Generalized roof duality

Kahl, Fredrik; Strandmark, Petter

doi:10.1016/j.dam.2012.06.009

Cited by 27 publications

(58 citation statements)

References 34 publications

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“…We propose to circumvent this problem by optimizing over positive linear combinations of a set of generators (or extreme rays). The idea of generators was first introduced in [12]. In this paper, even though the resulting function space is only a subset of all submodular functions, we show for 4th order models that optimizing over this subset yields significantly better results than previously published methods.…”

Section: Introductionmentioning

confidence: 67%

“…In this section we will briefly state some of the results from [12] that serve as a basis for our methods. The basic problem is that of minimizing a pseudo-boolean polynomial f : {0, 1} n → R of degree m. Since this problem is in general NP-hard the following family of relaxations is considered:…”

Section: Generalized Roof Dualitymentioning

confidence: 99%

“…In [20], a roof duality framework is presented based on reduction, but at the same time, the authors note that their roof duality bound depends on which reductions are applied. In contrast our work is based on [10,12] which introduce a framework that works directly on the higher-order potentials.…”

Section: Introductionmentioning

confidence: 99%

“…Specifically, we will consider the quartic case (4th order energies), but in principle our methods can be applied to interactions of any order. Our work builds on the generalized roof duality methods presented in [10][11][12]. Here it is shown that a lower bound on the minimum of an nth order pseudo-boolean function can be found by performing a maximization over a set of nth order symmetric submodular pseudo-boolean functions.…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Introductionmentioning

confidence: 67%

Section: Generalized Roof Dualitymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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

Fredriksson

Strandmark

Kahl

2012

Computer Vision – ECCV 2012. Workshops and Demonstrations

Self Cite

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“…It is related closely to an approximation tequnique previously known as the roof duality. It has been applied to more general approximation called generalized roof duality in optimization and inference [41]. Because variables in a relaxed problem take {0, 1/2, 1}, the LP-IP model and its analyses in this paper are applicable to the relaxation.…”

Section: Summary and Discussionmentioning

confidence: 99%

Statistical mechanical analysis of linear programming relaxation for combinatorial optimization problems

Takabe

Hukushima

2016

Phys. Rev. E

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Typical behavior of the linear programming (LP) problem is studied as a relaxation of the minimum vertex cover, a type of integer programming (IP) problem. A lattice-gas model on the Erdös-Rényi random graphs of α-uniform hyperedges is proposed to express both the LP and IP problems of the min-VC in the common statistical-mechanical model with a one-parameter family. Statisticalmechanical analyses reveal for α = 2 that the LP optimal solution is typically equal to that given by the IP below the critical average degree c = e in the thermodynamic limit. The critical threshold for good accuracy of the relaxation extends the mathematical result c = 1, and coincides with the replica symmetry-breaking threshold of the IP. The LP relaxation for the minimum hitting sets with α ≥ 3, minimum vertex covers on α-uniform random graphs, is also studied. Analytic and numerical results strongly suggest that the LP relaxation fails to estimate optimal values above the critical average degree c = e/(α − 1) where the replica symmetry is broken.

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On the Lifted Multicut Polytope for Trees

Lange

Andres

2021

Lecture Notes in Computer Science

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We study the lifted multicut problem restricted to trees, which is np -hard in general and solvable in polynomial time for paths. In particular, we characterize facets of the lifted multicut polytope for trees defined by the inequalities of a canonical relaxation. Moreover, we present an additional class of inequalities associated with paths that are facet-defining. Taken together, our facets yield a complete totally dual integral description of the lifted multicut polytope for paths. This description establishes a connection to the combinatorial properties of alternative formulations such as sequential set partitioning.

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Generalized roof duality

Cited by 27 publications

References 34 publications

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

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

Statistical mechanical analysis of linear programming relaxation for combinatorial optimization problems

On the Lifted Multicut Polytope for Trees

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