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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.
Lower Bounds for the Quadratic Assignment Problem Based Upon a Dual Formulation
"... A new bounding procedure for the Quadratic Assignment Problem (QAP) is described which extends the Hungarian method for the Linear Assignment Problem (LAP) to QAPs, operating on the four dimensional cost array of the QAP objective function. The QAP is iteratively transformed in a series of equivalen ..."
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Cited by 39 (7 self)
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A new bounding procedure for the Quadratic Assignment Problem (QAP) is described which extends the Hungarian method for the Linear Assignment Problem (LAP) to QAPs, operating on the four dimensional cost array of the QAP objective function. The QAP is iteratively transformed in a series of equivalent QAPs leading to an increasing sequence of lower bounds for the original problem. To this end, two classes of operations which transform the four dimensional cost array are defined. These have the property that the values of the transformed objective function Z' are the corresponding values of the old objective function Z, shifted by some amount C. In the case that all entries of the transformed cost array are nonnegative, then C is a lower bound for the initial QAP. If, moreover, there exists a feasible solution U to the QAP, such that its value in the transformed problem is zero, then C is the optimal value of Z and U is an optimal solution for the original QAP. The transformations are iteratively applied until no significant increase in constant C as above is found, resulting in the so called Dual Procedure (DP). Several strategies are listed for appropriately determining C, or equivalently, transforming the cost array. The goal is the modification of the elements in the cost array so as to obtain new equivalent problems which bring the QAP closer to solution. In some cases the QAP is actually solved, though solution is not guaranteed. The close relationship between the DP and the Linear Programming formulation of Adams and Johnson is presented. The DP attempts to solve Adams and Johnsons CLP, a continuous relaxation of a linearization of the QAP. This explains why the DP produces bounds close to the optimum values for CLP calculated by Johnson in her dissertation and by...
Selected Topics on Assignment Problems
, 1999
"... We survey recent developments in the fields of bipartite matchings, linear sum assignment and bottleneck assignment problems and applications, multidimensional assignment problems, quadratic assignment problems, in particular lower bounds, special cases and asymptotic results, biquadratic and co ..."
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Cited by 34 (1 self)
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We survey recent developments in the fields of bipartite matchings, linear sum assignment and bottleneck assignment problems and applications, multidimensional assignment problems, quadratic assignment problems, in particular lower bounds, special cases and asymptotic results, biquadratic and communication assignment problems.
A BranchandBound Algorithm for the Quadratic Assignment Problem Based on the Hungarian Method
 European Journal of Operational Research
, 1996
"... This paper presents a new branchandbound algorithm for solving the Quadratic Assignment Problem (QAP). The algorithm is based on a Dual Procedure (DP) similar to the Hungarian method for solving the Linear Assignment Problem. Our DP solves the QAP in certain cases, i.e., for some small problems (N ..."
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Cited by 28 (5 self)
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This paper presents a new branchandbound algorithm for solving the Quadratic Assignment Problem (QAP). The algorithm is based on a Dual Procedure (DP) similar to the Hungarian method for solving the Linear Assignment Problem. Our DP solves the QAP in certain cases, i.e., for some small problems (N<7) and for numerous larger problems (7N16) that arise as subproblems of a larger QAP such as the Nugent 20. The DP, however, does not guarantee a solution. It is used in our algorithm to calculate lower bounds on solutions to the QAP. As a result of a number of recently developed improvements, the DP produces lower bounds that are as tight as any which might be useful in a branchandbound algorithm. These are produced relatively cheaply, especially on larger problems. Experimental results show that the computational complexity of our algorithm is lower than known methods, and that its actual runtime is significantly shorter than the best known algorithms for QAPLIB test instances of size 16 through 22. Our method has the potential for being improved and therefore can be expected to aid in solving even larger problems. Keywords Quadratic Assignment Problem, Branchandbound, Quadratic Programming, Integer Programming, Mathematical Programming. 2 1.
Lower bounds for the quadratic assignment problem
 University of Munich
, 1994
"... Abstract. We investigate the classical GilmoreLawler lower bound for the quadratic assignment problem. We provide evidence of the difficulty of improving the GilmoreLawler Bound and develop new bounds by means of optimal reduction schemes. Computational results are reported indicating that the new ..."
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Cited by 21 (5 self)
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Abstract. We investigate the classical GilmoreLawler lower bound for the quadratic assignment problem. We provide evidence of the difficulty of improving the GilmoreLawler Bound and develop new bounds by means of optimal reduction schemes. Computational results are reported indicating that the new lower bounds have advantages over previous bounds and can be used in a branchandbound type algorithm for the quadratic assignment problem. 1.
A Hospital Facility Layout Problem Finally Solved
, 2000
"... This paper presents a history of a difficult facility layout problem that falls into the category of the KoopmansBeckmann variant of the Quadratic Assignment Problem (QAP), wherein 30 facilities are to be assigned to 30 locations. The problem arose in 1972 as part of the design of a German unive ..."
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Cited by 10 (1 self)
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This paper presents a history of a difficult facility layout problem that falls into the category of the KoopmansBeckmann variant of the Quadratic Assignment Problem (QAP), wherein 30 facilities are to be assigned to 30 locations. The problem arose in 1972 as part of the design of a German university hospital, Klinikum Regensburg. This problem, known as the Krarup30a upon its inclusion in the QAPLIB library of QAP instances, has remained an important example of one of the most difficult to solve. In 1999, two approaches provided multiple optimum solutions.
2000): Progress in solving the Nugent instances of the quadratic assignment problem. Working
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Semidefinite Programming Relaxation of Quadratic Assignment Problems based on Nonredundant Matrix Splitting
, 2013
"... Quadratic Assignment Problems (QAPs) are known to be among the most challenging discrete optimization problems. Recently, a new class of semidefinite relaxation (SDR) models for QAPs based on matrix splitting has been proposed [25, 28]. In this paper, we consider the issue of how to choose an appro ..."
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Quadratic Assignment Problems (QAPs) are known to be among the most challenging discrete optimization problems. Recently, a new class of semidefinite relaxation (SDR) models for QAPs based on matrix splitting has been proposed [25, 28]. In this paper, we consider the issue of how to choose an appropriate matrix splitting scheme so that the resulting relaxation model is easy to solve and able to provide a strong bound. For this, we first introduce a new notion of the socalled redundant and nonredundant matrix splitting and show that the relaxation based on a nonredundant matrix splitting can provide a stronger bound than a redundant one. Then we propose to follow the minimal trace principle to find a nonredundant matrix splitting via solving an auxiliary semidefinite programming problem (SDP). We show that applying the minimal trace principle directly leads to the socalled orthogonal matrix splitting introduced in [28]. To find other nonredundant matrix splitting schemes whose resulting relaxation models are relatively easy to solve, we elaborate on two splitting schemes based on the socalled onematrix and the summatrix. We analyze the solutions from the auxiliary problems for these two cases and characterize when they can pro