The Quadratic Assignment Problem, QAP, was studied using an algebraic approach through a linear relaxation, the Linear Assignment Problem, LAP. The reason for this approach is the Inversion Theorem demonstrated by Rangel [Ran00]. In this theorem, the QAP solution cost is associated to the number of inversions of the linear correspondent. Although recognizing if a linear solution correspond to a QAP solution is polynomial, there are much more LAP solutions than QAP solutions, and therefore to find them is a hard work. We construct a matrix that stores information about LAP solutions that are able to generate QAP solutions. The Inversion Theorem in conjuction with this matrix permitted us to present a constructive method that generates good initial solutions. The great advantage of this matrix is the low computational cost of time and memory. A parallel version of this algorithm is proposed and implemented in this work.
quadratic assignment problem; linear assignment problem; inversion theorem
