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Quadratic Assignment Problems

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Handbook of Combinatorial Optimization

Abstract

This chapter introduces quadratic assignment problems (QAP) as models for finding an optimal assignment among two sets of interrelated objects. References to many applications of QAPs, like in location theory, are given. Moreover, several classical combinatorial optimization problems are identified as special cases of the QAP, like the traveling salesman problem, graph partitioning, max clique problem, minimum-weight feedback arc set problem, (FASP) and packing problems in graphs. These subproblems show that the QAP is \(\mathcal{N}\mathcal{P}\)-hard.Several different formulations of the QAP are presented in detail as being basic for different solution approaches. Special emphasis is laid to the different possibilities for describing QAPs as (mixed-)integer linear programs. Moreover, QAP polyhedra are discussed.Crucial for exact solution approaches are lower bounding procedures. The paradigm of admissible transformations is introduced for describing several variants of Gilmore–Lawler type bounds. Moreover, bounds based on eigenvalues and bounds found by semidefinite programming are discussed in detail. Among exact solution methods, special emphasis is laid on branch-and-bound approaches. With respect to heuristics, not only simple construction heuristics and improvement heuristics are described, but also several metaheuristics like simulated annealing, tabu search, greedy randomized search, genetic algorithms, and ant systems are discussed. Links to available computer programs are delivered.As QAPs are \(\mathcal{N}\mathcal{P}\)-hard problems, there is a special interest in polynomially solvable special cases. A comprehensive survey of results in this direction is given. Moreover, QAPs show an interesting asymptotic behavior, as the ratio of best and worst solution value tends to one in probability. This phenomenon is thoroughly discussed and implications of this strange behavior are shown.Finally, related problems are shortly treated. So, the quadratic bottleneck assignment problem is introduced and lower bounds, efficiently solvable special cases, as well as asymptotic results are reported. Moreover, cubic assignment problems, biquadratic assignment problems, and the quadratic semi-assignment problem are surveyed. An extensive bibliography provides the most important links to the literature in this field.

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  1. Institute of Optimization and Discrete Mathematics, Graz University of Technology, Steyrergasse 30, 8010, Graz, Austria

    Rainer E. burkard

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  1. Department of Industrial and Systems Eng, University of Florida, Gainesville, Florida, USA

    Panos M. Pardalos

  2. Department of Computer Science, University of Texas, Dallas, Richardson, Texas, USA

    Ding-Zhu Du

  3. Dept. Comp. Sci. & Engineering, University of California, San Diego, La Jolla, California, USA

    Ronald L. Graham

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burkard, R.E. (2013). Quadratic Assignment Problems. In: Pardalos, P., Du, DZ., Graham, R. (eds) Handbook of Combinatorial Optimization. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-7997-1_22

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