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Combinatorial Structure of Polytopesassociated to Fuzzy Measures

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UNIVERSIDAD COMPLUTENSE DE MADRID FACULTAD DE CIENCIAS MATEMÁTICAS TESIS DOCTORAL Combinatorial Structure of Polytopesassociated to Fuzzy Measures Estructura Combinatoria de Politopos asociados a Medidas Difusas MEMORIA PARA OPTAR AL GRADO DE DOCTOR PRESENTADA POR Pedro García Segador Director Pedro Miranda Menéndez Madrid © Pedro García Segador, 2020 UNIVERSIDAD COMPLUTENSE DE MADRID FACULTAD DE CIENCIAS MATEMATICAS´ TESIS DOCTORAL Combinatorial Structure of Polytopes associated to Fuzzy Measures Estructura Combinatoria de Politopos asociados a Medidas Difusas MEMORIA PARA OPTAR AL GRADO DE DOCTOR PRESENTADA POR Pedro Garc´ıaSegador DIRECTOR Pedro Miranda Men´endez Programa de Doctorado en Ingenier´ıaMatem´atica, Estad´ısticae Investigaci´onOperativa por la Universidad Complutense de Madrid y la Universidad Polit´ecnicade Madrid Combinatorial Structure of Polytopes associated to Fuzzy Measures Estructura Combinatoria de Politopos asociados a Medidas Difusas Tesis Doctoral Pedro Garc´ıaSegador Director Pedro Miranda Men´endez A~no2020 \Es ist wahr, ein Mathematiker, der nicht etwas Poet ist, wird nimmer ein vollkommener Mathematiker sein." \It is true that a mathematician who is not somewhat of a poet, will never be a perfect mathematician." Karl Weierstraß ABSTRACT Combinatorial Structure of Polytopes associated to Fuzzy Measures by PEDRO GARC´IA SEGADOR Abstract in English This PhD thesis is devoted to the study of geometric and combinatorial aspects of polytopes associated to fuzzy measures. Fuzzy measures are an essential tool, since they generalize the concept of probability. This greater generality allows applications to be developed in various fields, from the Decision Theory to the Game Theory. The set formed by all fuzzy measures on a referential set is a polytope. In the same way, many of the most relevant subfamilies of fuzzy measures are also polytopes. Studying the combinatorial structure of these polytopes arises as a natural problem that allows us to better understand the properties of the associated fuzzy measures. Knowing the combinatorial structure of these polytopes helps us to develop algorithms to generate points uniformly at random inside these polytopes. Generating points uniform- ly inside a polytope is a complex problem from both a theoretical and a computational point of view. Having algorithms that allow us to sample uniformly in polytopes associ- ated to fuzzy measures allows us to solve many problems, among them the identification problem, i.e. estimate the fuzzy measure that underlies an observed data set. Many of these polytopes associated with subfamilies of fuzzy measures are order poly- topes and their combinatorial structure depends on a partially ordered set (poset). In this way we can transform problems of a geometric nature into problems of Combina- torics and Order Theory. In this thesis, we start by introducing the most important results about posets and order polytopes. Next, we focus on the problem of generating uniformly at random linear extensions in a poset. To this end, we have developed a method, which we have called Bottom-Up, which allows generating linear extensions for several posets quickly and easily. The main disadvantage of this method is that it is not applicable to every poset. The posets for which we can apply Bottom-Up are called BU-feasibles. For this reason we study another method that we call Ideal-based method. This method is more general than Bottom-Up and is applicable to any poset, however its computational cost is much higher. Among the posets that are not BU-feasible there are important cases that deserve an individualized treatment, as is the case of the Ferrers posets. Ferrers posets are asso- ciated to 2-symmetric measures. In this thesis, we study in detail the geometric and combinatorial properties of the 2-symmetric measures. For this purpose we make use of Young diagrams, which are well-known objects in combinatorics. Young diagrams allow us to develop algorithms to generate uniformly at random 2-symmetric measures. Next, we study the subfamily of 2-additive measures. The polytopes in this subfamily are not order polytopes, and therefore we cannot apply the techniques we have developed related to posets. As we have done with 2-symmetric measures, we study the faces of these polytopes and develop a triangulation that allows us to uniformly sample 2-additive measures. Monotone games emerge as the non-normalized version of fuzzy measures. As they are non-normalized, the associated polyhedra are not bounded and do not have a polytope structure. In this case these sets form convex cones whose structure also depends on a poset. We have called them order cones. In this thesis, we explain the link between order cones and order polytopes. We also solve the problem of obtaining the extreme rays of the set of monotone games using order cones. Later in the thesis we study integration techniques to count linear extensions. These techniques can be applied to know the volume of the associated order polytopes. In the same way, it allows us to solve specific combinatorial problems such as counting the number of 2-alternating permutations. Finally, the thesis ends with a chapter on conclusions and open problems. It is con- cluded that, in view of the results obtained in the thesis, there is a very close link between Combinatorics, mainly Order Theory, and the geometry of the polytopes asso- ciated with fuzzy measures. The more we improve our knowledge about the underlying combinatorics, the better we will understand these polytopes. Resumen en Espa~nol La presente tesis doctoral est´adedicada al estudio de distintas propiedades geom´etricasy combinatorias de politopos de medidas difusas. Las medidas difusas son una herramienta esencial puesto que generalizan el concepto de probabilidad. Esta mayor generalidad permite desarrollar aplicaciones en diversos campos, desde la Teor´ıade la Decisi´ona la Teor´ıade Juegos. El conjunto formado por todas las medidas difusas sobre un referencial tiene estructura de politopo. De la misma forma, la mayor´ıade las subfamilias m´asrelevantes de medidas difusas son tambi´enpolitopos. Estudiar la estructura combinatoria de estos politopos surge como un problema natural que nos permite comprender mejor las propiedades de las medidas difusas asociadas. Conocer la estructura combinatoria de estos politopos tambi´ennos ayuda a desarrollar algoritmos para generar aleatoria y uniformemente puntos dentro de estos politopos. Generar puntos de forma uniforme dentro de un politopo es un problema complejo desde el punto de vista tanto te´oricocomo computacional. Disponer de algoritmos que nos permitan generar uniformemente en politopos asociados a medidas difusas nos permite resolver muchos problemas, entre ellos el problema de identificaci´onque trata de estimar la medida difusa que subyace a un conjunto de datos observado. Muchos de los politopos asociados a subfamilias de medidas difusas son politopos de orden y su estructura combinatoria depende de un orden parcial (poset). De esta forma podemos transformar problemas de naturaleza geom´etricaen problemas de Combinato- ria y Teor´ıadel Orden. En esta tesis, empezamos introduciendo los resultados m´asimportantes sobre posets y politopos de orden. A continuaci´onnos centramos en el problema de generar extensiones lineales de forma aleatoria para un poset. A tal fin hemos desarrollado un m´etodo, al que hemos llamado Bottom-Up, que permite generar extensiones lineales para diversos posets de forma r´apiday sencilla. La principal desventaja de este m´etodo es que no es aplicable a cualquier poset. A los posets que se les puede aplicar este m´etodo los llamamos BU- factibles. Por este motivo estudiamos otro m´etodo al que llamamos m´etodo basado en ideales (Ideal-based method). Este m´etodo es m´asgeneral que el Bottom-Up y es aplicable a cualquier poset. Sin embargo, su coste computacional es mucho mayor. Dentro de los posets que no son BU-factibles hay casos importantes que merecen un estudio individualizado, como es el caso de los posets de Ferrers. Los posets de Ferrers est´anasociados a las medidas 2-sim´etricas. En esta tesis estudiamos en detalle las propiedades geom´etricasy combinatorias de las medidas 2-sim´etricas.Para ello hacemos uso de unos objetos muy conocidos en Combinatoria, que son los diagramas de Young. Estos objetos nos permiten desarrollar algoritmos para generar aleatoriamente medidas 2-sim´etricasde forma uniforme. A continuaci´onestudiamos la subfamilia de medidas 2-aditivas. Los politopos de esta subfamilia no son politopos de orden, y por tanto no podemos aplicar las t´ecnicas que hemos desarrollado relacionadas con posets. Al igual que hemos hecho con las medidas 2-sim´etricas,estudiamos las caras de estos politopos y desarrollamos una triangulaci´on que nos permite simular uniformemente medidas 2-aditivas. Los juegos mon´otonossurgen como la versi´onno normalizada de las medidas difusas. Al no estar normalizados, los poliedros asociados no est´anacotados y no tienen estructura de politopo. En este caso estos conjuntos forman conos convexos cuya estructura tambi´en depende de un poset. A estos conos los hemos llamado conos de orden (Order Cones). En la tesis explicamos la relaci´onexistente entre conos de orden y politopos de orden. Tambi´enresolvemos el problema de obtener los rayos extremos de los juegos mon´otonos usando resultados de conos de orden. Posteriormente en la tesis estudiamos t´ecnicasde integraci´onpara contar extensiones lineales. Estas t´ecnicasse pueden aplicar para conocer el volumen de los politopos de orden asociados. De la misma forma permiten resolver problemas concretos de Combi- natoria como el de contar el n´umerode permutaciones 2-alternantes. Finalmente la tesis acaba con un cap´ıtulosobre conclusiones y problemas abiertos. Se concluye que, a la vista de los resultados obtenidos en la tesis, hay una relaci´onmuy estrecha entre la Combinatoria, principalmente aquella asociada a la Teor´ıadel Orden, y la geometr´ıade los politopos asociados a medidas difusas. Seg´unpodamos avanzar en nuestros conocimientos te´oricossobre estos objetos combinatorios podremos conocer mejor estos politopos.
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  • Sorting and a Tale of Two Polytopes

    Sorting and a Tale of Two Polytopes

    Sorting and a Tale of Two Polytopes Jean Cardinal ULB, Brussels, Belgium Algorithms & Permutations, Paris, 2012 Sorting by Comparisons Input: a set V , totally ordered by an unknown order 6 Goal: Discover 6 by making queries \is x 6 y?", for some x; y V 2 Objective function: #queries I Calssical problem in algorithms I Θ( V log V ) queries necessary and sufficient (Heap Sort, Mergej j Sort)j j Sorting by Comparisons under Partial Information Input: I a set V , totally ordered by an unknown order 6 I a partial order P = (V ; 6P ) compatible with 6 Goal: Discover 6 by making queries \is x 6 y?", for some x; y V 2 Objective function: #queries Sorting by Comparisons under Partial Information Input: I a set V , totally ordered by an unknown order 6 I a partial order P = (V ; 6P ) compatible with 6 Goal: Discover 6 by making queries \is x 6 y?", for some x; y V 2 Objective function: #queries Sorting by Comparisons under Partial Information Input: I a set V , totally ordered by an unknown order 6 I a partial order P = (V ; 6P ) compatible with 6 Goal: Discover 6 by making queries \is x 6 y?", for some x; y V 2 Objective function: #queries Sorting by Comparisons under Partial Information Input: I a set V , totally ordered by an unknown order 6 I a partial order P = (V ; 6P ) compatible with 6 Goal: Discover 6 by making queries \is x 6 y?", for some x; y V 2 Objective function: #queries OR ? Sorting by Comparisons under Partial Information Input: I a set V , totally ordered by an unknown order 6 I a partial order P = (V ; 6P ) compatible with 6