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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.
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 148 (13 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 114 (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...
Fortran Subroutines For Approximate Solution Of Dense Quadratic Assignment Problems Using Grasp
, 1994
"... In the NPcomplete quadratic assignment problem (QAP), n facilities are to be assigned to n sites at minimum cost. The contribution of assigning facility i to site k and facility j to site l to the total cost is f ij \Delta d kl , where f ij is the flow between facilites i and j, and d kl is the di ..."
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Cited by 13 (6 self)
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In the NPcomplete quadratic assignment problem (QAP), n facilities are to be assigned to n sites at minimum cost. The contribution of assigning facility i to site k and facility j to site l to the total cost is f ij \Delta d kl , where f ij is the flow between facilites i and j, and d kl is the distance between sites k and l. Only very small (n 20) instances of the QAP have been solved exactly, and heuristics are therefore used to produce approximate solutions. This paper describes a set of FORTRAN subroutines to find approximatesolutions to dense quadratic assignment problems, having at least one symmetric flow or distance matrix. A greedy randomized adaptive search procedure (GRASP) is used to produce the solutions. The design and implementation of the code are described in detail, and extensive computational experiments are reported, illustrating solution quality as a function of running time.
Index Sorting: Programs and Theory for Nonredundant Channel Coding . . .
, 1992
"... this report we present six programs for nonredundant, fixed length channel coding. The program codes were written for use with Vector Quantization (VQ) codebooks to provide robustness to channel errors. More Generally, the codes implement heuristic algorithm for embedding a hypercube graph  an NP ..."
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Cited by 1 (1 self)
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this report we present six programs for nonredundant, fixed length channel coding. The program codes were written for use with Vector Quantization (VQ) codebooks to provide robustness to channel errors. More Generally, the codes implement heuristic algorithm for embedding a hypercube graph  an NPhard combinatorial optimization problem [1]. The program are intended for use as part of an image compression scheme diagrammed in Figure 1.1. A VQ algorithm, perhaps predictive or differential, performs source coding on input image to achieve data compression. Then, the channel coding task is to assign binary indices to VQ codevectors so as to minimize the distortion introduced in the received image when transmitted indices are corrupted by channel noises. By arranging the indices such that channel errors cause incorrectly received codevectors to be close, on average, to the original codevectors, channel distortion can be reduced. In Section 2 we give description of the six programs adapted from SPANN manual pages, which are available online on the SPANN Laboratory computer network. In Section 3 we review background theory for development of the algorithms, and in Section 4 we provide detailed descriptions of the algorithms implemented in the six programs. 2. Codebook Sorting Programs Image VQ Encode Transmitter Receiver Decode Codebook Channel Coding Image Channel 01 Figure 1.1: Basic VQ system Six programs are available on the SPANN Laboratory network for the codebook index assignment task. These programs are listed in Table 2.1 in order of increasing computational complexity. In this section, we describe each program and its usage. We have observed that for differential VQ with 256 codevectors, the splitsort algorithm, though computationally cheapest, has typically yie...
Algorithm ____: FORTRAN Subroutines for . . .
"... In the NPcomplete quadratic assignment problem (QAP), n facilities are to be assigned to n sites at minimum cost. The contribution of assigning facility i to site k and facility j to site l to the total cost is fij · dkl, where fij is the flow between facilites i and j, and dkl is the distance betw ..."
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In the NPcomplete quadratic assignment problem (QAP), n facilities are to be assigned to n sites at minimum cost. The contribution of assigning facility i to site k and facility j to site l to the total cost is fij · dkl, where fij is the flow between facilites i and j, and dkl is the distance between sites k and l. Only very small (n ≤ 20) instances of the QAP have been solved exactly, and heuristics are therefore used to produce approximate solutions. This paper describes a set of FORTRAN subroutines to find approximate solutions to dense quadratic assignment problems, having at least one symmetric flow or distance matrix. A greedy randomized adaptive search procedure (GRASP) is used to produce the solutions. The design and implementation of the code are described in detail, and extensive computational experiments are reported, illustrating solution quality as a function of running time.
Universidade Federal do Rio de Janeiro Rio de Janeiro – RJ
, 2002
"... This work discusses the use of a neighbouring structure in the design of specific heuristics for the Quadratic Assignment Problem (QAP). This structure is formed by the 4 and 6cycles adjacent to a vertex in the Hasse diagram of the permutation lattice and it can be adequately partitioned in subset ..."
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This work discusses the use of a neighbouring structure in the design of specific heuristics for the Quadratic Assignment Problem (QAP). This structure is formed by the 4 and 6cycles adjacent to a vertex in the Hasse diagram of the permutation lattice and it can be adequately partitioned in subsets of linear and quadratic cardinalities, a characteristics which frequently allows an economy in the processing time. We propose also a restart strategy and a mechanism for generating initial solutions which constitute, together with the neighbouring structure, a possible QAPspecific heuristic proposal. For the construction of these instruments we used the relaxed ordered set of QAP solutions.
Vector Quantization for Noisy Channels: A Guide to Performance and Computation
, 1995
"... In this paper, we present vector quantization (VQ) design approaches for reliable communication over noisy channels. The VQ design problems we consider differ in the channel characterization known to the VQ designer, and in the computational resources available. For known channel behavior and availa ..."
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In this paper, we present vector quantization (VQ) design approaches for reliable communication over noisy channels. The VQ design problems we consider differ in the channel characterization known to the VQ designer, and in the computational resources available. For known channel behavior and availability of source samples for offline batch training, Channel Optimized Vector Quantization (COVQ) minimizes the expected distortion in the decoded message. For a timevarying bit error rate known only to the receiver, an adaptive VQ decoder minimizes channel distortion. For any VQ communication scenario, index assignment (IA) is an indispensable computational tool. IA provides improved training for COVQ, offers error mitigation when using preexisting codebooks designed for noiseless channels, and yields computationally attractive error protection for adaptive VQ encoders. We evaluate the tradeoff between performance and computational cost for seven IA algorithms. These comparisons yield r...