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Very LargeScale Neighborhood Search for the Quadratic Assignment Problem
 DISCRETE APPLIED MATHEMATICS
, 2002
"... The Quadratic Assignment Problem (QAP) consists of assigning n facilities to n locations so as to minimize the total weighted cost of interactions between facilities. The QAP arises in many diverse settings, is known to be NPhard, and can be solved to optimality only for fairly small size instances ..."
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Cited by 108 (11 self)
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The Quadratic Assignment Problem (QAP) consists of assigning n facilities to n locations so as to minimize the total weighted cost of interactions between facilities. The QAP arises in many diverse settings, is known to be NPhard, and can be solved to optimality only for fairly small size instances (typically, n < 25). Neighborhood search algorithms are the most popular heuristic algorithms to solve larger size instances of the QAP. The most extensively used neighborhood structure for the QAP is the 2exchange neighborhood. This neighborhood is obtained by swapping the locations of two facilities and thus has size O(n²). Previous efforts to explore larger size neighborhoods (such as 3exchange or 4exchange neighborhoods) were not very successful, as it took too long to evaluate the larger set of neighbors. In this paper, we propose very largescale neighborhood (VLSN) search algorithms where the size of the neighborhood is very large and we propose a novel search procedure to heuristically enumerate good neighbors. Our search procedure relies on the concept of improvement graph which allows us to evaluate neighbors much faster than the existing methods. We present extensive computational results of our algorithms on standard benchmark instances. These investigations reveal that very largescale neighborhood search algorithms give consistently better solutions compared the popular 2exchange neighborhood algorithms considering both the solution time and solution accuracy.
The Quadratic Assignment Problem: A Survey and Recent Developments
 In Proceedings of the DIMACS Workshop on Quadratic Assignment Problems, volume 16 of DIMACS Series in Discrete Mathematics and Theoretical Computer Science
, 1994
"... . Quadratic Assignment Problems model many applications in diverse areas such as operations research, parallel and distributed computing, and combinatorial data analysis. In this paper we survey some of the most important techniques, applications, and methods regarding the quadratic assignment probl ..."
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Cited by 91 (16 self)
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. Quadratic Assignment Problems model many applications in diverse areas such as operations research, parallel and distributed computing, and combinatorial data analysis. In this paper we survey some of the most important techniques, applications, and methods regarding the quadratic assignment problem. We focus our attention on recent developments. 1. Introduction Given a set N = f1; 2; : : : ; ng and n \Theta n matrices F = (f ij ) and D = (d kl ), the quadratic assignment problem (QAP) can be stated as follows: min p2\Pi N n X i=1 n X j=1 f ij d p(i)p(j) + n X i=1 c ip(i) ; where \Pi N is the set of all permutations of N . One of the major applications of the QAP is in location theory where the matrix F = (f ij ) is the flow matrix, i.e. f ij is the flow of materials from facility i to facility j, and D = (d kl ) is the distance matrix, i.e. d kl represents the distance from location k to location l [62, 67, 137]. The cost of simultaneously assigning facility i to locat...
A Genetic Approach to the Quadratic Assignment Problem
, 1995
"... The Quadratic Assignment Problem (QAP) is a wellknown combinatorial optimization problem with a wide variety of practical applications. Although many heuristics and semienumerative procedures for QAP have been proposed, no dominant algorithm has emerged. In this paper, we describe a Genetic Algori ..."
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Cited by 54 (7 self)
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The Quadratic Assignment Problem (QAP) is a wellknown combinatorial optimization problem with a wide variety of practical applications. Although many heuristics and semienumerative procedures for QAP have been proposed, no dominant algorithm has emerged. In this paper, we describe a Genetic Algorithm (GA) approach to QAP. Genetic algorithms are a class of randomized parallel search heuristics which emulate biological natural selection on a population of feasible solutions. We present computational results which show that this GA approach finds solutions competitive with those of the best previouslyknown heuristics, and argue that genetic algorithms provide a particularly robust method for QAP and its more complex extensions. 5 A Genetic Approach to the Quadratic Assignment Problem David M. Tate and Alice E. Smith Department of Industrial Engineering 1048 Benedum Hall University of Pittsburgh Pittsburgh, PA 15261 4126249837 4126249831 (Fax) 1. Introduction The Quadrat...
A Greedy Genetic Algorithm for the Quadratic Assignment Problem
 Computers and Operations Research
, 1997
"... The Quadratic Assignment Problem (QAP) is one of the classical combinatorial optimization problems and is known for its diverse applications. In this paper, we suggest a genetic algorithm for the QAP and report its computational behavior. The genetic algorithm incorporates many greedy principles in ..."
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Cited by 43 (2 self)
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The Quadratic Assignment Problem (QAP) is one of the classical combinatorial optimization problems and is known for its diverse applications. In this paper, we suggest a genetic algorithm for the QAP and report its computational behavior. The genetic algorithm incorporates many greedy principles in its design and, hence, is called the greedy genetic algorithm. The ideas we incorporate in the greedy genetic algorithm include (i) generating the initial population using a randomized construction heuristic; (ii) new crossover schemes; (iii) a special purpose immigration scheme that promotes diversity; (iv) periodic local optimization of a subset of the population; (v) tournamenting among different populations; and (vi) an overall design that attempts to strike a balance between diversity and a bias towards fitter individuals. We test our algorithm on all the benchmark instances of QAPLIB, a wellknown library of QAP instances. Out of the 132 total instances in QAPLIB of varied sizes, the g...
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 28 (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...
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 20 (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 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 20 (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.
Data Flow Partitioning for Clock Period and Latency Minimization
, 1993
"... Abstract  We propose an e cient performancedriven twoway partitioning algorithm to take into account clock cycle period and latency with retiming. We model the problem with a Quadratic Programming formulation to minimize the crossing edge count with nonlinear timing constraints. By using Lagrangia ..."
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Cited by 5 (0 self)
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Abstract  We propose an e cient performancedriven twoway partitioning algorithm to take into account clock cycle period and latency with retiming. We model the problem with a Quadratic Programming formulation to minimize the crossing edge count with nonlinear timing constraints. By using Lagrangian Approach on Modular Partitioning (LAMP), we merge nonlinear constraints to the objective function. The problem is then decomposed into primal and dual two subprograms. The primal and dual problems are solved by a Quadratic Boolean Programming approach and by a subgradient method using cycle mean method, respectively. Experimental results show our algorithm achieves an average of 23.25 % clock cycle period and 19.54 % latency reductions compared to the Fiduccia
Implementation Of A Variance Reduction Based Lower Bound In A Branch And Bound Algorithm For The Quadratic Assignment Problem
, 1997
"... . The efficient implementation of a branch and bound algorithm for the quadratic assignment problem (QAP), incorporating the lower bound, based on variance reduction, of Li, Pardalos, Ramakrishnan, and Resende (1994), is presented. A new data structure for efficient implementation of branch and boun ..."
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Cited by 4 (1 self)
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. The efficient implementation of a branch and bound algorithm for the quadratic assignment problem (QAP), incorporating the lower bound, based on variance reduction, of Li, Pardalos, Ramakrishnan, and Resende (1994), is presented. A new data structure for efficient implementation of branch and bound algorithms for the QAP is introduced. Computational experiments with the branch and bound algorithm on different classes of QAP test problems are reported. The branch and bound algorithm using the new lower bounds is compared with the same algorithm utilizing the commonly applied GilmoreLawler lower bound. Both implementations use a greedy randomized adaptive search procedure for obtaining initial upper bounds. The algorithms report all optimal permutations. Optimal solutions for previously unsolved instances from the literature, of dimensions n = 16 and n = 20, have been found with the new algorithm. In addition, the new algorithm has been tested on a class of large data variance problem...