Results 1  10
of
22
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 ..."
Abstract

Cited by 182 (3 self)
 Add to MetaCart
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.
Exact And Approximate Nondeterministic TreeSearch Procedures For The Quadratic Assignment Problem
, 1998
"... This paper introduces two new techniques for solving the Quadratic Assignment Problem. The first is a heuristic technique, defined in accordance to the Ant System metaphor, and includes as a distinctive feature the use of a new lower bound at each constructive step. The second is a branch and bound ..."
Abstract

Cited by 61 (5 self)
 Add to MetaCart
This paper introduces two new techniques for solving the Quadratic Assignment Problem. The first is a heuristic technique, defined in accordance to the Ant System metaphor, and includes as a distinctive feature the use of a new lower bound at each constructive step. The second is a branch and bound exact approach, containing some elements introduced in the Ant algorithm. Computational results prove the effectiveness of both approaches.
A New Lower Bound via Projection for the Quadratic Assignment Problem
 Mathematics of Operations Research
, 1992
"... New lower bounds for the quadratic assignment problem QAP are presented. These bounds are based on the orthogonal relaxation of QAP. The additional improvement is obtained by making efficient use of a tractable representation of orthogonal matrices having constant row and column sums. The new bound ..."
Abstract

Cited by 58 (16 self)
 Add to MetaCart
New lower bounds for the quadratic assignment problem QAP are presented. These bounds are based on the orthogonal relaxation of QAP. The additional improvement is obtained by making efficient use of a tractable representation of orthogonal matrices having constant row and column sums. The new bound is easy to implement and often provides high quality bounds under an acceptable computational effort. Key Words: quadratic assignment problem, lower bounds, relaxations, orthogonal projection, eigenvalue bounds. 0 The authors would like to thank the Natural Sciences and Engineering Research Council of Canada and the Austrian Science Foundatation (FWF) for their support. 1 Introduction The Quadratic Assignment Problem QAP is a generic model for various problems arising e.g. in location theory, VLSI design, facility layout, keyboard design and many other areas, see [1] for a recent survey on the QAP. Formally the QAP consists of minimizing f(X) = tr(AXB t + C)X t over the set of permu...
Solving Large Quadratic Assignment Problems in Parallel.
 Computational Optimization and Applications
, 1994
"... . Quadratic Assignment problems are in practice among the most difficult to solve in the class of NPcomplete problems. The only successful approach hitherto has been BranchandBound based algorithms, but such algorithms are crucially dependent on good bound functions to limit the size of the space ..."
Abstract

Cited by 29 (6 self)
 Add to MetaCart
(Show Context)
. Quadratic Assignment problems are in practice among the most difficult to solve in the class of NPcomplete problems. The only successful approach hitherto has been BranchandBound based algorithms, but such algorithms are crucially dependent on good bound functions to limit the size of the space searched. Much work has been done to identify such functions for the QAP, but with limited success. Parallel processing has also been used in order to increase the size of problems solvable to optimality. The systems used have, however, often been systems with relatively few, but very powerful vector processors, and have hence not been ideally suited for computations essentially involving nonvectorizable computations on integers. In this paper we investigate the combination of one of the best bound functions for a Branchand Bound algorithm (the GilmoreLawler bound) and various testing, variable binding and recalculation of bounds between branchings when used in a parallel BranchandBo...
A Parallel GRASP Implementation for the Quadratic Assignment Problem
 Parallel Algorithms for Irregularly Structured Problems – Irregular’94
, 1995
"... In this paper we present a parallel implementation of a Greedy Randomized Adaptive Search Procedure (grasp) for finding approximate solutions to the quadratic assignment problem. In particular, we discuss efficient techniques for largescale sparse quadratic assignment problems on an MIMD parallel c ..."
Abstract

Cited by 23 (15 self)
 Add to MetaCart
(Show Context)
In this paper we present a parallel implementation of a Greedy Randomized Adaptive Search Procedure (grasp) for finding approximate solutions to the quadratic assignment problem. In particular, we discuss efficient techniques for largescale sparse quadratic assignment problems on an MIMD parallel computer. We report computational experience on a collection of quadratic assignment problems. The code was run on a Kendall Square Research KSR1 parallel computer, using 1, 4, 14, 24, 34, 44, 54, and 64 processors, and achieves an average speedup that is almost linear in the number of processors. 1 Introduction Nonlinear assignment problems, such as quadratic, cubic, and Nadic assignment problems were formulated by Lawler [11]. One of the most extensively studied nonlinear assignment problems is the quadratic assignment problem (QAP). The QAP was first introduced by Koopmans and Beckmann in 1957 as a mathematical model for locating a set of indivisible economic activities [9]. Consider th...
Parallel Processing of Discrete Optimization Problems
 IN ENCYCLOPEDIA OF MICROCOMPUTERS
, 1993
"... Discrete optimization problems (DOPs) arise in various applications such as planning, scheduling, computer aided design, robotics, game playing and constraint directed reasoning. Often, a DOP is formulated in terms of finding a (minimum cost) solution path in a graph from an initial node to a goa ..."
Abstract

Cited by 21 (6 self)
 Add to MetaCart
Discrete optimization problems (DOPs) arise in various applications such as planning, scheduling, computer aided design, robotics, game playing and constraint directed reasoning. Often, a DOP is formulated in terms of finding a (minimum cost) solution path in a graph from an initial node to a goal node and solved by graph/tree search methods such as branchandbound and dynamic programming. Availability of parallel computers has created substantial interest in exploring the use of parallel processing for solving discrete optimization problems. This article provides an overview of parallel search algorithms for solving discrete optimization problems.
Bounds for the Quadratic Assignment Problem Using Continuous Optimization Techniques
, 1990
"... The quadratic assignment problem (denoted QAP ), in the trace formulation over the permutation matrices, is min X2\Pi tr(AXB +C)X t : Several recent lower bounds for QAP are discussed. These bounds are obtained by applying continuous optimization techniques to approximations of this combinator ..."
Abstract

Cited by 18 (5 self)
 Add to MetaCart
The quadratic assignment problem (denoted QAP ), in the trace formulation over the permutation matrices, is min X2\Pi tr(AXB +C)X t : Several recent lower bounds for QAP are discussed. These bounds are obtained by applying continuous optimization techniques to approximations of this combinatorial optimization problem, as well as by exploiting the special matrix structure of the problem. In particular, we apply constrained eigenvalue techniques, reduced gradient methods, subdifferential calculus, generalizations of trust region methods, and sequential quadratic programming. Keywords : Quadratic Assignment Problem, Bounds, Constrained Eigenvalues, Reduced Gradient, Trust Regions, Sequential Quadratic Programming. 1 Introduction The quadratic assignment problem, denoted QAP , is a generalization of the linear sum assignment problem, i.e. given the set N = f1; 2; : : : ; ng and three n by n matrices A = (a ik ); B = (B jl ); and C = (c ij ); find a permutation ß of the set N which m...
A Branch and Bound Algorithm for the Quadratic Assignment Problem using a Lower Bound Based on Linear Programming
 In C. Floudas and P.M. Pardalos, editors, State of the Art in Global Optimization: Computational Methods and Applications
, 1995
"... In this paper, we study a branch and bound algorithm for the quadratic assignment problem (QAP) that uses a lower bound based on the linear programming (LP) relaxation of a classical integer programming formulation of the QAP. Computational experience with the branch and bound algorithm on several Q ..."
Abstract

Cited by 10 (2 self)
 Add to MetaCart
In this paper, we study a branch and bound algorithm for the quadratic assignment problem (QAP) that uses a lower bound based on the linear programming (LP) relaxation of a classical integer programming formulation of the QAP. Computational experience with the branch and bound algorithm on several QAP test problems is reported. The linear programming relaxations are solved with an implementation of an interior point algorithm that uses a preconditioned conjugate gradient algorithm to compute directions. The branch and bound algorithm is compared with a similar branch and bound algorithm that uses the GilmoreLawler lower bound (GLB) instead of the LPbased bound. The LPbased algorithm examines a small portion of the nodes explored by the GLBbased algorithm. 1 Introduction The quadratic assignment problem (QAP), first proposed by Koopmans and Beckmann [16], can be stated as min p2\Pi n X i=1 n X j=1 a ij b p(i)p(j) ; To appear in Proceedings of State of the Art in Global Opti...
Parallel BestFirst BranchandBound in Discrete Optimization: a Framework
 IN SOLVING COMBINATORIAL OPTIMIZATION PROBLEMS IN PARALLEL
, 1995
"... In discrete optimization problems, we search for an optimal solution among all vectors in a discrete solution space that satisfy a set of constraints, and the search efficiency is of considerable importance since exhaustive search is often impracticable. The method called branchandbound (noted B&a ..."
Abstract

Cited by 8 (1 self)
 Add to MetaCart
In discrete optimization problems, we search for an optimal solution among all vectors in a discrete solution space that satisfy a set of constraints, and the search efficiency is of considerable importance since exhaustive search is often impracticable. The method called branchandbound (noted B&B) is a heuristic tree search algorithm used in this context. Its principle lies in successive decompositions of the original problem in smaller disjoint subproblems until an optimal solution is found, and the search avoids visiting some subproblems which are known not to contain an optimal solution. Given that disjoint subproblems can be decomposed simultaneously and independently, parallel processing has been widely considered as an additional source of improvement in search efficiency, using the set of processors to concurrently decompose several subproblems at each iteration. Parallel B&B is traditionally considered as an irregular parallel algorithm due to the fact that the structure o...
On the Biquadratic Assignment Problem
 in Quadratic Assignment and Related Problems
, 1993
"... Motivated by a problem arising in VLSI synthesis we introduce the biquadratic assignment problem (BQAP) which generalizes the wellknown quadratic assignment problem (QAP). The BQAP is to minimize a weighted sum of products of four variables subject to assignment constraints on the variables. We give ..."
Abstract

Cited by 8 (2 self)
 Add to MetaCart
Motivated by a problem arising in VLSI synthesis we introduce the biquadratic assignment problem (BQAP) which generalizes the wellknown quadratic assignment problem (QAP). The BQAP is to minimize a weighted sum of products of four variables subject to assignment constraints on the variables. We give two integer programming formulations for the problem and design lower bounds for the optimal solution value. These lower bounds are tested computationally on BQAP instances with known objective function value. Finally the asymptotic behaviour of BQAPs is analyzed. It turns out that the ratio between the best and the worst objective function values tends in probability to one when the size of the problem tends to infinity.