We give an overview of recent progress in the research of hypergraph jumps -- a problem from extremal combinatorics. The number $\alpha \in [0, 1)$ is a jump for $r$ if for any $\epsilon > 0$ and any integer $m \ge r$ any $r$-graph with $N > N(\epsilon, m)$ vertices and at least $(\alpha + \epsilon) {N \choose r}$ edges contains a subgraph with $m$ vertices and at least $(\alpha + c) {m \choose r}$ edges, where $c := c(\alpha)$ does depend only on $\alpha$. Baber and Talbot \cite{Baber} recently gave first examples of jumps for $r = 3$ in the interval $[2/9, 1)$. Their result uses the framework of flag algebras \cite{Raz07} and involves solving a semidefinite optimization problem. A software implementation of their method is a part of this work.

本文藉構造bipartite graph 的形式討論超圖與對偶超圖的transversal number，進而探討最小著色數的上界，以及證明出此兩圖的最小著色數可差異很大，也可用此方法構造出想要的最小著色數之差異。最後探討在某些情形下，超圖與其對偶超圖的同構性，再則整理出其必要條件。 / H=(X,D) is called a hypergraph, where X is the vertex set and D is a family of subsets of X, denoted as D-edges, and we assume that every D-edges have at least two elements. A strict t-coloring is a onto mapping from X to {1,2,....,t} such that each D contained in D-edge set has two vertices having distinct values. The maximum(minimum) number of colors over all strict k-coloring is called the upper(lower) chromatic number and is denoted as .

Zugl.: Kassel, University, Diss., 2007.

Zugl.: Kassel, Univ., Diss., 2007 / Download lizenzpflichtig

abstract: Multi-label learning, which deals with data associated with multiple labels simultaneously, is ubiquitous in real-world applications. To overcome the curse of dimensionality in multi-label learning, in this thesis I study multi-label dimensionality reduction, which extracts a small number of features by removing the irrelevant, redundant, and noisy information while considering the correlation among different labels in multi-label learning. Specifically, I propose Hypergraph Spectral Learning (HSL) to perform dimensionality reduction for multi-label data by exploiting correlations among different labels using a hypergraph. The regularization effect on the classical dimensionality reduction algorithm known as Canonical Correlation Analysis (CCA) is elucidated in this thesis. The relationship between CCA and Orthonormalized Partial Least Squares (OPLS) is also investigated. To perform dimensionality reduction efficiently for large-scale problems, two efficient implementations are proposed for a class of dimensionality reduction algorithms, including canonical correlation analysis, orthonormalized partial least squares, linear discriminant analysis, and hypergraph spectral learning. The first approach is a direct least squares approach which allows the use of different regularization penalties, but is applicable under a certain assumption; the second one is a two-stage approach which can be applied in the regularization setting without any assumption. Furthermore, an online implementation for the same class of dimensionality reduction algorithms is proposed when the data comes sequentially. A Matlab toolbox for multi-label dimensionality reduction has been developed and released. The proposed algorithms have been applied successfully in the Drosophila gene expression pattern image annotation. The experimental results on some benchmark data sets in multi-label learning also demonstrate the effectiveness and efficiency of the proposed algorithms. / Dissertation/Thesis / Ph.D. Computer Science 2011

Cette thèse en informatique s'intéresse à la structuration et à l'exploration de collections journalistiques. Elle fait appel à plusieurs domaines de recherche : sciences sociales, à travers l'étude de la production journalistique ; ergonomie ; traitement des langues et la recherche d'information ; multimédia et notamment la recherche d'information multimédia. Une branche de la recherche d'information multimédia, appelée hyperliage, constitue la base sur laquelle cette thèse est construite. L'hyperliage consiste à construire automatiquement des liens entre documents multimédias. Nous étendons ce concept en l'appliquant à l'entièreté d'une collection afin d'obtenir un hypergraphe, et nous intéressons notamment à ses caractéristiques topologiques et à leurs conséquences sur l'explorabilité de la structure construite. Nous proposons dans cette thèse des améliorations de l'état de l'art selon trois axes principaux : une structuration de collections d'actualités à l'aide de graphes mutli-sources et multimodaux fondée sur la création de liens inter-documents, son association à une diversité importante des liens permettant de représenter la grande variété des intérêts que peuvent avoir différents utilisateurs, et enfin l'ajout d'un typage des liens créés permettant d'expliciter la relation existant entre deux documents. Ces différents apports sont renforcés par des études utilisateurs démontrant leurs intérêts respectifs. / This thesis studies the structuring and exploration of news collections. While its main focus is on natural language processing and multimedia retrieval, it also deals with social studies through the study of the production of news and ergonomy through the conduct of user tests. The task of hyperlinking, which was recently put forward by the multimedia retrieval community, is at the center of this thesis. Hyperlinking consists in automatically finding relevant links between multimedia segments. We apply this concept to whole news collections, resulting in the creation of a hypergraph, and study the topological properties and their influence on the explorability of the resulting structure. In this thesis, we provide improvements beyond the state of the art along three main {axes:} a structuring of news collections by means of mutli-sources and multimodal graphs based on the creation of inter-document links, its association with a large diversity of links allowing to represent the variety of interests that different users may have, and a typing of the created links in order to make the nature of the relation between two documents explicit. Extensive user studies confirm the interest of the methods developped in this thesis.

In recent years, the popularity of graph databases has grown rapidly. This paper focuses on single-graph as an effective model to represent information and its related graph mining techniques. In frequent pattern mining in a single-graph setting, there are two main problems: support measure and search scheme. In this paper, we propose a novel framework for constructing support measures that brings together existing minimum-image-based and overlap-graph-based support measures. Our framework is built on the concept of occurrence / instance hypergraphs. Based on that, we present two new support measures: minimum instance (MI) measure and minimum vertex cover (MVC) measure, that combine the advantages of existing measures. In particular, we show that the existing minimum-image-based support measure is an upper bound of the MI measure, which is also linear-time computable and results in counts that are close to number of instances of a pattern. Although the MVC measure is NP-hard, it can be approximated to a constant factor in polynomial time. We also provide polynomial-time relaxations for both measures and bounding theorems for all presented support measures in the hypergraph setting. We further show that the hypergraph-based framework can unify all support measures studied in this paper. This framework is also flexible in that more variants of support measures can be defined and profiled in it.

La visualisation d'information est une discipline récente en pleine expansion et qui a pour objet l'étude des méthodes de représentation visuelle de données abstraites, c'est-à-dire non géolocalisées. La sémiotique est quant à elle une discipline beaucoup plus ancienne (fin du XIXième siècle) qui s'intéresse aux divers systèmes de signes nécessaires aux processusde communication. A ce jour, peu de travaux ont été réalisés pour mettre en parallèle ces deux disciplines. C'est pourquoi le premier chapitre de cette thèse est dédié à l'étude de la visualisation d'information selon les paradigmes élaborés par son ainée tout au long du XXième siècle. Nous montrons en particulier comment l'un des modèles les plus aboutis de validation de visualisations (modèle imbriqué de Tamara Munzner) correspond au processus d'étude sémiotique d'énoncés. Le second chapitre est consacré à la visualisation de graphe, outil de modélisation puissant de divers ensembles de données abstraites. Nous proposons d'une part une application permettant de visualiser et de naviguer à travers les pages Internet retournées par un moteur de recherche et d'autre part un algorithme de visualisation de hiérarchies dynamiques sous forme de "cartes géographiques". Enfin, nous évoquons dans le troisième chapitre un autre outil de modélisation de donnéesabstraites : les hypergraphes. Nous proposons des résultats théoriques concernant leur représentation et donnons une ébauche de solution permettant de les visualiser. / Information visualization aims at designing visual representations of abstract data, furthermore relying on interaction as a mean to discover knowledge. The first part of this thesis challenges Information Visualization by drawing a parallel with semiotics, a 19th century research field focusing on systems of signs required for communication. We develop a point of view on Information Visualization based on the paradigms developed by semioticians during the 20th century. In particular, we show how the visualization validation model proposed by Tamara Munzner is related to the process used by semioticians for utterance analysis. The second part of the thesis focuses on graph visualization and describes two techniques and system prototypes targeting specific application domains. The first one is an interactive technique to visualize and navigate through Web search results. The second one is an algorithm for the visualization of dynamic hierarchies exploiting the analogy with “geographical maps”. Finally, the third chapter is devoted to another model used to structure abstract data : hypergraphs. We propose theoretical results on hypergraph drawing and a preliminary technique to visualize hypergraphs.

In this work, new definitions of hypergraph products are presented. The main focus is on the generalization of the commutative standard graph products: the Cartesian, the direct and the strong graph product. We will generalize these well-known graph products to products of hypergraphs and show several properties like associativity, commutativity and distributivity w.r.t. the disjoint union of hypergraphs. Moreover, we show that all defined products of simple (hyper)graphs result in a simple (hyper)graph. We will see, for what kind of product the projections into the factors are (at least weak) homomorphisms and for which products there are similar connections between the hypergraph products as there are for graphs. Last, we give a new and more constructive proof for the uniqueness of prime factorization w.r.t. the Cartesian product than in [Studia Sci. Math. Hungar. 2: 285–290 (1967)] and moreover, a product relation according to such a decomposition. That might help to find efficient algorithms for the decomposition of hypergraphs w.r.t. the Cartesian product.

This PhD project is on the algorithmic aspects of synthesis planning and mass spectrometry; two separate chemical problems concerning the understanding of molecules and how these behave. Part I: In synthesis planning, the goal is to synthesize a target molecule from available starting materials, possibly optimizing costs such as price or environmental impact of the process. Current algorithmic approaches to synthesis planning are usually based on selecting a bond set and finding a single good plan among those induced by it. We demonstrate that synthesis planning can be phrased as a combinatorial optimization problem on hypergraphs, not necessarily using a pre-defined bond set. For this, individual synthesis plans are modeled as directed hyperpaths embedded in a hypergraph of reactions (HoR) representing the chemistry of interest. As a consequence, application of a know polynomial time algorithm to find the K shortest hyperpaths yields the K best synthesis plans for a given target molecule. To this end, classical quality measures are discussed. Having K good plans to choose from has several benefits: It makes the synthesis planning process much more robust when in later stages adding further chemical details, it allows one to combine several notions of cost, and it provides a way to deal with imprecise yield estimates. An empirical study of our method illustrates the limitations of what a chemist can expect is feasible to compute, as well as the practical value of our method for cases where yield estimates are imprecise or unknown. To illustrate the realism of the approach, synthesis plans from our abstraction level are compared with detailed chemical synthesis plans from the literature. For this, a synthesis plan for Wieland-Miescher ketone and a synthesis plan for lysergic acid are used. In addition, equivalence of our structural definition of a hyperpath and two definitions from the hypergraph literature is shown. Part II: Mass spectrometry is an analytic technique for characterizing molecules and molecular mixtures, by gaining knowledge of their structure and composition from the way they fragment. In a mass spectrometer, molecules or molecular mixtures are ionized and fragmented, and the abundances of the different fragment masses are measured, resulting in so-called mass spectra. We suggest a new road map improving the current state-of-the art in computational methods for mass spectrometry. The main focus is on increasing the chemical realism of the modeling of the fragmentation process. Two core ingredients for this are i) describing the individual fragmentation reactions via graph transformation rules and ii) expressing the dynamics of the system via reaction rates and quasi-equilibrium theory. Graph transformation rules are used both for specifying the possible core fragmentation reactions, and for characterizing the reaction sites when learning values for the rates. We believe that this model describes chemical mechanisms more accurately than previous ones, and that this can lead to both better spectrum prediction and more explanatory power. Our modeling of system dynamics also allows better separation of instrument dependent and instrument independent parameters of the model.