Eigenvalue Bounds Versus Semidefinite Relaxations for the Quadratic Assignment Problem

It was recently demonstrated that a well-known eigenvalue bound for the Quadratic Assignment Problem (QAP) actually corresponds to a semidefinite programming (SDP) relaxation. However, for this bound to be computationally useful the assignment constraints of the QAP must first be eliminated, and the bound then applied to a lowerdimensional problem. The resulting "projected eigenvalue bound" is one of the best available bounds for the QAP, especially when considering the quality of bounds relative to the complexity of obtaining them. In this paper we show that the projected eigenvalue bound also corresponds to an SDP relaxation of the original QAP. Keywords: Quadratic Assignment Problem, Eigenvalue Bounds, Semidefinite Programming. 1 Introduction The Quadratic Assignment Problem (QAP) is a well-studied problem in discrete optimization. For recent surveys see for example [6], [7], and [16]. In this paper we consider the "KoopmansBeckmann " form of the problem, which can be written QAP..