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The quadratic assignment problem is a well-known optimization problem withnumerous applications. A common strategy to solve it is to use one of itslinearizations and then apply the toolbox of mixed integer linear programmingmethods. One measure of quality of a mixed integer formulation is the qualityof its linear relaxation. In this paper, we compare two linearizations of thequadratic assignment problem and prove that the linear relaxation of thelinearization of Adams and Johnson is contained in the linear relaxation of thelinearization of Xia and Yuan. We furthermore develop a Branch and Cut approachusing the insights obtained in the proof that enhances the linearization of Xiaand Yuan via a new family of cuts called ab-cuts.

The quadratic assignment problem: the linearization of Xia and Yuan is  weaker than the linearization of Adams and Johnson and a family of cuts to  narrow the gap