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The Quadratic Assignment Problem
 TO APPEAR IN THE HANDBOOK OF COMBINATORIAL OPTIMIZATION
"... This paper aims at describing the state of the art on quadratic assignment problems (QAPs). It discusses the most important developments in all aspects of the QAP such as linearizations, QAP polyhedra, algorithms to solve the problem to optimality, heuristics, polynomially solvable special cases, an ..."
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Cited by 182 (3 self)
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This paper aims at describing the state of the art on quadratic assignment problems (QAPs). It discusses the most important developments in all aspects of the QAP such as linearizations, QAP polyhedra, algorithms to solve the problem to optimality, heuristics, polynomially solvable special cases, and asymptotic behavior. Moreover, it also considers problems related to the QAP, e.g. the biquadratic assignment problem, and discusses the relationship between the QAP and other well known combinatorial optimization problems, e.g. the traveling salesman problem, the graph partitioning problem, etc.
Polyhedral Methods for the QAP
, 1999
"... For many combinatorial optimization problems investigations of associated polyhedra have led to enormous successes with respect to both theoretical insights into the structures of the problems as well as to their algorithmic solvability. Among these problems are quite prominent NPhard ones, like, e ..."
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Cited by 5 (0 self)
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For many combinatorial optimization problems investigations of associated polyhedra have led to enormous successes with respect to both theoretical insights into the structures of the problems as well as to their algorithmic solvability. Among these problems are quite prominent NPhard ones, like, e.g., the traveling salesman problem, the stable set problem, or the maximum cut problem. In this chapter we overview the polyhedral work that has been done on the quadratic assignment problem (QAP). Our treatment includes a brief introduction to the methods of polyhedral combinatorics in general, descriptions of the most important polyhedral results that are known about the QAP, explanations of the techniques that are used to prove such results, and a discussion of the practical results obtained by cutting plane algorithms that exploit the polyhedral knowledge. We close by some remarks on the perspectives of this kind of approach to the QAP.
A Basic Study of the QAPPolytope
 INSTITUT FÜR INFORMATIK, UNIVERSITÄT ZU KÖLN, POHLIGSTRASSE 1, D50969
, 1996
"... We investigate a polytope (the QAPPolytope) beyond a "natural" integer programming formulation of the Quadratic Assignment Problem (QAP) that has been used successfully in order to compute good lower bounds for the QAP in the very recent years. We present basic structural properties of th ..."
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Cited by 5 (0 self)
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We investigate a polytope (the QAPPolytope) beyond a "natural" integer programming formulation of the Quadratic Assignment Problem (QAP) that has been used successfully in order to compute good lower bounds for the QAP in the very recent years. We present basic structural properties of the QAPPolytope, partially independently also obtained by Rijal (1995). The main original contribution of this work is the representation of the QAPPolytope in a space different from the one in which it is defined naturally. This representation provides us with a much simpler way to derive the dimension of the QAPPolytope, as well as to investigate the facial structure of it. Furthermore, it leads to some interesting observations concerning the combinatorial structure of the QAPPolytope.
Polyhedral Combinatorics of QAPs with Less Objects than Locations
 LECTURE NOTES IN COMPUT. SCI.
, 1998
"... For the classical quadratic assignment problem (QAP) that requires n objects to be assigned to n locations (the nx ncase), polyhedral studies have been started in the very recent years by several authors. In this paper, we investigate the variant of the QAP, where the number of locations may ex ..."
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Cited by 2 (1 self)
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For the classical quadratic assignment problem (QAP) that requires n objects to be assigned to n locations (the nx ncase), polyhedral studies have been started in the very recent years by several authors. In this paper, we investigate the variant of the QAP, where the number of locations may exceed the number of objects (the m x ncase). It turns out that one can obtain structural results on the m x npolytopes by exploiting knowledge on the nncase, since the first ones are certain projections of the latter ones. Besides answering the basic questions for the ane hulls, the dimensions, and the trivial facets of the m x npolytopes, we present a large class of facet dening inequalities. Employed into a cutting plane procedure, these polyhedral results enable us to compute optimal solutions for some hard instances from the QAPLIB for the first time without using branchandbound. Moreover, we can calculate for several yet unsolved instances signicantly improved lower bounds.