A max-flow approach to improved lower bounds for quadratic unconstrained binary optimization (QUBO)

Boros, Endre; Hammer, Peter L.; Sun, Richard; Tavares, Gabriel

doi:10.1016/j.disopt.2007.02.001

Cited by 43 publications

(40 citation statements)

References 30 publications

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“…Among them are several exact methods using branch and bound or branch and cut (see, e.g., [6,17,30]). Due to the fact that the exact methods become prohibitively expensive to apply for solving large instances, various metaheuristic algorithms have been extensively used to find high-quality solutions to large UBQP instances in an acceptable time.…”

Section: Introductionmentioning

confidence: 99%

Probabilistic GRASP-Tabu Search algorithms for the UBQP problem

Wang¹,

Lu²,

Glover³

et al. 2013

Computers & Operations Research

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This paper presents two algorithms combining GRASP and Tabu Search for solving the Unconstrained Binary Quadratic Programming (UBQP) problem. We first propose a simple GRASP-Tabu Search algorithm working with a single solution and then reinforce it by introducing a population management strategy. Both algorithms are based on a dedicated randomized greedy construction heuristic and a tabu search procedure. We show extensive computational results on two sets of 31 large random UBQP instances and one set of 54 structured instances derived from the MaxCut problem. Comparisons with state-of-the-art algorithms demonstrate the efficacy of our proposed algorithms in terms of both solution quality and computational efficiency. It is noteworthy that the reinforced GRASP-Tabu Search algorithm is able to improve the previous best known results for 19 MaxCut instances.

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Section: Introductionmentioning

confidence: 99%

Probabilistic GRASP-Tabu Search algorithms for the UBQP problem

Wang¹,

Lu²,

Glover³

et al. 2013

Computers & Operations Research

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show abstract

“…Bounds of the beasley and some of the bqpgka data sets have also been calculated by Boros, Hammer, Sun, and Tavares [13]. They use the so-called iterated roof dual as bounding routine and report that, applying this bound computation in a Branch-and-Bound framework, they could solve the four sparsest instances of the bqpgka-d data sets and two of the instances beasley-250.…”

Section: Beasleymentioning

confidence: 99%

“…We are going to use the semidefinite relaxation (13). Instead of solving this relaxation with a limited number of inequality constraints by interior point methods, as done in [27], we use the bundle approach suggested in [19].…”

Section: Further Approachesmentioning

confidence: 99%

Solving Max-Cut to optimality by intersecting semidefinite and polyhedral relaxations

2008

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We present a method for finding exact solutions of Max-Cut, the problem of finding a cut of maximum weight in a weighted graph. We use a Branch-and-Bound setting, that applies a dynamic version of the bundle method as bounding procedure. This approach uses Lagrangian duality to obtain a "nearly optimal" solution of the basic semidefinite Max-Cut relaxation, strengthened by triangle inequalities. The expensive part of our bounding procedure is solving the basic semidefinite relaxation of the Max-Cut problem, which has to be done several times during the bounding process.We review other solution approaches and compare the numerical results with our method. We also extend our experiments to instances of unconstrained quadratic 0-1 optimization and to instances of the graph equipartition problem.The experiments show, that our method nearly always outperforms all other approaches. In particular, for dense graphs, where linear programming based methods fail, our method performs very well. Exact solutions are obtained in a reasonable time for any instance of size up to n = 100, independent of the density. For some problems of special structure we can solve even larger problem classes. We could prove optimality for several problems of the literature where, to the best of our knowledge, no other method is able to do so. The Max-Cut ProblemThe Max-Cut problem is one of the basic NP-hard combinatorial optimization problems and has attracted scientific interest from both the discrete (see, e.g.,

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“…The approach is iterative and is guaranteed to produce at least the lower bound of the roof dual. A similar approach was developed in [14] for the special case of quadratic pseudo-boolean functions.…”

Section: Introductionmentioning

confidence: 99%

Tighter Relaxations for Higher-Order Models Based on Generalized Roof Duality

Fredriksson

Strandmark

Kahl

2012

Computer Vision – ECCV 2012. Workshops and Demonstrations

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Abstract. Many problems in computer vision can be turned into a large-scale boolean optimization problem, which is in general NP-hard. In this paper, we further develop one of the most successful approaches, namely roof duality, for approximately solving such problems for higherorder models. Two new methods that can be applied independently or in combination are investigated. The first one is based on constructing relaxations using generators of the submodular function cone. In the second method, it is shown that the roof dual bound can be applied in an iterated way in order to obtain a tighter relaxation. We also provide experimental results that demonstrate better performance with respect to the state-of-the-art, both in terms of improved bounds and the number of optimally assigned variables.

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A max-flow approach to improved lower bounds for quadratic unconstrained binary optimization (QUBO)

Cited by 43 publications

References 30 publications

Probabilistic GRASP-Tabu Search algorithms for the UBQP problem

Probabilistic GRASP-Tabu Search algorithms for the UBQP problem

Solving Max-Cut to optimality by intersecting semidefinite and polyhedral relaxations

Tighter Relaxations for Higher-Order Models Based on Generalized Roof Duality

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