. 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...

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 NP-hard, 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 2-exchange 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 3-exchange or 4-exchange 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 large-scale neighborhood search algorithms give consistently better solutions compared the popular 2-exchange neighborhood algorithms considering both the solution time and solution accuracy.

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. The paper will appear in the Handbook of Combinatorial Optimization to be published by Kluwer Academic Publishers, P. Pardalos and D.-Z. Du, eds.

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 well-known library of QAP instances. Out of the 132 total instances in QAPLIB of varied sizes, the g...

. In the NP-complete 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. Key words. Combinatorial optimization, quadratic assignment problem, local search, G...

Financial support by the &quot;Fonds zur F&quot;orderung der wissenschaftlichen Forschung, Projekt P8971-PHY&quot;, by the &quot;Christian Doppler Laboratorium f&quot;ur Diskrete Optimierung &quot; and by the &quot;Spezialforschungsbereich, Projektbereich Diskrete Optimierung &quot; is gratefully acknowledged. Prind&quot;erve t&quot;e mi Acknowledgments First of all, I express my deepest gratitude to my supervisor Prof. Rainer E. Burkard for his manifold help and encouragement during the last three years. He motivated me to investigate the quadratic assignment problem (QAP) and its generalizations. I appreciate very much the opportunity to benefit from his long and rich experience with the QAP. Prof. Burkard's support was decisive for providing the financial basis which enabled my graduate studies. Next, I would like to thank Franz Rendl for his willingness to be the second referee of this thesis.

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-hard 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 on-line 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...

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 6-cycles 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 QAP-specific heuristic proposal. For the construction of these instruments we used the relaxed ordered set of QAP solutions.

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 off-line batch training, Channel Optimized Vector Quantization (COVQ) minimizes the expected distortion in the decoded message. For a time-varying 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 pre-existing codebooks designed for noiseless channels, and yields computationally attractive error protection for adaptive VQ encoders. We evaluate the trade-off between performance and computational cost for seven IA algorithms. These comparisons yield r...

Abstract not found