On the Exactness of SDP Relaxation for Quadratic Assignment Problem

TL;DR

This paper investigates when semidefinite relaxation (SDR) can exactly solve the quadratic assignment problem (QAP), showing that under certain algebraic conditions with low noise, SDR recovers the global minimizer, supported by empirical results.

Contribution

It provides a purely algebraic sufficient condition for SDR exactness in QAP, applicable under a simple signal-plus-noise model, advancing understanding of SDR's capabilities.

Findings

SDR recovers the global minimizer when noise is sufficiently small

The algebraic condition for SDR exactness is independent of statistical assumptions

Empirical results demonstrate SDR's strong performance despite theoretical sub-optimality

Abstract

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.