The Expressive Power of Binary Submodular Functions

Živný, Stanislav; Cohen, David A.; Jeavons, Peter

doi:10.1007/978-3-642-03816-7_63

Cited by 6 publications

(5 citation statements)

References 49 publications

(42 reference statements)

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“…Similar properties have been studied by Del Pia and Khajavirad [48], who considered cases in which the configuration of the monomials of the hypergraph allows the computation of the convex hull P * S L by decomposing the set of monomials into simpler subsets. The submodularity property mentioned in the previous subsection is also an aspect of the structure of pseudo-Boolean functions, but as proved byŽivný, Cohen and Jeavons [103] it cannot always be carried over to quadratic reformulations, even for functions of degree four.…”

Section: Introduction To Quadratizationsmentioning

confidence: 95%

“…However, it is difficult to identify which submodular functions can be quadratized keeping this property. In fact,Živný, Cohen and Jeavons [103] proved that there exist submodular pseudo-Boolean functions of degree four that do not admit a submodular quadratization. Exploiting structural properties of multilinear polynomials Another interesting property of quadratic reformulations is their ability of better exploiting structural properties of the original multilinear polynomials and the underlying applications.…”

Section: Introduction To Quadratizationsmentioning

confidence: 99%

See 1 more Smart Citation

Linear and quadratic reformulations of nonlinear optimization problems in binary variables

Rodríguez-Heck¹

2018

4OR-Q J Oper Res

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Section: Introduction To Quadratizationsmentioning

confidence: 95%

Section: Introduction To Quadratizationsmentioning

confidence: 99%

Linear and quadratic reformulations of nonlinear optimization problems in binary variables

Rodríguez-Heck¹

2018

4OR-Q J Oper Res

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“…In a related but different direction, (Cohen, Jeavons, and Živný 2008) studied which valued constraint languages can be transformed to binary valued constraint languages over the same domain. It was shown in ( Živný, Cohen, and Jeavons 2009) that there are submodular valued constraint languages which cannot be expressed (using min and sum) by binary submodular languages over the same domain. We call D the domain, the elements of D labels (for variables), and we say that the weighted relations in Φ D take values (which are elements of Q).…”

Section: Introductionmentioning

confidence: 99%

Binarisation via Dualisation for Valued Constraints

Cohen

Cooper

Jeavons

et al. 2015

AAAI

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Constraint programming is a natural paradigm for many combinatorial optimisation problems. The complexity of constraint satisfaction for various forms of constraints has been widely-studied, both to inform the choice of appropriate algorithms, and to understand better the boundary between polynomial-time complexity and NP-hardness. In constraint programming it is well-known that any constraint satisfaction problem can be converted to an equivalent binary problem using the so-called dual encoding. Using this standard approach any fixed collection of constraints, of arbitrary arity, can be converted to an equivalent set of constraints of arity at most two. Here we show that this transformation, although it changes the domain of the constraints, preserves all the relevant algebraic properties that determine the complexity. Moreover, we show that the dual encoding preserves many of the key algorithmic properties of the original instance. We also show that this remains true for more general valued constraint languages, where constraints may assign different cost values to different assignments. Hence, we obtain a simple proof of the fact that to classify the computational complexity of all valued constraint languages it suffices to classify only binary valued constraint languages.

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“…En 2008 Živnỳ et al [153,154] demuestran que toda función 2-monótona es representable, al comprobar que la siguiente energía es un gadget:…”

Section: Funciones 2-monótonasunclassified

“…La caracterización de las energías (potenciales) dependientes de cuatro variables que son representables se debe a Živnỳ et al [153]: son las funciones submodulares que satisfacen ciertas condiciones denominadas condiciones de representabilidad. Dichas condiciones se traducen en seis restricciones que deben cumplir los coeficientes del desarrollo polinomial de la energía.…”

Section: Potenciales De Orden Cuatrounclassified

Optimización de una energía mediante cortes de grafos. Segmentación de imágenes

Puente¹

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The image segmentation can be described as the problem of minimizing a discrete energy. We face two problems: first, to define an energy whose minimum provides the desired segmentation and, second, once the energy is defined we must find its global minimum. The first part of this thesis addresses the second problem, and the second part, in a more applied context, the first problem.Minimization techniques based on graph cuts find the minimum of a discrete energy in polynomial time via min-cut/max-flow algorithms. Nevertheless, these techniques can only be applied to graph-representable energies. An important challenge is to study which energies are graph-representable and to construct graphs which represent these energies. This is the same as finding a gadget function with additional variables. In the first part there are studied the properties of gadget functions which allow the number of additional variables to be bounded from above. Moreover, the graph-representable energies with four variables are characterised and gadgets with two additional variables are defined for these.The second part addresses the application of these ideas to medical image segmentation. This is often the first step in computer-assisted diagnosis and monitoring therapy. Multiatlas segmentation is a powerful automatic segmentation technique for medical images, with three important aspects that are highlighted here: the registration between the atlas and the target image, the atlas selection, and the label fusion method. We formulate the label fusion method as a minimization problem and we introduce two new graph-representable energies. The first is a second order energy and it is used for the segmentation of the liver in computed tomography (CT) images. The second energy is a higher order energy and it is used for the segmentation of the hippocampus in magnetic resonance images (MRI). III AgradecimientosDeseo expresar mi sincero agradecimiento a mi director de tesis, Carlos Platero Dueñas, por orientarme, ayudarme y estimularme en todos los aspectos de la tesis. Agradecerle la confianza que siempre me ha demostrado, así como la dedicación y la atención que en todo momento me ha ofrecido.Mi más profundo agradecimiento a Pedro Gonzalez Manchón. Sin su ayuda, paciencia e interés, esta tesis difícilmente habría llegado a concluirse en la forma que hoy tiene.Quisiera dar las gracias a mis compañeros de departamento, y muy especialmente a mis compañeros del extinguido grupo de Bioingeniería Aplicada de la UPM, por su apoyo y disponibilidad durante todos estos años.Por último, dar las gracias a mis amigos que de un modo u otro han respaldado este esfuerzo, y a mi familia por su apoyo incondicional.

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The Expressive Power of Binary Submodular Functions

Cited by 6 publications

References 49 publications

Linear and quadratic reformulations of nonlinear optimization problems in binary variables

Linear and quadratic reformulations of nonlinear optimization problems in binary variables

Binarisation via Dualisation for Valued Constraints

Optimización de una energía mediante cortes de grafos. Segmentación de imágenes

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