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.

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...

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.

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 large-scale 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 KSR-1 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 N-adic 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...

. Quadratic Assignment problems are in practice among the most difficult to solve in the class of NP-complete problems. The only successful approach hitherto has been Branch-andBound -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 non-vectorizable computations on integers. In this paper we investigate the combination of one of the best bound functions for a Branchand -Bound algorithm (the Gilmore-Lawler bound) and various testing, variable binding and recalculation of bounds between branchings when used in a parallel Branch-and-Bo...

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 branch-and-bound 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.

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...

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 Gilmore-Lawler lower bound (GLB) instead of the LP-based bound. The LP-based algorithm examines a small portion of the nodes explored by the GLB-based 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...

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 branch-andbound (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...

. General quadratic matrix minimization problems, with orthogonal constraints, arise in continuous relaxations for the (discrete) quadratic assignment problem (QAP). Currently, bounds for QAP are obtained by treating the quadratic and linear parts of the objective function, of the relaxations, separately. This paper handles general objectives as one function. The objectives can be both nonhomogeneous and nonconvex. The constraints are orthogonal or Loewner partial order (positive semidefinite) constraints. Comparisons are made to standard trust region subproblems. Numerical results are obtained using a parametric eigenvalue technique. Contents 1. Introduction 1 2. Preliminary Notations and Motivation 2 2.1. Notations 2.2. A Survey on Eigenvalue Bounds for the QAP 2.3. Loewner Partial Order 3. Optimality Conditions 6 3.1. First Order Conditions 3.2. Second Order Conditions 1991 Mathematics Subject Classification. Primary 90B80, 90C20, 90C35, 90C27; Secondary 65H20, 65K05. Key words...