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24
Solving Large Quadratic Assignment Problems on Computational Grids
, 2000
"... The quadratic assignment problem (QAP) is among the hardest combinatorial optimization problems. Some instances of size n = 30 have remained unsolved for decades. The solution of these problems requires both improvements in mathematical programming algorithms and the utilization of powerful computat ..."
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Cited by 82 (7 self)
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The quadratic assignment problem (QAP) is among the hardest combinatorial optimization problems. Some instances of size n = 30 have remained unsolved for decades. The solution of these problems requires both improvements in mathematical programming algorithms and the utilization of powerful computational platforms. In this article we describe a novel approach to solve QAPs using a stateoftheart branchandbound algorithm running on a federation of geographically distributed resources known as a computational grid. Solution of QAPs of unprecedented complexity, including the nug30, kra30b, and tho30 instances, is reported.
An Integer Projected Fixed Point Method for Graph Matching and MAP Inference
"... Graph matching and MAP inference are essential problems in computer vision and machine learning. We introduce a novel algorithm that can accommodate both problems and solve them efficiently. Recent graph matching algorithms are based on a general quadratic programming formulation, which takes in con ..."
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Cited by 48 (3 self)
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Graph matching and MAP inference are essential problems in computer vision and machine learning. We introduce a novel algorithm that can accommodate both problems and solve them efficiently. Recent graph matching algorithms are based on a general quadratic programming formulation, which takes in consideration both unary and secondorder terms reflecting the similarities in local appearance as well as in the pairwise geometric relationships between the matched features. This problem is NPhard, therefore most algorithms find approximate solutions by relaxing the original problem. They find the optimal continuous solution of the modified problem, ignoring during optimization the original discrete constraints. Then the continuous solution is quickly binarized at the end, but very little attention is put into this final discretization step. In this paper we argue that the stage in which a discrete solution is found is crucial for good performance. We propose an efficient algorithm, with climbing and convergence properties, that optimizes in the discrete domain the quadratic score, and it gives excellent results either by itself or by starting from the solution returned by any graph matching algorithm. In practice it outperforms stateorthe art graph matching algorithms and it also significantly improves their performance if used in combination. When applied to MAP inference, the algorithm is a parallel extension of Iterated Conditional Modes (ICM) with climbing and convergence properties that make it a compelling alternative to the sequential ICM. In our experiments on MAP inference our algorithm proved its effectiveness by significantly outperforming [13], ICM and MaxProduct Belief Propagation. 1
MultiStart Tabu Search and Diversification Strategies for the Quadratic Assignment Problem
, 2006
"... The quadratic assignment problem (QAP) is a well known combinatorial optimization problem most commonly used to model the facilitylocation problem. The widely acknowledged difficulty of the QAP has made it the focus of many metaheuristic solution approaches. In this study, we introduce several mul ..."
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Cited by 15 (1 self)
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The quadratic assignment problem (QAP) is a well known combinatorial optimization problem most commonly used to model the facilitylocation problem. The widely acknowledged difficulty of the QAP has made it the focus of many metaheuristic solution approaches. In this study, we introduce several multistart tabu search variants and show the benefit of utilizing strategic diversification within the tabu search framework for the QAP. Computational results for a set of problems obtained from QAPLIB demonstrate the ability of our TS multistart variants to improve on the classic tabu search approach that is one of the principal and most widely used methods for the QAP. We also show that our new procedures are highly competitive with the best recently introduced methods from the literature, including more complex hybrid approaches that incorporate a classic tabu search method as a subroutine.
Recent Advances for the Quadratic Assignment Problem with Special Emphasis on Instances that are Difficult for MetaHeuristic Methods
 Annals of OR
, 2005
"... This paper reports heuristic and exact solution advances for the Quadratic Assignment Problem (QAP). QAP instances frequently used in the literature are relatively well solved by heuristic approaches. Indeed, solutions at a fraction of one percent from the best known solution values are rapidly foun ..."
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Cited by 14 (1 self)
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This paper reports heuristic and exact solution advances for the Quadratic Assignment Problem (QAP). QAP instances frequently used in the literature are relatively well solved by heuristic approaches. Indeed, solutions at a fraction of one percent from the best known solution values are rapidly found by most heuristic methods. Exact methods are not able to prove optimality for these instances as soon as the problem size approaches 30 to 40. This article presents new QAP instances that are ill conditioned for many metaheuristicbased methods. However, these new instances are shown to be solved relatively well by some exact methods, since problem instances up to a size of 75 have been exactly solved. Key words: Quadratic assignment problem, local search, branch & bound, benchmarks. 1. Introduction. 1.1 The quadratic assignment problem (QAP). The QAP is a combinatorial optimization problem stated for the first time by Koopmans and Beckmann in 1957. It can be described as follows: Given two n × n matrices (aij) and(bkl), find a permutation ππππ minimizing:
Tree Elaboration Strategies In Branch and Bound Algorithms For Solving the Quadratic Assignment Problem
, 1999
"... This paper presents a new strategy for selecting nodes in a branchandbound algorithm for solving exactly the Quadratic Assignment Problem (QAP). It was developed when it was learned that older strategies failed on the larger size problems. The strategy is a variation of polytomic depthfirst searc ..."
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Cited by 12 (3 self)
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This paper presents a new strategy for selecting nodes in a branchandbound algorithm for solving exactly the Quadratic Assignment Problem (QAP). It was developed when it was learned that older strategies failed on the larger size problems. The strategy is a variation of polytomic depthfirst search of Mautor and Roucairol which extends a node by all assignments of an unassigned facility to unassigned locations based upon the counting of 'forbidden' locations. A forbidden location is one where the addition of the corresponding leader (linear cost) element would increase the lower bound beyond the upper bound. We learned that this fortuitous situation never occurs near the root on Nugent problems larger than 15. One has to make better estimates of the bound if the strategy is to work. We have, therefore, designed and implemented an increasingly improved set of bound calculations. The better of these bound calculations to be utilized near the root and the less accurate (poorer bounds) utilized further into the tree. The result is an effective and powerful technique for shortening the run times of problem instances in the range of size 16 to 25. Run times were decreased generally by five or sixtoone and the number of nodes evaluated was decreased as much as 10toone. Later improvements in our strategy produced a better than 3to1 reduction in runtime so that overall improvement in run time was as high as 20to1 as compared to our earlier results. At the end of our paper, we compare the performance of the four most successful algorithms for exact solution of the QAP.
A Hospital Facility Layout Problem Finally Solved
, 2000
"... This paper presents a history of a difficult facility layout problem that falls into the category of the KoopmansBeckmann variant of the Quadratic Assignment Problem (QAP), wherein 30 facilities are to be assigned to 30 locations. The problem arose in 1972 as part of the design of a German unive ..."
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Cited by 11 (1 self)
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This paper presents a history of a difficult facility layout problem that falls into the category of the KoopmansBeckmann variant of the Quadratic Assignment Problem (QAP), wherein 30 facilities are to be assigned to 30 locations. The problem arose in 1972 as part of the design of a German university hospital, Klinikum Regensburg. This problem, known as the Krarup30a upon its inclusion in the QAPLIB library of QAP instances, has remained an important example of one of the most difficult to solve. In 1999, two approaches provided multiple optimum solutions.
A Study of Quadratic Assignment Problem Instances That Are Difficult for MetaHeuristic Methods
"... The quadratic assignment instances frequently used in the literature are relatively well solved by heuristic approaches. Indeed, solutions at a fraction of one percent from the best known solution values are rapidly found by most heuristic methods. Exact methods are not able to prove optimality for ..."
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Cited by 6 (0 self)
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The quadratic assignment instances frequently used in the literature are relatively well solved by heuristic approaches. Indeed, solutions at a fraction of one percent from the best known solution values are rapidly found by most heuristic methods. Exact methods are not able to prove optimality for these instances as soon as the problem size approaches 30 to 40. This article presents new QAP instances that are ill conditioned for many metaheuristicbased methods. However, these new instances are shown to be solved relatively well by some exact methods since problem instances up to a size of 75 have been exactly solved.
SDP relaxations for some combinatorial optimization problems
, 2010
"... In this chapter we present recent developments on solving various combinatorial optimization problems by using semidefinite programming (SDP). We present several SDP relaxations of the quadratic assignment problem and the traveling salesman problem. Further, we show the equivalence of several known ..."
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Cited by 5 (4 self)
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In this chapter we present recent developments on solving various combinatorial optimization problems by using semidefinite programming (SDP). We present several SDP relaxations of the quadratic assignment problem and the traveling salesman problem. Further, we show the equivalence of several known SDP relaxations of the graph equipartition problem, and present recent results on the bandwidth problem.
The Steinberg Wiring Problem
, 2001
"... this paper was written we learned of a previously unreleased technical report by M. Nystrom [35] that describes the solution of the ste36b/c problems. Nystrom used a distributed B&B algorithm based on the GLB, implemented on 22 200 MHz Pentium Pro CPUs. The serial time to solve the ste36b/c inst ..."
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Cited by 5 (0 self)
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this paper was written we learned of a previously unreleased technical report by M. Nystrom [35] that describes the solution of the ste36b/c problems. Nystrom used a distributed B&B algorithm based on the GLB, implemented on 22 200 MHz Pentium Pro CPUs. The serial time to solve the ste36b/c instances on one of these CPUs is estimated to be approximately 60 days/200 days, respectively. (The time for ste36c is substantially higher because this problem was solved using an initial incumbent value of +1.) "wiring" 2001/12/19 page 13 i i i i i i i i 13 1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 1.E+08 0 3 6 9 12 15 18 21 24 27 30 33 Level Figure 4. Distribution of nodes in solution of ste36a 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0 3 6 9 12 15 18 21 24 27 30 33 Level 0.0 20.0 40.0 60.0 80.0 100.0 120.0 140.0 160.0 180.0 200.0 Rel. Gap Cum. Hrs
Seeded graph matching for correlated ErdösRenyi graphs
, 2014
"... We study the consistency of graph matching for estimating a latent alignment function between the vertex sets of two graphs, as well as subsequent algorithmic implications when the latent alignment is partially observed. In the correlated ErdosRenyi graph setting, we prove that graph matching provi ..."
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Cited by 4 (2 self)
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We study the consistency of graph matching for estimating a latent alignment function between the vertex sets of two graphs, as well as subsequent algorithmic implications when the latent alignment is partially observed. In the correlated ErdosRenyi graph setting, we prove that graph matching provides a strongly consistent estimate of the latent alignment in the presence of even modest correlation. We then investigate a tractable, restrictedfocus version of graph matching, which is only concerned with adjacency involving vertices in a partial observation of the latent alignment; we prove that a logarithmic number of vertices in the known partial latent alignment is sufficient for this restrictedfocus version of graph matching to yield a strongly consistent estimate of the latent alignment of the remaining vertices. Lastly, we show how FrankWolfe methodology for approximate graph matching, when there is a known partial latent alignment, inherently incorporates this restrictedfocus graph matching.