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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.
Genetic Hybrids for the Quadratic Assignment Problem
 DIMACS Series in Mathematics and Theoretical Computer Science
, 1993
"... . A new hybrid procedure that combines genetic operators to existing heuristics is proposed to solve the Quadratic Assignment Problem (QAP). Genetic operators are found to improve the performance of both local search and tabu search. Some guidelines are also given to design good hybrid schemes. Thes ..."
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Cited by 89 (0 self)
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. A new hybrid procedure that combines genetic operators to existing heuristics is proposed to solve the Quadratic Assignment Problem (QAP). Genetic operators are found to improve the performance of both local search and tabu search. Some guidelines are also given to design good hybrid schemes. These hybrid algorithms are then used to improve on the best known solutions of many test problems in the literature. 1. Introduction The quadratic assignment problem (QAP) can be stated as: min OE2P (n) n X i=1 n X j=1 a ij b OE(i)OE(j) ; where A = (a ij ) and B = (b kl ) are two n \Theta n matrices and P (n) is the set of all permutations of f1; :::; ng. Matrix A is often referred to as a distance matrix between sites, and B as a flow matrix between objects. In most cases, the matrices A and B are symmetrical with a null diagonal. A permutation may then be interpreted as an assignment of objects to sites with a quadratic cost associated to it. There are many applications that can be fo...
Edward,“Guided Local Search
, 1995
"... Abstract Combinatorial explosion problem is a well known phenomenon that prevents complete algorithms from solving many reallife combinatorial optimization problems. In many situations, heuristic search methods are needed. This chapter describes the principles of Guided Local Search (GLS) and Fast ..."
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Cited by 56 (5 self)
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Abstract Combinatorial explosion problem is a well known phenomenon that prevents complete algorithms from solving many reallife combinatorial optimization problems. In many situations, heuristic search methods are needed. This chapter describes the principles of Guided Local Search (GLS) and Fast Local Search (FLS) and surveys their applications. GLS is a penaltybased metaheuristic algorithm that sits on top of other local search algorithms, with the aim to improve their efficiency and robustness. FLS is a way of reducing the size of the neighbourhood to improve the efficiency of local search. The chapter also provides guidance for implementing and using GLS and FLS. Four problems, representative of general application categories, are examined with detailed information provided on how to build a GLSbased method in each case.
A global solution to sparse correspondence problems
 IEEE Transactions on pattern Analysis and Machine Intelligence
, 2003
"... Abstract—We propose a new methodology for reliably solving the correspondence problem between sparse sets of points of two or more images. This is a key step in most problems of computer vision and, so far, no general method exists to solve it. Our methodology is able to handle most of the commonly ..."
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Cited by 54 (3 self)
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Abstract—We propose a new methodology for reliably solving the correspondence problem between sparse sets of points of two or more images. This is a key step in most problems of computer vision and, so far, no general method exists to solve it. Our methodology is able to handle most of the commonly used assumptions in a unique formulation, independent of the domain of application and type of features. It performs correspondence and outlier rejection in a single step and achieves global optimality with feasible computation. Feature selection and correspondence are first formulated as an integer optimization problem. This is a blunt formulation, which considers the whole combinatorial space of possible point selections and correspondences. To find its global optimal solution, we build a concave objective function and relax the search domain into its convexhull. The special structure of this extended problem assures its equivalence to the original one, but it can be optimally solved by efficient algorithms that avoid combinatorial search. This methodology can use any criterion provided it can be translated into cost functions with continuous second derivatives. Index Terms—Correspondence problem, linear and concave programming, sparse stereo. 1
On Lagrangian relaxation of quadratic matrix constraints
 SIAM J. Matrix Anal. Appl
, 2000
"... Abstract. Quadratically constrained quadratic programs (QQPs) play an important modeling role for many diverse problems. These problems are in general NP hard and numerically intractable. Lagrangian relaxations often provide good approximate solutions to these hard problems. Such relaxations are equ ..."
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Cited by 50 (19 self)
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Abstract. Quadratically constrained quadratic programs (QQPs) play an important modeling role for many diverse problems. These problems are in general NP hard and numerically intractable. Lagrangian relaxations often provide good approximate solutions to these hard problems. Such relaxations are equivalent to semidefinite programming relaxations. For several special cases of QQP, e.g., convex programs and trust region subproblems, the Lagrangian relaxation provides the exact optimal value, i.e., there is a zero duality gap. However, this is not true for the general QQP, or even the QQP with two convex constraints, but a nonconvex objective. In this paper we consider a certain QQP where the quadratic constraints correspond to the matrix orthogonality condition XXT = I. For this problem we show that the Lagrangian dual based on relaxing the constraints XXT = I and the seemingly redundant constraints XT X = I has a zero duality gap. This result has natural applications to quadratic assignment and graph partitioning problems, as well as the problem of minimizing the weighted sum of the largest eigenvalues of a matrix. We also show that the technique of relaxing quadratic matrix constraints can be used to obtain a strengthened semidefinite relaxation for the maxcut problem. Key words. Lagrangian relaxations, quadratically constrained quadratic programs, semidefinite programming, quadratic assignment, graph partitioning, maxcut problems
Eigenvalues in combinatorial optimization
, 1993
"... In the last decade many important applications of eigenvalues and eigenvectors of graphs in combinatorial optimization were discovered. The number and importance of these results is so fascinating that it makes sense to present this survey. ..."
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Cited by 42 (0 self)
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In the last decade many important applications of eigenvalues and eigenvectors of graphs in combinatorial optimization were discovered. The number and importance of these results is so fascinating that it makes sense to present this survey.
A New Bound for the Quadratic Assignment Problem Based on Convex Quadratic Programming
 MATHEMATICAL PROGRAMMING
, 1999
"... We describe a new convex quadratic programming bound for the quadratic assignment problem (QAP). The construction of the bound uses a semidefinite programming representation of a basic eigenvalue bound for QAP. The new bound dominates the wellknown projected eigenvalue bound, and appears to be comp ..."
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Cited by 33 (4 self)
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We describe a new convex quadratic programming bound for the quadratic assignment problem (QAP). The construction of the bound uses a semidefinite programming representation of a basic eigenvalue bound for QAP. The new bound dominates the wellknown projected eigenvalue bound, and appears to be competitive with existing bounds in the tradeoff between bound quality and computational effort.
A Comparison of Memetic Algorithms, Tabu Search, and Ant Colonies for the Quadratic Assignment Problem
 Proc. Congress on Evolutionary Computation, IEEE
, 1999
"... A memetic algorithm (MA), i.e. an evolutionary algorithm making use of local search, for the quadratic assignment problem is presented. A new recombination operator for realizing the approach is described, and the behavior of the MA is investigated on a set of problem instances containing between 25 ..."
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Cited by 32 (4 self)
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A memetic algorithm (MA), i.e. an evolutionary algorithm making use of local search, for the quadratic assignment problem is presented. A new recombination operator for realizing the approach is described, and the behavior of the MA is investigated on a set of problem instances containing between 25 and 100 facilities/locations. The results indicate that the proposed MA is able to produce high quality solutions quickly. A comparison of the MA with some of the currently best alternative approaches  reactive tabu search, robust tabu search and the fast ant colony system  demonstrates that the MA outperforms its competitors on all studied problem instances of practical interest. 1 Introduction The problem of assigning a set of facilities (with given flows between them) to a set of locations (with given distances between them) in such a way that the sum of the product between flows and distances is minimized is known as the facilities location problem [1] or the quadratic assignment ...
Probabilistic graph and hypergraph matching
 in IEEE Conf. Comput. Vis. Pattern Recognit
"... We consider the problem of finding a matching between two sets of features, given complex relations among them, going beyond pairwise. Each feature set is modeled by a hypergraph where the complex relations are represented by hyperedges. A match between the feature sets is then modeled as a hypergr ..."
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Cited by 31 (0 self)
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We consider the problem of finding a matching between two sets of features, given complex relations among them, going beyond pairwise. Each feature set is modeled by a hypergraph where the complex relations are represented by hyperedges. A match between the feature sets is then modeled as a hypergraph matching problem. We derive the hypergraph matching problem in a probabilistic setting represented by a convex optimization. First, we formalize a soft matching criterion that emerges from a probabilistic interpretation of the problem input and output, as opposed to previous methods that treat soft matching as a mere relaxation of the hard matching problem. Second, the model induces an algebraic relation between the hyperedge weight matrix and the desired vertextovertex probabilistic matching. Third, the model explains some of the graph matching normalization proposed in the past on a heuristic basis such as doubly stochastic normalizations of the edge weights. A key benefit of the model is that the global optimum of the matching criteria can be found via an iterative successive projection algorithm. The algorithm reduces to the well known Sinkhorn [15] row/column matrix normalization procedure in the special case when the two graphs have the same number of vertices and a complete matching is desired. Another benefit of our model is the straightforward scalability from graphs to hypergraphs. 1.
A Unifying Objective Function for Topographic Mappings
, 1997
"... Many different algorithms and objective functions for topographic mappings have been proposed. We show that several of these approaches can be seen as particular cases of a more general objective function. Consideration of a very simple mapping problem reveals large differences in the form of the ma ..."
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Cited by 30 (4 self)
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Many different algorithms and objective functions for topographic mappings have been proposed. We show that several of these approaches can be seen as particular cases of a more general objective function. Consideration of a very simple mapping problem reveals large differences in the form of the map that each particular case favors. These differences have important consequences for the practical application of topographic mapping methods.