Semi-definite Relaxation of Quadratic Assignment Problems based on Nonredundant Matrix Splitting

Quadratic Assignment Problems (QAPs) are known to be among the most challenging discrete optimization problems. Recently, a new class of semi-definite relaxation models for QAPs based on matrix splitting has been proposed [17, 20]. In this paper, we consider the issue of how to choose an appropriate matrix splitting scheme so that the resulting relaxation model can provide a strong bound. For this, we introduce a new notion of the so-called redundant and non-redundant matrix splitting. We show that the relaxation based on a non-redundant matrix splitting can provide a stronger bound than a redundant one. We then propose to follow the minimal trace principle to find such a non-redundant matrix splitting. Based on the minimal trace principle, three matrix splitting schemes are derived and coincidentally, two of them had already been used in [17, 20]. A new matrix splitting, called sum-matrix splitting, is introduced. A similar procedure as in [20] is used to construct the SDP relaxation model. Numerical experiments show the bound based on the sum-matrix splitting is very competitive with existing bounds including the bounds based on other matrix splitting schemes.