Problem Statement and Complexity Aspects

Abstract

In this chapter we introduce the quadratic assignment problem (QAP) . We start with the definition of the QAP and some frequently used formulations for it. Then, we consider the applications of the QAP and discuss in more details the most celebrated ones: applications in facility location and applications in wiring problems. Further, three mixed integer linear programming (MILP) formulations of the QAP are introduced: the Kaufman and Broeckx linearization [141], the Frieze and Yadegar linearization [83], and the linearization of Padberg and Rijal [177]. These formulations are chosen among numerous MILP formulation proposed in the literature in the hope that they can better help for a deep understanding of the QAP and its combinatorial structure. Consider that the linearization of Kaufman and Broeckx is perhaps the smallest QAP linearizations in terms of the number of variables and constraints, whereas some of the best existing bounding procedures for the QAP are obtained by building upon the linearization of Frieze and Yadegar. Furthermore, the linearization of Padberg and Rijal is the basis of recent significant results on the QAP polytope. Based on this linearization, the affine hull and the dimension of the QAP polytope (symmetric QAP polytope) have been computed. Moreover, some valid inequalities for the QAP polytope and some facet defining equalities for the symmetric QAP polytope have been identified. Finally, we consider computational complexity aspects of the QAP, discussing among others the complexity of approximating the problem, and the complexity of the local search.