Quadratic assignment problems

This chapter discusses the quadratic assignment problems (QAPs). The benefit or cost resulting from an economic activity at some location is dependent on the locations of other facilities. To model such location problems, QAP is introduced. The objective assigns the plants to the possible sites such that the total cost of building and operating the plants becomes minimal. There are many application areas that are modeled by QAPs. The chapter surveys the theory and the solution procedures for QAPs and outlines the various exact and approximate solution methods. There is only one successful exact solution technique available that is based on matrix reductions and on the Gilmore–Lawler bounds. It is demonstrated that the trace form provides a very convenient tool to derive the theory of this method. There is also a second use of the trace formulation. This approach yields completely new competitive bounds for the symmetric case based on the eigenvalues of the underlying matrices. The eigenvalue related bounds give access to an optimal reduction procedure and also help to characterize QAPs that are almost linear. The chapter describes some preliminary numerical comparisons with the classical lower bounds.