Thesis (MSc (Mathematical Sciences))--University of Stellenbosch, 2011. / Includes bibliography. / Please refer to full text to view abstract.

The <i>mixed postman problem</i> consists of finding a minimum cost tour of a mixed graph <i>M</i> = (<i>V</i>,<i>E</i>,<i>A</i>) traversing all its edges and arcs at least once. We prove that two well-known linear programming relaxations of this problem are equivalent. The <i>extra cost</i> of a mixed postman tour <i>T</i> is the cost of <i>T</i> minus the cost of the edges and arcs of <i>M</i>. We prove that it is <i>NP</i>-hard to approximate the minimum extra cost of a mixed postman tour. A related problem, known as the <i>windy postman problem</i>, consists of finding a minimum cost tour of an undirected graph <i>G</i>=(<i>V</i>,<i>E</i>) traversing all its edges at least once, where the cost of an edge depends on the direction of traversal. We say that <i>G</i> is <i>windy postman perfect</i> if a certain <i>windy postman polyhedron O</i> (<i>G</i>) is integral. We prove that series-parallel undirected graphs are windy postman perfect, therefore solving a conjecture of Win. Given a mixed graph <i>M</i> = (<i>V</i>,<i>E</i>,<i>A</i>) and a subset <i>R</i> &#8838; <i>E</i> &#8746; <i>A</i>, we say that a mixed postman tour of <i>M</i> is <i>restricted</i> if it traverses the elements of <i>R</i> exactly once. The <i>restricted mixed postman problem</i> consists of finding a minimum cost restricted tour. We prove that this problem is <i>NP</i>-hard even if <i>R</i>=<i>A</i> and we restrict <i>M</i> to be planar, hence solving a conjecture of Veerasamy. We also prove that it is <i>NP</i>-complete to decide whether there exists a restricted tour even if <i>R</i>=<i>E</i> and we restrict <i>M</i> to be planar. The <i>edges postman problem</i> is the special case of the restricted mixed postman problem when <i>R</i>=<i>A</i>. We give a new class of valid inequalities for this problem. We introduce a relaxation of this problem, called the <i>b-join problem</i>, that can be solved in polynomial time. We give an algorithm which is simultaneously a 4/3-approximation algorithm for the edges postman problem, and a 2-approximation algorithm for the extra cost of a tour. The <i>arcs postman problem</i> is the special case of the restricted mixed postman problem when <i>R</i>=<i>E</i>. We introduce a class of necessary conditions for <i>M</i> to have an arcs postman tour, and we give a polynomial-time algorithm to decide whether one of these conditions holds. We give linear programming formulations of this problem for mixed graphs arising from windy postman perfect graphs, and mixed graphs whose arcs form a forest.

This thesis will study a variety of problems in graph theory. Initially, the focus will be on finding minimal degree conditions which guarantee the existence of various subgraphs. These subgraphs will all be formed of cycles, and this area of work will fall broadly into two main categories. First to be considered are cycles in edge-coloured graphs and, in particular, two questions of Li, Nikiforov and Schelp. It will be shown that a 2-edge-coloured graph with minimal degree at least 3n/4 either is isomorphic to the complete 4-partite graph with classes of order n/4, or contains monochromatic cycles of all lengths between 4 and n/2 (rounded up). This answers a conjecture of Li, Nikiforov and Schelp. Attention will then turn to the length of the longest monochromatic cycle in a 2-edge-coloured graph with minimal degree at least cn. In particular, a lower bound for this quantity will be proved which is asymptotically best possible. The next chapter of the thesis then shows that a hamiltonian graph with minimal degree at least (5-sqrt7)n/6 contains a 2-factor with two components. The thesis then concludes with a chapter about X_H, which is the number of copies of a graph H in the random graph G(n,p). In particular, it will be shown that, for a connected graph H, the value of X_H modulo k is approximately uniformly distributed, provided that k is not too large a function of n.

In this dissertation the results of Jerrum and Sinclair on the conductance of Markov chains are used to prove that almost all generalized Steinhaus graphs are rapidly mixing and an algorithm for the uniform generation of 2 - (4k + 1,4,1) cyclic Mendelsohn designs is developed.

This investigation proposes a powerful formalization of common sense knowledge based on function-free normal deduction graphs (NDGs) which form a powerful tool for deriving Horn and non-Horn clauses without functions. Such formalization allows common sense reasoning since it has the ability to handle not only negative but also incomplete information.

We study the existence of expander graphs with a focus on odd and unique expanders. The main goal is to describe configurations of arguments for which there is no infinite family of expanders. The most imporant result is that for every graph there is a nonempty subset of at most half of its vertices, such that every other vertex is connected at least twice to the subset or not connected to the subset at all. It follows that certain classes of unique expanders cannot exist. On the other hand we present some configurations for which there are families of expanders. Powered by TCPDF (www.tcpdf.org)

Dans cette thèse, nous présentons une approche algorithmique permettant la génération de guides de construction de cages moléculaires organiques. Il s'agit d'architectures semi-moléculaires possédant un espace interne défini capable de piéger une molécule cible appelée substrat. De nombreuses œuvres proposent de générer des cages organiques moléculaires obtenues à partir de structures symétriques, qui ont une bonne complexité, mais elles ne sont pas spécifiques car elles ne prennent pas en compte des cibles précises. L'approche proposée permet de générer des guides de construction de cages moléculaires organiques spécifiques à un substrat donné. Afin de garantir la spécificité de la cage moléculaire pour le substrat cible, une structure intermédiaire, qui est une expansion de l'enveloppe du substrat cible, est utilisée. Cette structure définie la forme de l'espace dans lequel est piégé le substrat. Des petits ensembles d'atomes, appelés motifs moléculaires liants, sont ensuite intégrés à cette structure intermédiaire. Ces motifs moléculaires sont les ensembles d'atomes nécessaires aux cages moléculaires pour leur permettre d’interagir avec le substrat afin de le capturer. / In this thesis, we present an algorithmic approach allowing the generation of construction guides of organic molecular cages. These semi-molecular architectures have a defined internal space capable of trapping a target molecule called substrate. Many works propose to generate molecular organic cages obtained from symmetrical structures, which have a good complexity, but they are not specific because they do not take into account precise targets. The proposed approach makes it possible to generate guides for the construction of organic molecular cages specific to a given substrate. In order to ensure the specificity of the molecular cage for the target substrate, an intermediate structure, which is an expansion of the envelope of the target substrate, is used. This structure defines the shape of the space in which the substrate is trapped. Small sets of atoms, called molecular binding patterns, are then integrated into this intermediate structure. These molecular patterns are the sets of atoms needed by molecular cages to allow them to interact with the substrate to capture it.

O grande volume de informação textual sendo gerado a todo momento torna necessário o aprimoramento constante de sistemas capazes de classificar textos em categorias específicas. Essa categorização visa, por exemplo, separar notícias indexadas por mecanismos de buscas, identificar a autoria de livros e cartas antigas ou detectar plágio em artigos científicos. As técnicas de classificação textual existentes, baseadas em conteúdo, apesar de conseguirem uma boa performance quantitativamente, ainda apresentam dificuldades em lidar com aspectos semânticos presentes nos textos escritos em língua natural. Neste sentido, abordagens alternativas vem sendo propostas, como as baseadas em redes complexas, que levam em consideração apenas o relacionamento entre as palavras. Neste estudo, aplicamos a modelagem de textos como redes complexas e utilizamos as métricas extraídas como atributos para classificação, utilizando um problema de reconhecimento de autoria para ilustrar a aplicação das técnicas descritas ao longo deste texto / The large volume of textual information being generated at all times makes it necessary to constantly improve systems capable of classifying texts into specific categories. This categorization aims, for example, to separate news items indexed by search engines, identify authorship of old books and letters, or detect plagiarism in scientific articles. Existing textual classification techniques, based on content, despite achieving good quantitative performance, still present difficulties in dealing with semantic aspects present in texts written in natural language. In this sense, alternative approaches have been proposed, such as those based on complex networks, which take into account only the relationship between words. In this study, we applied text modeling as graphs and extracted metrics typically used in the study of complex networks to be used as classifier attributes. To illustrate these techniques, a problem of authorship recognition in small texts was chosen as an example

Dans cette thèse nous explorons différentes facettes des modèles de tenseurs aléatoires. Les modèles de tenseurs aléatoires ont été introduits en physique dans le cadre de l'étude de la gravité quantique. En effet les modèles de matrices aléatoires, qui sont un cas particuliers de modèles de tenseurs, en sont une des origines. Ces modèles de matrices sont connus pour leur riche combinatoire et l'incroyable diversité de leurs propriétés qui les font toucher tous les domaines de l'analyse, la géométrie et des probabilités. De plus leur étude par les physiciens ont prouvé leur efficacité en ce qui concerne l'étude de la gravité quantique à deux dimensions. Les modèles de tenseurs aléatoires incarnent une généralisation possible des modèles de matrices. Comme leurs cousins, les modèles de matrices, ils posent questions dans les domaines de la combinatoire (comment traiter les cartes combinatoires d dimensionnelles ?), de la géométrie (comment contrôler la géométrie des triangulations générées ?) et de la physique (quel type d'espace-temps produisent-ils ? Quels sont leurs différentes phases ?). Cette thèse espère établir des pistes ainsi que des techniques d'études de ces modèles. Dans une première partie nous donnons une vue d'ensemble des modèles de matrices. Puis, nous discutons la combinatoire des triangulations en dimensions supérieures ou égales à trois en nous concentrant sur le cas tridimensionnelle (lequel est plus simple à visualiser). Nous définissons ces modèles et étudions certaines de leurs propriétés à l'aide de techniques combinatoires permettant de traiter les cartes d dimensionnelles. Enfin nous nous concentrons sur la généralisation de techniques issues des modèles de matrices dans le cas d'une famille particulières de modèles de tenseurs aléatoires. Ceci culmine avec le dernier chapitre de la thèse donnant des résultats partiels concernant la généralisation de la récurrence topologique de Eynard et Orantin à cette famille de modèles de tenseurs. / In this thesis manuscript we explore different facets of random tensor models. These models have been introduced to mimic the incredible successes of random matrix models in physics, mathematics and combinatorics. After giving a very short introduction to few aspects of random matrix models and recalling a physical motivation called Group Field Theory, we start exploring the world of random tensor models and its relation to geometry, quantum gravity and combinatorics. We first define these models in a natural way and discuss their geometry and combinatorics. After these first explorations we start generalizing random matrix methods to random tensors in order to describes the mathematical and physical properties of random tensor models, at least in some specific cases.

Projeções Multidimensionais são úteis para gerar visualizações adequadas para apoiar a análise exploratória de uma grande variedade de dados complexos e de alta dimensionalidade. Tarefas de análise visual de dados têm se beneficiado de projeções para explorar dados textuais, de imagens, de sensores, entre outros. Porém, diferentes técnicas de projeção e diferentes parametrizações de uma mesma técnica produzem resultados distintos para um mesmo conjunto de dados, pois as técnicas adotam estratégias distintas para representar os dados originais em um espaço cuja dimensionalidade permite sua visualização. Atualmente, ainda há poucos recursos para avaliar a qualidade dessas projeções e, em geral, as soluções existentes avaliam propriedades específicas, demandando grande esforço do analista para uma avaliação mais abrangente. Neste trabalho, introduzimos um arcabouço para computar medidas de avaliação de projeções com enfoque em análise de vizinhanças e de agrupamentos. Para elaborar este arcabouço, foi realizado um estudo de percepção para entender melhor como os usuários observam as projeções e foi conduzida uma investigação de representações dos dados capazes de favorecer a identificação de vizinhanças e de agrupamentos. Os padrões identificados no estudo de percepção auxiliaram a validar a representação dos dados, em que foi proposto um modelo de grafo, chamado de Extended Minimum Spanning Tree (EMST), capaz de capturar características condizentes com as observações dos participantes no estudo. O grafo EMST também foi validado por meio de dois estudos comparativos de identificação de vizinhanças e de agrupamentos. Com base neste arcabouço foram propostas cinco medidas de qualidade de projeções multidimensionais, duas delas para avaliar características relacionadas à separação visual das classes, e outras três para avaliar a preservação de propriedades do espaço original, especificamente a separação das classes, as vizinhanças e os agrupamentos. As medidas são ilustradas por meio de sua aplicação a conjuntos de dados artificiais, favorecendo a sua interpretação, bem como a conjuntos de dados reais, evidenciando a sua potencial utilidade em cenários reais. Também é apresentada uma comparação das novas medidas de preservação de vizinhanças com medidas similares descritas na literatura, permitindo identificar diferenças e semelhanças entre elas. / Multidimensional projections are valuable tools to generate visualizations that support exploratory analysis of a wide variety of complex high-dimensional data. Many examples are found in the literature of visual data analysis tasks that employ projections to explore, for instance, text, image, network and sensor data. Nonetheless, dierent projection techniques applied to a particular data set, or even alternative parameterizations of a single technique, can produce very distinct outcomes, as techniques adopt different strategies to reduce data dimensionality. Few resources are available to support assessing projection quality and, in general, existing solutions focus on specific properties. Thus, a broader assessment typically requires considerable human effort. In this work we introduce a framework to compute projection evaluation measures that focus on neighborhoods and clusters. To elaborate this framework we conducted (i) an experimental study to better understand how users perceive projections and (ii) an investigation of possible data representations capable of favoring the identification of neighborhoods and clusters. The observations resulting from the experimental study have been considered to propose and validate a novel graph data model, called Extended Minimum Spanning Tree (EMST), which captures data properties shown to be consistent with the observations by the participants in the study. The EMST graph has been validated by means of two comparative studies conducted to identify neighborhoods and clusters in multidimensional data. Under this framework, five novel measures of projection quality are introduced, two of them to assess properties related to the visual separation of classes, and three to assess the preservation of data properties in the original space, in particular the preservation of class separation, the preservation of neighborhoods and the preservation of groups. The quality measures have been applied to projections of synthetic data sets, favoring their understanding and interpretation, and also to projections of real data sets, illustrating their potential applicability in real scenarios. The newly introduced neighborhood preservation measures are also compared with existing methods in order to illustrate their differences and similarities.