## A New Lower Bound via Projection for the Quadratic Assignment Problem (1992)

Venue: | Mathematics of Operations Research |

Citations: | 51 - 16 self |

### BibTeX

@ARTICLE{Hadley92anew,

author = {S.W. Hadley and F. Rendl and H. Wolkowicz},

title = {A New Lower Bound via Projection for the Quadratic Assignment Problem},

journal = {Mathematics of Operations Research},

year = {1992},

volume = {17},

pages = {727--739}

}

### Years of Citing Articles

### OpenURL

### Abstract

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...

### Citations

98 |
The quadratic assignment problem
- Lawler
- 1963
(Show Context)
Citation Context ... number of nodes in the branching tree becomes excessively large. All these approaches use, as a basic bounding procedure for the QAP, a technique proposed by Gilmore and Lawler in the early sixties, =-=[6, 8]-=-. This bounding technique is combinatorial in nature and requires the solution of a LAP, besides sorting the rows of both A and B . Table 5.1 contains the Gilmore-Lawler-Bound, denoted GLB, for QAPs o... |

82 |
An Experimental Comparison of Techniques for the Assignment of Facilities to Locations
- Vollman, Ruml
- 1968
(Show Context)
Citation Context ...xperiments and Discussion We conclude with some numerical experiments comparing the new bound PB with several existing bounding strategies. The first group of QAP instances in Table 5.1 is taken from =-=[9]-=-. These data are commonly used in the literature. The second group of instances, all of size n = 10 , is taken from [3]. According to the authors, these problems were generated as follows: A is symmet... |

78 |
Optimal and Suboptimal Algorithms for the Quadratic Assignment Problem
- Gilmore
- 1962
(Show Context)
Citation Context ... number of nodes in the branching tree becomes excessively large. All these approaches use, as a basic bounding procedure for the QAP, a technique proposed by Gilmore and Lawler in the early sixties, =-=[6, 8]-=-. This bounding technique is combinatorial in nature and requires the solution of a LAP, besides sorting the rows of both A and B . Table 5.1 contains the Gilmore-Lawler-Bound, denoted GLB, for QAPs o... |

51 |
Quadratic Assignment Problems
- Finke, Burkard, et al.
- 1987
(Show Context)
Citation Context ...is indicates that progress to solve larger QAPs is unlikely unless stronger bounding rules are used. Recently a lower bound for symmetric QAPs was introduced based on the eigenvalues of A and B , see =-=[4, 11]. The basi-=-c idea to derive this bound consists in minimizing f(X) over orthogonal rather than just permutation matrices. In the present paper this "orthogonal relaxation" of the QAP will be further im... |

39 |
Applications of parametric programming and eigenvalue maximization to the quadratic assignment problem
- Rendl, Wolkowicz
- 1992
(Show Context)
Citation Context ...is indicates that progress to solve larger QAPs is unlikely unless stronger bounding rules are used. Recently a lower bound for symmetric QAPs was introduced based on the eigenvalues of A and B , see =-=[4, 11]. The basi-=-c idea to derive this bound consists in minimizing f(X) over orthogonal rather than just permutation matrices. In the present paper this "orthogonal relaxation" of the QAP will be further im... |

33 |
Assignment and Matching Problems: Solution Methods with Fortran Programs. Volume 184
- Burkard, Derigs
- 1980
(Show Context)
Citation Context ...tational effort to solve QAP is very likely to grow exponentially with the problem size. There are several Branch and Bound based solution procedures described in the literature to solve the QAP, see =-=[2, 10, 12]-=-. All these approaches seem to break down on problems of sizes around n = 15 , because the number of nodes in the branching tree becomes excessively large. All these approaches use, as a basic boundin... |

31 |
Locations with spatial interactions: the quadratic assignment problem
- BURKARD
- 1991
(Show Context)
Citation Context ...t. 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 permutation matrices. A , B and C are given (real) matrices defining the QAP. (Througho... |

27 |
A New Lower Bound for the Quadratic Assignment Problem
- Carraresi, Malucelli
- 1992
(Show Context)
Citation Context ...bounding strategies. The first group of QAP instances in Table 5.1 is taken from [9]. These data are commonly used in the literature. The second group of instances, all of size n = 10 , is taken from =-=[3]-=-. According to the authors, these problems were generated as follows: A is symmetric with entries drawn uniformly from the integers 0; 1; : : : ; 10 ; matrix C also has entries drawn at random from 0;... |

26 |
A parallel algorithm for the quadratic assignment problem
- PM, Crouse
- 1989
(Show Context)
Citation Context ...tational effort to solve QAP is very likely to grow exponentially with the problem size. There are several Branch and Bound based solution procedures described in the literature to solve the QAP, see =-=[2, 10, 12]-=-. All these approaches seem to break down on problems of sizes around n = 15 , because the number of nodes in the branching tree becomes excessively large. All these approaches use, as a basic boundin... |

22 | On the quadratic assignment problem - Frieze, Yadegar - 1983 |

19 | A parallel branch and bound algorithm for the quadratic assignment problem, Discrete Applied Mathematics 18 - Roucairol - 1987 |

11 | Symmetrization of nonsymmetric quadratic assignment problems and the Hoffman-Wielandt inequality, Linear Algebra Appl
- Hadley, Rendl, et al.
- 1992
(Show Context)
Citation Context ... matrices A and B are symmetric. From now on we consider only symmetric QAPs and observe that it is possible to transform an arbitrary QAP into an Hermitian QAP with (complex) Hermitian A and B , see =-=[7]-=-. For the sake of simplicity of presentation we consider just the real symmetric case, even though the results carry over also to the Hermitian case, but become more complicated. The following notatio... |