Center for Data Science, Peking University（北京大学数据科学中心）；Wangxuan Institute of Computer Technology, Peking University（北京大学王选计算机技术研究院）

Zhentao Tan, Yadong Mu

备注：
Accepted by ICML 2024
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AI总结：

针对二次分配问题（QAP）强NP难、现有求解方法规模受限且效率低的问题，提出解感知Transformer（SAWT）架构，通过分别编码设施与位置节点、融合当前解矩阵与注意力得分捕捉高阶信息，在自生成实例及QAPLIB基准上验证了有效性与可扩展性。

摘要：

近年来，借助机器学习的能力，混合整数线性规划问题（MILPs）等各类优化问题已得到广泛研究。本工作聚焦于基于学习的高效求解二次分配问题（QAPs）的方案，该问题是组合优化领域的一项艰巨挑战。许多更简单的问题实例存在完全多项式时间近似解（FPTAS），但QAP已被证明是强NP难问题；就连为QAP找到FPTAS都十分困难，因为FPTAS的存在意味着$P = NP$。当前针对QAP的研究存在规模受限与计算效率低下的问题。为解决上述问题，本文提出了首个属于“学习改进”类别的QAP求解方案。本工作对设施节点和位置节点分别进行编码，而非采用当前方法中普遍存在的、计算密集的关联图构建方式，这一设计选择使其可扩展至更大规模的问题。此外，**解感知Transformer（Solution AWare Transformer，SAWT）**架构将当前解矩阵与注意力得分相结合，可有效捕捉QAP的高阶信息。我们在不同规模的自生成QAP实例以及QAPLIB基准上开展了大量实验，验证了该模型的有效性。

英文摘要：

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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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