59 research outputs found

    Efficient computation of rank probabilities in posets

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    As the title of this work indicates, the central theme in this work is the computation of rank probabilities of posets. Since the probability space consists of the set of all linear extensions of a given poset equipped with the uniform probability measure, in first instance we develop algorithms to explore this probability space efficiently. We consider in particular the problem of counting the number of linear extensions and the ability to generate extensions uniformly at random. Algorithms based on the lattice of ideals representation of a poset are developed. Since a weak order extension of a poset can be regarded as an order on the equivalence classes of a partition of the given poset not contradicting the underlying order, and thus as a generalization of the concept of a linear extension, algorithms are developed to count and generate weak order extensions uniformly at random as well. However, in order to reduce the inherent complexity of the problem, the cardinalities of the equivalence classes is fixed a priori. Due to the exponential nature of these algorithms this approach is still not always feasible, forcing one to resort to approximative algorithms if this is the case. It is well known that Markov chain Monte Carlo methods can be used to generate linear extensions uniformly at random, but no such approaches have been used to generate weak order extensions. Therefore, an algorithm that can be used to sample weak order extensions uniformly at random is introduced. A monotone assignment of labels to objects from a poset corresponds to the choice of a weak order extension of the poset. Since the random monotone assignment of such labels is a step in the generation process of random monotone data sets, the ability to generate random weak order extensions clearly is of great importance. The contributions from this part therefore prove useful in e.g. the field of supervised classification, where a need for synthetic random monotone data sets is present. The second part focuses on the ranking of the elements of a partially ordered set. Algorithms for the computation of the (mutual) rank probabilities that avoid having to enumerate all linear extensions are suggested and applied to a real-world data set containing pollution data of several regions in Baden-Württemberg (Germany). With the emergence of several initiatives aimed at protecting the environment like the REACH (Registration, Evaluation, Authorisation and Restriction of Chemicals) project of the European Union, the need for objective methods to rank chemicals, regions, etc. on the basis of several criteria still increases. Additionally, an interesting relation between the mutual rank probabilities and the average rank probabilities is proven. The third and last part studies the transitivity properties of the mutual rank probabilities and the closely related linear extension majority cycles or LEM cycles for short. The type of transitivity is translated into the cycle-transitivity framework, which has been tailor-made for characterizing transitivity of reciprocal relations, and is proven to be situated between strong stochastic transitivity and a new type of transitivity called delta*-transitivity. It is shown that the latter type is situated between strong stochastic transitivity and a kind of product transitivity. Furthermore, theoretical upper bounds for the minimum cutting level to avoid LEM cycles are found. Cutting levels for posets on up to 13 elements are obtained experimentally and a theoretic lower bound for the cutting level to avoid LEM cycles of length 4 is computed. The research presented in this work has been published in international peer-reviewed journals and has been presented on international conferences. A Java implementation of several of the algorithms presented in this work, as well as binary files containing all posets on up to 13 elements with LEM cycles, can be downloaded from the website http://www.kermit.ugent.be

    Estructura Combinatoria de Politopos asociados a Medidas Difusas

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    Tesis inédita de la Universidad Complutense de Madrid, Facultad de Ciencias Matemáticas, leída el 23-11-2020This 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 elds, 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 uniformly 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 associated to fuzzy measures allows us to solve many problems, among them the identi cation problem, i.e. estimate the fuzzy measure that underlies an observed data set...La presente tesis doctoral esta dedicada al estudio de distintas propiedades geometricas y 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 Teoría de la Decision a laTeoría de Juegos. El conjunto formado por todas las medidas difusas sobre un referencial tiene estructura de politopo. De la misma forma, la mayora de las subfamilias mas relevantes de medidas difusas son tambien politopos. Estudiar la estructura combinatoria de estos politopos surge como un problema natural que nos permite comprender mejor las propiedades delas medidas difusas asociadas. Conocer la estructura combinatoria de estos politopos tambien nos 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 teorico como 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 identificacion que trata de estimarla medida difusa que subyace a un conjunto de datos observado...Fac. de Ciencias MatemáticasTRUEunpu

    CP-nets: From Theory to Practice

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    Conditional preference networks (CP-nets) exploit the power of ceteris paribus rules to represent preferences over combinatorial decision domains compactly. CP-nets have much appeal. However, their study has not yet advanced sufficiently for their widespread use in real-world applications. Known algorithms for deciding dominance---whether one outcome is better than another with respect to a CP-net---require exponential time. Data for CP-nets are difficult to obtain: human subjects data over combinatorial domains are not readily available, and earlier work on random generation is also problematic. Also, much of the research on CP-nets makes strong, often unrealistic assumptions, such as that decision variables must be binary or that only strict preferences are permitted. In this thesis, I address such limitations to make CP-nets more useful. I show how: to generate CP-nets uniformly randomly; to limit search depth in dominance testing given expectations about sets of CP-nets; and to use local search for learning restricted classes of CP-nets from choice data

    Set Theory

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    This workshop included selected talks on pure set theory and its applications, simultaneously showing diversity and coherence of the subject

    Optimal resolution of reversed preference in multi-criteria data sets

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    A structural approach to matching problems with preferences

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    This thesis is a study of a number of matching problems that seek to match together pairs or groups of agents subject to the preferences of some or all of the agents. We present a number of new algorithmic results for five specific problem domains. Each of these results is derived with the aid of some structural properties implicitly embedded in the problem. We begin by describing an approximation algorithm for the problem of finding a maximum stable matching for an instance of the stable marriage problem with ties and incomplete lists (MAX-SMTI). Our polynomial time approximation algorithm provides a performance guarantee of 3/2 for the general version of MAX-SMTI, improving upon the previous best approximation algorithm, which gave a performance guarantee of 5/3. Next, we study the sex-equal stable marriage problem (SESM). We show that SESM is W[1]-hard, even if the men's and women's preference lists are both of length at most three. This improves upon the previously known hardness results. We contrast this with an exact, low-order exponential time algorithm. This is the first non-trivial exponential time algorithm known for this problem, or indeed for any hard stable matching problem. Turning our attention to the hospitals / residents problem with couples (HRC), we show that HRC is NP-complete, even if very severe restrictions are placed on the input. By contrast, we give a linear-time algorithm to find a stable matching with couples (or report that none exists) when stability is defined in terms of the classical Gale-Shapley concept. This result represents the most general polynomial time solvable restriction of HRC that we are aware of. We then explore the three dimensional stable matching problem (3DSM), in which we seek to find stable matchings across three sets of agents, rather than two (as in the classical case). We show that under two natural definitions of stability, finding a stable matching for a 3DSM instance is NP-complete. These hardness results resolve some open questions in the literature. Finally, we study the popular matching problem (POP-M) in the context of matching a set of applicants to a set of posts. We provide a characterization of the set of popular matchings for an arbitrary POP-M instance in terms of a new structure called the switching graph exploited to yield efficient algorithms for a range of associated problems, extending and improving upon the previously best-known results for this problem

    Some contributions to decision making in complex information settings with imprecise probabilities and incomplete preferences

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    The Discrete Acyclic Digraph Markov Model in Data Mining

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    Graphical Markov models are a powerful tool for the description of complex interactions between the variables of a domain. They provide a succinct description of the joint distribution of the variables. This feature has led to the most successful application of graphical Markov models, that is as the core component of probabilistic expert systems. The fascinating theory behind this type of models arises from three different disciplines, viz., Statistics, Graph Theory and Artificial Intelligence. This interdisciplinary origin has given rich insight from different perspectives. There are two main ways to find the qualitative structure of graphical Markov models. Either the structure is specified by a domain expert or ``structural learning'' is applied, i.e., the structure is automatically recovered from data. For structural learning, one has to compare how well different models describe the data. This is easy for, e.g., acyclic digraph Markov models. However, structural learning is still a hard problem because the number of possible models grows exponentially with the number of variables. The main contributions of this thesis are as follows. Firstly, a new class of graphical Markov models, called TCI models, is introduced. These models can be represented by labeled trees and form the intersection of two previously well-known classes. Secondly, the inclusion order of graphical Markov models is studied. From this study, two new learning algorithms are derived. One for heuristic search and the other for the Markov Chain Monte Carlo Method. Both algorithms improve the results of previous approaches without compromising the computational cost of the learning process. Finally, new diagnostics for convergence assessment of the Markov Chain Monte Carlo Method in structural learning are introduced. The results of this thesis are illustrated using both synthetic and real world datasets
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