Solution Approaches

Abstract

The quadratic assignment problem (QAP) has attracted a surpassing algorithmic research interest since its introduction in 1957 by Koopmans and Beckmann. A wide variety of algorithms and heuristics have been developed to solve the QAP exactly or approximately. Moreover, since all the problems described in Chapter 2 are closely related to the QAP, one could modify the available exact and approximate techniques for the QAP and utilize them to “solve” every one of these problems. While this is conceptually correct, we do not recommend to solve e.g. traveling salesman problems this way, because the largest size QAP solved to optimality, so far, has n = 30; see Clausen [1994], Mans et al. [1992], Pardalos et al. [1994], and Resende et al. [1994]. More to the point, this means that existing algorithms for QAPs are nowhere close to solving practical problems arising from real-life applications to optimality. This state of affairs is unsatisfactory, but not surprising since very little is known about the mathematical properties of QAPs. A straight-forward application of the appropriately modified QAP algorithms to solve its variants can thus not be expected to solve large-scale instances of these problems. While many authors propose (different) mixed zero-one formulations of QAPs, they are hardly exploited in the numerical computations and the facial structure of the associated integer polyhedra has not been studied in any detail.