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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 37 (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.
Selected Topics on Assignment Problems
, 1999
"... We survey recent developments in the fields of bipartite matchings, linear sum assignment and bottleneck assignment problems and applications, multidimensional assignment problems, quadratic assignment problems, in particular lower bounds, special cases and asymptotic results, biquadratic and co ..."
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Cited by 34 (1 self)
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We survey recent developments in the fields of bipartite matchings, linear sum assignment and bottleneck assignment problems and applications, multidimensional assignment problems, quadratic assignment problems, in particular lower bounds, special cases and asymptotic results, biquadratic and communication assignment problems.
Solving liftandproject relaxations of binary integer programs
 SIAM Journal on Optimization
"... Abstract. We propose a method for optimizing the liftandproject relaxations of binary integer programs introduced by Lovász and Schrijver. In particular, we study both linear and semidefinite relaxations. The key idea is a restructuring of the relaxations, which isolates the complicating constrain ..."
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Cited by 26 (1 self)
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Abstract. We propose a method for optimizing the liftandproject relaxations of binary integer programs introduced by Lovász and Schrijver. In particular, we study both linear and semidefinite relaxations. The key idea is a restructuring of the relaxations, which isolates the complicating constraints and allows for a Lagrangian approach. We detail an enhanced subgradient method and discuss its efficient implementation. Computational results illustrate that our algorithm produces tight bounds more quickly than stateoftheart linear and semidefinite solvers.
Solving Quadratic Assignment Problems Using Convex Quadratic Programming Relaxations
, 2000
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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 LowDimensional Semidefinite Relaxation for the Quadratic Assignment Problem
"... doi 10.1287/moor.1090.0419 ..."
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A Matrixlifting Semidefinite Relaxation for the Quadratic Assignment Problem
, 2006
"... The quadratic assignment problem (QAP) is arguably one of the hardest of the NPhard discrete optimization problems. Problems of dimension greater than 20 are considered to be large scale. Current successful solution techniques depend on branch and bound methods. However, it is di#cult to get strong ..."
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Cited by 7 (0 self)
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The quadratic assignment problem (QAP) is arguably one of the hardest of the NPhard discrete optimization problems. Problems of dimension greater than 20 are considered to be large scale. Current successful solution techniques depend on branch and bound methods. However, it is di#cult to get strong and inexpensive bounds. In this paper we introduce a new semidefinite programming (SDP) relaxation for generating lower bounds for the QAP. The bound exploits the matrix structure of QAP and uses O(n²) variables, a much smaller dimension than other current SDP relaxations, and the same order of dimension as the original QAP . We compare this method with several other computationally inexpensive bounds such as the convex quadratic programming relaxation (QPB). We find that our method provides stronger bounds on average and is adaptable for branch and bound methods.
A Level3 ReformulationLinearization Technique Based Bound for the Quadratic Assignment Problem
"... We apply the level3 Reformulation Linearization Technique (RLT3) to the Quadratic Assignment Problem (QAP). We then present our experience in calculating lower bounds using an essentially new algorithm, based on this RLT3 formulation. This algorithm is not guaranteed to calculate the RLT3 lower bou ..."
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Cited by 6 (0 self)
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We apply the level3 Reformulation Linearization Technique (RLT3) to the Quadratic Assignment Problem (QAP). We then present our experience in calculating lower bounds using an essentially new algorithm, based on this RLT3 formulation. This algorithm is not guaranteed to calculate the RLT3 lower bound exactly, but approximates it very closely and reaches it in some instances. For Nugent problem instances up to size 24, our RLT3based lower bound calculation solves these problem instances exactly or serves to verify the optimal value. Calculating lower bounds for problems sizes larger than size 25 still presents a challenge due to the large memory needed to implement the RLT3 formulation. Our presentation emphasizes the steps taken to significantly conserve memory by using the numerous problem symmetries in the RLT3 formulation of the QAP.
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
Graph Modeling for Quadratic Assignment Problem Associated with the Hypercube
, 2007
"... Abstract. In the paper we consider the quadratic assignment problem arising from channel coding in communications where one coefficient matrix is the adjacency matrix of a hypercube in a finite dimensional space. By using the geometric structure of the hypercube, we first show that there exist at le ..."
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Cited by 4 (3 self)
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Abstract. In the paper we consider the quadratic assignment problem arising from channel coding in communications where one coefficient matrix is the adjacency matrix of a hypercube in a finite dimensional space. By using the geometric structure of the hypercube, we first show that there exist at least n different optimal solutions to the underlying QAPs. Moreover, the inherent symmetries in the associated hypercube allow us to obtain partial information regarding the optimal solutions and thus shrink the search space and improve all the existing QAP solvers for the underlying QAPs. Secondly, we use graph modeling technique to derive a new integer linear program (ILP) models for the underlying QAPs. The new ILP model has n(n − 1) binary variables and O(n 3 log(n)) linear constraints. This yields the smallest known number of binary variables for the ILP reformulation of QAPs. Various relaxations of the new ILP model are obtained based on the graphical characterization of the hypercube, and the lower bounds provided by the LP relaxations of the new model are analyzed and compared with what provided by several classical LP relaxations of QAPs in the literature.