Copositive and semidefinite relaxations of the quadratic assignment problem

Semidefinite relaxations of the quadratic assignment problem ($QAP$) have recently turned out to provide good approximations to the optimal value of $QAP$. We take a systematic look at various conic relaxations of $QAP$. We first show that $QAP$ can equivalently be formulated as a linear program over the cone of completely positive matrices. Since it is hard to optimize over this cone, we also look at tractable approximations and compare with several relaxations from the literature. We show that several of the well-studied models are in fact equivalent. It is still a challenging task to solve the strongest of these models to reasonable accuracy on instances of moderate size.