A Análise de Conceitos Formais (FCA) é uma teoria matemática que formaliza a noção de conceitos e hierarquias conceituais. De importância central a esta teoria é uma estrutura algébrica denominada reticulado de conceitos. Esta estrutura é definida em função de um conjunto de objetos, outro de atributos e uma relação que indica os atributos apresentados por cada objeto. Uma representação gráfica de um reticulado de conceitos, por meio de uma interface computacional, é capaz de expor regularidades presentes em dados a um usuário, e este pode então realizar tarefas de análise exploratória de dados. Este tipo de aplicação de FCA vem sendo empregado em dezenas de projetos pertencentes a áreas diversas, como medicina, serviços de inteligência, engenharia de software e bioinformática. Mostramos neste trabalho um sistema de análise exploratória de dados baseado em FCA, e sua utilização sobre dados reais. Também é mostrado como reticulados de conceitos podem ser empregados em interfaces de recuperação de informação. Do ponto de vista algorítmico, analisamos métodos computacionais para a determinação do reticulado de conceitos, e também de uma subestrutura simplificada, o conjunto de conceitos. O tamanho de um reticulado de conceitos pode ser exponencial em função dos tamanhos dos conjuntos de objetos e de atributos. Assim, é de vital interesse o estabelecimento de cotas superiores para o número de conceitos de um reticulado. Neste trabalho, apresentamos as cotas já conhecidas presentes na literatura. Também estabelecemos uma nova cota superior, e mostramos famílias de casos em que nossa cota superior é mais justa que as demais. Para algumas famílias particulares, nossa cota é polinomial, enquanto que as demais são exponenciais. / Formal Concept Analysis (FCA) is a mathematical theory that formalizes the notion of concepts and conceptual hierarchies. Of central importance to this theory is an algebraic structure termed concept lattice. Such structure becomes defined after being given one set of objects, one of attributes, and an incidence relation describing the attributes held by each object. A graphical representation of a concept lattice, by means of a computational interface, is capable of unfolding regularities present in data to an user, who is then able to conduct exploratory data analysis tasks. This sort of FCA application is currently deployed in tens of projects belonging to a wide range of areas, such as medicine, intelligence services, software engineering and bioinformatics. We show in this work an FCA-based system of exploratory data analysis, and its use over real data. Moreover, it is shown how concept lattices can be employed in information retrieval interfaces. From the algorithmic viewpoint, we analyse computational methods for the determination of a concept lattice, and also of a simplified substructure, the concept set. The size of a concept lattice can be exponential when compared to the size of the objects and the attributes sets. Therefore, it is of paramount interest the establishment of upper bounds for the number of concepts of a lattice. In this work, we present the upper bounds already known in the literature. We also establish a new upper bound, and show families of cases in which our bound is sharper than the others. For particular families, our bound is polynomial, whereas the other bounds are exponential.

A Análise de Conceitos Formais (FCA) é uma teoria matemática que formaliza a noção de conceitos e hierarquias conceituais. De importância central a esta teoria é uma estrutura algébrica denominada reticulado de conceitos. Esta estrutura é definida em função de um conjunto de objetos, outro de atributos e uma relação que indica os atributos apresentados por cada objeto. Uma representação gráfica de um reticulado de conceitos, por meio de uma interface computacional, é capaz de expor regularidades presentes em dados a um usuário, e este pode então realizar tarefas de análise exploratória de dados. Este tipo de aplicação de FCA vem sendo empregado em dezenas de projetos pertencentes a áreas diversas, como medicina, serviços de inteligência, engenharia de software e bioinformática. Mostramos neste trabalho um sistema de análise exploratória de dados baseado em FCA, e sua utilização sobre dados reais. Também é mostrado como reticulados de conceitos podem ser empregados em interfaces de recuperação de informação. Do ponto de vista algorítmico, analisamos métodos computacionais para a determinação do reticulado de conceitos, e também de uma subestrutura simplificada, o conjunto de conceitos. O tamanho de um reticulado de conceitos pode ser exponencial em função dos tamanhos dos conjuntos de objetos e de atributos. Assim, é de vital interesse o estabelecimento de cotas superiores para o número de conceitos de um reticulado. Neste trabalho, apresentamos as cotas já conhecidas presentes na literatura. Também estabelecemos uma nova cota superior, e mostramos famílias de casos em que nossa cota superior é mais justa que as demais. Para algumas famílias particulares, nossa cota é polinomial, enquanto que as demais são exponenciais. / Formal Concept Analysis (FCA) is a mathematical theory that formalizes the notion of concepts and conceptual hierarchies. Of central importance to this theory is an algebraic structure termed concept lattice. Such structure becomes defined after being given one set of objects, one of attributes, and an incidence relation describing the attributes held by each object. A graphical representation of a concept lattice, by means of a computational interface, is capable of unfolding regularities present in data to an user, who is then able to conduct exploratory data analysis tasks. This sort of FCA application is currently deployed in tens of projects belonging to a wide range of areas, such as medicine, intelligence services, software engineering and bioinformatics. We show in this work an FCA-based system of exploratory data analysis, and its use over real data. Moreover, it is shown how concept lattices can be employed in information retrieval interfaces. From the algorithmic viewpoint, we analyse computational methods for the determination of a concept lattice, and also of a simplified substructure, the concept set. The size of a concept lattice can be exponential when compared to the size of the objects and the attributes sets. Therefore, it is of paramount interest the establishment of upper bounds for the number of concepts of a lattice. In this work, we present the upper bounds already known in the literature. We also establish a new upper bound, and show families of cases in which our bound is sharper than the others. For particular families, our bound is polynomial, whereas the other bounds are exponential.

Nesta tese, estudamos o problema de Empacotamento de Bicliques. Um biclique é um grafo bipartido completo. No problema de Empacotamento de Bicliques são dados um inteiro k e um grafo bipartido G e deseja-se encontrar um conjunto de k bicliques, subgrafos de G, dois a dois disjuntos nos vértices, tal que a quantidade total de arestas dos bicliques escolhidos seja máxima. No caso em que k=1, temos o problema de Biclique máximo. Esses dois problemas possuem aplicações na área de Bioinformática. Mantemos neste trabalho um enfoque prático, no sentido de que nosso interesse é resolver instâncias desses dois problemas com tamanho razoavelmente grande. Para isso, utilizamos técnicas de Programação Linear Inteira. Para avaliar os métodos propostos aqui, mostramos resultados de experimentos computacionais feitos com instâncias vindas de aplicações e também com instâncias geradas aleatoriamente. / In this thesis, we study the Biclique Packing problem. A biclique is a complete bipartite graph. In the Biclique Packing problem we are given an integer k and a bipartite graph G and we want to find a set of k vertex disjoint bicliques of G, such that the total number of biclique\'s edges is maximum. When k=1, we have the Maximum Biclique problem. These two problems have applications in Bioinformatics. In this work we keep a practical focus, in the sense that we are interested in solving large size instances of these problems. To tackle these problems, we use Integer Linear Programming techniques. In order to evaluate the methods proposed here, we show results of computational experiments carried out with practical application\'s instances and also with randomly generated ones.

Nesta tese, estudamos o problema de Empacotamento de Bicliques. Um biclique é um grafo bipartido completo. No problema de Empacotamento de Bicliques são dados um inteiro k e um grafo bipartido G e deseja-se encontrar um conjunto de k bicliques, subgrafos de G, dois a dois disjuntos nos vértices, tal que a quantidade total de arestas dos bicliques escolhidos seja máxima. No caso em que k=1, temos o problema de Biclique máximo. Esses dois problemas possuem aplicações na área de Bioinformática. Mantemos neste trabalho um enfoque prático, no sentido de que nosso interesse é resolver instâncias desses dois problemas com tamanho razoavelmente grande. Para isso, utilizamos técnicas de Programação Linear Inteira. Para avaliar os métodos propostos aqui, mostramos resultados de experimentos computacionais feitos com instâncias vindas de aplicações e também com instâncias geradas aleatoriamente. / In this thesis, we study the Biclique Packing problem. A biclique is a complete bipartite graph. In the Biclique Packing problem we are given an integer k and a bipartite graph G and we want to find a set of k vertex disjoint bicliques of G, such that the total number of biclique\'s edges is maximum. When k=1, we have the Maximum Biclique problem. These two problems have applications in Bioinformatics. In this work we keep a practical focus, in the sense that we are interested in solving large size instances of these problems. To tackle these problems, we use Integer Linear Programming techniques. In order to evaluate the methods proposed here, we show results of computational experiments carried out with practical application\'s instances and also with randomly generated ones.

Die vorliegende Arbeit besteht aus zwei Teilen. Der erste Teil der Arbeit befasst sich mit unteren Schranken für approximative Steinerbaumalgorithmen. Ein Steinerbaum ist ein kürzester Teilgraph, der eine gegebene Teilmenge der Knoten eines Graphen spannt. Das Berechnen eines Steinerbaumes ist ein klassisches NP-schweres Problem, und es existieren mehrere Approximationsalgorithmen, wobei bei den meisten Algorithmen die Approximationsgüte nur durch untere und obere Schranken eingegrenzt werden kann. Für einige dieser Algorithmen werden in dieser Arbeit Instanzen vorgestellt, welche die unteren Schranken verbessern. Für den Relativen Greedy Algorithmus wird außerdem ein Verfahren vorgestellt, mit dem die Güte des Algorithmus eingeschränkt auf die Graphenklasse mit k Terminalen auf einen beliebigen Faktor genau bestimmt werden kann. Der zweite Teil der Arbeit widmet sich vollständig bipartiten Subgraphen mit gleicher Partitionsgrößse, sogenannten balancierten Bicliquen. Seit langem ist bekannt, dass in dichten bipartiten Graphen balancierte Bicliquen mit Omega(log(n)) Knoten existieren, aber es ist unbekannt, ob und wie diese in polynomieller Zeit konstruiert werden können. Der zweite Teil liefert dazu einen Beitrag, indem ein polynomieller Algorithmus vorgestellt wird, der in hinreichend großen dichten bipartiten Graphen eine balancierte Biclique mit Omega(sqrt(log(n))) Knoten konstruiert. / This thesis consists of two parts. The first part is concerned with lower bounds for approximating Steiner trees. The Steiner tree problem is to find a shortest subgraph that spans a given set of vertices in a graph and is a classical NP-hard problem. Several approximation algorithms exist, but for most algorithms only lower and upper bounds for the approximation ratio are known. For some of these algorithms we present instances which improve the lower bounds. When the problem is restricted to the class of graphs with k terminals, we also present a method which can be used to determine the approximation ratio of the Relative Greedy Algorithm with arbitrary precision. The second part is about balanced bicliques, i.e. complete bipartite subgraphs with equal partition sizes. It has been known for a long time that every dense bipartite graph contains a balanced biclique of size Omega(log(n)), but whether and how such a biclique can be constructed in polynomial time is still unknown. Our contribution to this problem is a polynomial time algorithm which constructs a balanced biclique of size Omega(sqrt(log(n))) in sufficiently large and dense bipartite graphs.

We prove that there exist three distinct, comprehensive classes of (formal) contexts with polynomially many concepts. Namely: contexts which are nowhere dense, of bounded breadth or highly convex. Already present in G. Birkhoff's classic monograph is the notion of breadth of a lattice; it equals the number of atoms of a largest boolean suborder. Even though it is natural to define the breadth of a context as being that of its concept lattice, this idea had not been exploited before. We do this and establish many equivalences. Amongst them, it is shown that the breadth of a context equals the size of its largest minimal generator, its largest contranominal-scale subcontext, as well as the Vapnik-Chervonenkis dimension of both its system of extents and of intents. The polynomiality of the aforementioned classes is proven via upper bounds (also known as majorants) for the number of maximal bipartite cliques in bipartite graphs. These are results obtained by various authors in the last decades. The fact that they yield statements about formal contexts is a reward for investigating how two established fields interact, specifically Formal Concept Analysis (FCA) and graph theory. We improve considerably the breadth bound. Such improvement is twofold: besides giving a much tighter expression, we prove that it limits the number of minimal generators. This is strictly more general than upper bounding the quantity of concepts. Indeed, it automatically implies a bound on these, as well as on the number of proper premises. A corollary is that this improved result is a bound for the number of implications in the canonical basis too. With respect to the quantity of concepts, this sharper majorant is shown to be best possible. Such fact is established by constructing contexts whose concept lattices exhibit exactly that many elements. These structures are termed, respectively, extremal contexts and extremal lattices. The usual procedure of taking the standard context allows one to work interchangeably with either one of these two extremal structures. Extremal lattices are equivalently defined as finite lattices which have as many elements as possible, under the condition that they obey two upper limits: one for its number of join-irreducibles, other for its breadth. Subsequently, these structures are characterized in two ways. Our first characterization is done using the lattice perspective. Initially, we construct extremal lattices by the iterated operation of finding smaller, extremal subsemilattices and duplicating their elements. Then, it is shown that every extremal lattice must be obtained through a recursive application of this construction principle. A byproduct of this contribution is that extremal lattices are always meet-distributive. Despite the fact that this approach is revealing, the vicinity of its findings contains unanswered combinatorial questions which are relevant. Most notably, the number of meet-irreducibles of extremal lattices escapes from control when this construction is conducted. Aiming to get a grip on the number of meet-irreducibles, we succeed at proving an alternative characterization of these structures. This second approach is based on implication logic, and exposes an interesting link between number of proper premises, pseudo-extents and concepts. A guiding idea in this scenario is to use implications to construct lattices. It turns out that constructing extremal structures with this method is simpler, in the sense that a recursive application of the construction principle is not needed. Moreover, we obtain with ease a general, explicit formula for the Whitney numbers of extremal lattices. This reveals that they are unimodal, too. Like the first, this second construction method is shown to be characteristic. A particular case of the construction is able to force - with precision - a high number of (in the sense of "exponentially many'') meet-irreducibles. Such occasional explosion of meet-irreducibles motivates a generalization of the notion of extremal lattices. This is done by means of considering a more refined partition of the class of all finite lattices. In this finer-grained setting, each extremal class consists of lattices with bounded breadth, number of join irreducibles and meet-irreducibles as well. The generalized problem of finding the maximum number of concepts reveals itself to be challenging. Instead of attempting to classify these structures completely, we pose questions inspired by Turán's seminal result in extremal combinatorics. Most prominently: do extremal lattices (in this more general sense) have the maximum permitted breadth? We show a general statement in this setting: for every choice of limits (breadth, number of join-irreducibles and meet-irreducibles), we produce some extremal lattice with the maximum permitted breadth. The tools which underpin all the intuitions in this scenario are hypergraphs and exact set covers. In a rather unexpected, but interesting turn of events, we obtain for free a simple and interesting theorem about the general existence of "rich'' subcontexts. Precisely: every context contains an object/attribute pair which, after removed, results in a context with at least half the original number of concepts.

We prove that there exist three distinct, comprehensive classes of (formal) contexts with polynomially many concepts. Namely: contexts which are nowhere dense, of bounded breadth or highly convex. Already present in G. Birkhoff's classic monograph is the notion of breadth of a lattice; it equals the number of atoms of a largest boolean suborder. Even though it is natural to define the breadth of a context as being that of its concept lattice, this idea had not been exploited before. We do this and establish many equivalences. Amongst them, it is shown that the breadth of a context equals the size of its largest minimal generator, its largest contranominal-scale subcontext, as well as the Vapnik-Chervonenkis dimension of both its system of extents and of intents. The polynomiality of the aforementioned classes is proven via upper bounds (also known as majorants) for the number of maximal bipartite cliques in bipartite graphs. These are results obtained by various authors in the last decades. The fact that they yield statements about formal contexts is a reward for investigating how two established fields interact, specifically Formal Concept Analysis (FCA) and graph theory. We improve considerably the breadth bound. Such improvement is twofold: besides giving a much tighter expression, we prove that it limits the number of minimal generators. This is strictly more general than upper bounding the quantity of concepts. Indeed, it automatically implies a bound on these, as well as on the number of proper premises. A corollary is that this improved result is a bound for the number of implications in the canonical basis too. With respect to the quantity of concepts, this sharper majorant is shown to be best possible. Such fact is established by constructing contexts whose concept lattices exhibit exactly that many elements. These structures are termed, respectively, extremal contexts and extremal lattices. The usual procedure of taking the standard context allows one to work interchangeably with either one of these two extremal structures. Extremal lattices are equivalently defined as finite lattices which have as many elements as possible, under the condition that they obey two upper limits: one for its number of join-irreducibles, other for its breadth. Subsequently, these structures are characterized in two ways. Our first characterization is done using the lattice perspective. Initially, we construct extremal lattices by the iterated operation of finding smaller, extremal subsemilattices and duplicating their elements. Then, it is shown that every extremal lattice must be obtained through a recursive application of this construction principle. A byproduct of this contribution is that extremal lattices are always meet-distributive. Despite the fact that this approach is revealing, the vicinity of its findings contains unanswered combinatorial questions which are relevant. Most notably, the number of meet-irreducibles of extremal lattices escapes from control when this construction is conducted. Aiming to get a grip on the number of meet-irreducibles, we succeed at proving an alternative characterization of these structures. This second approach is based on implication logic, and exposes an interesting link between number of proper premises, pseudo-extents and concepts. A guiding idea in this scenario is to use implications to construct lattices. It turns out that constructing extremal structures with this method is simpler, in the sense that a recursive application of the construction principle is not needed. Moreover, we obtain with ease a general, explicit formula for the Whitney numbers of extremal lattices. This reveals that they are unimodal, too. Like the first, this second construction method is shown to be characteristic. A particular case of the construction is able to force - with precision - a high number of (in the sense of "exponentially many'') meet-irreducibles. Such occasional explosion of meet-irreducibles motivates a generalization of the notion of extremal lattices. This is done by means of considering a more refined partition of the class of all finite lattices. In this finer-grained setting, each extremal class consists of lattices with bounded breadth, number of join irreducibles and meet-irreducibles as well. The generalized problem of finding the maximum number of concepts reveals itself to be challenging. Instead of attempting to classify these structures completely, we pose questions inspired by Turán's seminal result in extremal combinatorics. Most prominently: do extremal lattices (in this more general sense) have the maximum permitted breadth? We show a general statement in this setting: for every choice of limits (breadth, number of join-irreducibles and meet-irreducibles), we produce some extremal lattice with the maximum permitted breadth. The tools which underpin all the intuitions in this scenario are hypergraphs and exact set covers. In a rather unexpected, but interesting turn of events, we obtain for free a simple and interesting theorem about the general existence of "rich'' subcontexts. Precisely: every context contains an object/attribute pair which, after removed, results in a context with at least half the original number of concepts.

Le graphe du Web, plus précisément le crawl qui permet de l'obtenir et les communautés qu'il contient est le sujet de cette thèse, qui est divisée en deux parties.<br />La première partie fait une analyse des grand réseaux d'interactions et introduit un nouveau modèle de crawls du Web. Elle commence par définir les propriétés communes des réseaux d'interactions, puis donne quelques modèles graphes aléatoires générant des graphes semblables aux réseaux d'interactions. Pour finir, elle propose un nouveau modèle de crawls aléatoires.<br />La second partie propose deux modèles de calcul de communautés par émergence dans le graphe du Web. Après un rappel sur les mesures d'importances, PageRank et HITS est présenté le modèle gravitationnel dans lequel les nœuds d'un réseau sont mobile et interagissent entre eux grâce aux liens entre eux. Les communautés émergent rapidement au bout de quelques itérations. Le second modèle est une amélioration du premier, les nœuds du réseau sont dotés d'un objectif qui consiste à atteindre sa communautés.