Average-Case Analyse parametrisierter und probabilistischer Algorithmen

In both Theoretical Computer Science and practical work it is a disappointing outcome if the considered problem is NP complete. There is almost no hope for an efficient algorithm. However, many approaches have been developed to overcome this barrier: - The study of parameterized complexity allows in many cases the concentration of the explosion of the running time in a given parameter. - The behavior of problems not only in the worst case but also in average cases are studied. Or the data to work with is slightly perturbed. Then the concept of a smoothed analysis gives new insides - Also sometimes the use of randomness in the computing process can help to circumvent some obstacles. - And maybe an approximation is also nearly as good as an optimal solution. All these approaches are well studied on its own, but interactions between them, and the use of multiple approaches together, is a mostly unstudied field of research. In this thesis we study a part of these interactions for some test problems. We show that the reduction rules, given by Gramm et al., for the Clique-Cover problem with high probability not only reduce yes instances, but solve them entirely. We also consider the paradigm of bounded search trees, which is widely used for parameterizd problems. We find that the expected running time of a simple bounded search tree algorithm is much lower than the worst case bound for FPT problems Vertex-Cover and d-Hitting-Set. For certain sets of parameter values expected FPT running time for the W[1] and W[2] complete problems Clique and Hitting-Set is achieved, too. Furthermore, we study a simple probabilistic generalization of greedy approximation algorithms. For the Vertex-Cover, Hitting-Set, and the Triangle-Vertex-Deletion problem we find that the probabilistic algorithms we give have a substantially smaller expected approximation ratio than their deterministic equivalents. There is also a trade off: With more time one can expect better solutions

Métrologie des graphes de terrain, application à la construction de ressources lexicales et à la recherche d'information

This thesis is organized in two parts : the first part focuses on measures of similarity (or proximity) between vertices of a graph, the second part on clustering methods for bipartite graph. A new measure of similarity between vertices, based on short time random walks, is introduced. The main advantage of the method is that it is insensitive to the density of the graph. A broad state of the art of similarities between vertices is then proposed, as well as experimental comparisons of these measures. This is followed by the proposal of a robust method for comparing graphs sharing the same set of vertices. This measure is shown to be applicable to the comparison and merging of synonymy networks. Finally an application for the enrichment of lexical resources is presented. It consists in providing candidate synonyms on the basis of already existing links. In the second part, a parallel between formal concept analysis and clustering of bipartite graph is established. This parallel leads to the particular case where a partition of one of the vertex groups can be determined whereas there is no corresponding partition on the other group of vertices. A simple method that addresses this problem is proposed and evaluated. Finally, a system of automatic classification of search results (Kodex) is presented. This system is an application of previously seen clustering methods. An evaluation on a collection of two million web pages shows the benefits of the approach and also helps to understand some differences between clustering methods.Cette thèse s'organise en deux parties : une première partie s'intéresse aux mesures de similarité (ou de proximité) définies entre les sommets d'un graphe, une seconde aux méthodes de clustering de graphe biparti. Une nouvelle mesure de similarité entre sommets basée sur des marches aléatoires en temps courts est introduite. Cette méthode a l'avantage, en particulier, d'être insensible à la densité du graphe. Il est ensuite proposé un large état de l'art des similarités entre sommets, ainsi qu'une comparaison expérimentale de ces différentes mesures. Cette première partie se poursuit par la proposition d'une méthode robuste de comparaison de graphes partageant le même ensemble de sommets. Cette méthode est mise en application pour comparer et fusionner des graphes de synonymie. Enfin une application d'aide à la construction de ressources lexicales est présentée. Elle consiste à proposer de nouvelles relations de synonymie à partir de l'ensemble des relations de synonymie déjà existantes. Dans une seconde partie, un parallèle entre l'analyse formelle de concepts et le clustering de graphe biparti est établi. Ce parallèle conduit à l'étude d'un cas particulier pour lequel une partition d'un des groupes de sommets d'un graphe biparti peut-être déterminée alors qu'il n'existe pas de partitionnement correspondant sur l'autre type de sommets. Une méthode simple qui répond à ce problème est proposée et évaluée. Enfin Kodex, un système de classification automatique des résultats d'une recherche d'information est présenté. Ce système est une application en RI des méthodes de clustering vues précédemment. Une évaluation sur une collection de deux millions de pages web montre les avantages de l'approche et permet en outre de mieux comprendre certaines différences entre méthodes de clustering

Clique coverings of the edges of a random graph

The edges of the random graph (with the edge probability p=1/2) can be covered using O(n 2 lnln n/(ln n) 2 ) cliques. Hence this is an upper bound on the intersection number (also called clique cover number) of the random graph. A lower bound, obtained by counting arguments, is (1-e{open})n 2 /(2lg n) 2 . © 1993 Akadémiai Kiadó