Quadratic assignment problem (QAP) is a fundamental problem in combinatorial optimization and finds numerous applications in operation research, computer vision, and pattern recognition. However, it is a very well-known NP-hard problem to find the global minimizer to the QAP. In this work, we study the semidefinite relaxation (SDR) of the QAP and investigate when the SDR recovers the global minimizer. In particular, we consider the two input matrices satisfy a simple signal-plus-noise model, and show that when the noise is sufficiently smaller than the signal, then the SDR is exact, i.e., it recovers the global minimizer to the QAP. It is worth noting that this sufficient condition is purely algebraic and does not depend on any statistical assumption of the input data. We apply our bound to several statistical models such as correlated Gaussian Wigner model. Despite the sub-optimality in theory under those models, empirical studies show the remarkable performance of the SDR. Our work could be the first step towards a deeper understanding of the SDR exactness for the QAP.

翻译：二次分配问题（QAP）是组合优化中的一个基础问题，在运筹学、计算机视觉和模式识别等领域具有广泛应用。然而，寻找QAP的全局极小值是一个众所周知的NP难问题。本文研究了QAP的半定规划松弛（SDR），并探讨了SDR在何种条件下能够恢复全局极小值。特别地，我们考虑两个输入矩阵满足简单的信号加噪声模型，并证明当噪声充分小于信号时，SDR是精确的，即能够恢复QAP的全局极小值。值得注意的是，该充分条件是纯代数的，不依赖于输入数据的任何统计假设。我们将所提出的界应用于多种统计模型，如相关高斯维格纳模型。尽管在这些模型下理论结果存在次优性，实证研究显示了SDR的卓越性能。我们的工作可能是深入理解QAP的SDR精确性的第一步。