<div>Spectral Methods have had huge influence of modern algorithm design. For algorithmic problems on graphs, this is done by using a deep connection between random walks and the powers of various natural matrices associated with the graph. The major contribution</div><div>of this thesis initiates attempts to recover algorithmic results in Graph Minor Theory via spectral methods.</div><div><br></div><div>We make progress towards this goal by exploring these questions in the Property Testing Model for bounded degree graphs. Our main contributions are</div><div>1. The first result gives an almost query optimal one-sided tester for the property of H-minor-freeness. Benjamini-Schramm-Shapira (STOC 2008) conjectured that for fixed H, this can be done in time O(sqrt n). Our algorithm solves this in time n^{1/2+o(1)} which nearly resolves this upto n^{o(1)} factors.</div><div><br></div><div>2. BSS also conjectured that in the two-sided model, H-minor-freeness can be tested in time poly(1/eps). We resolve this conjecture in the affirmative.</div><div><br></div><div>3.Lastly, in a previous work on the two-sided-question above, Hassidim-Kelner-Nguyen-Onak (FOCS 2009) introduced a tool they call partition oracle. They conjectured that partition oracles could be implemented in time poly(1/eps) and gave an implementation which took exp(poly(1/eps)) time. In this work, we resolve this conjecture and produce such an oracle.</div><div><br></div><div><br></div><div>Additionally, this work also presents an algorithm which can recover a planted 3-coloring in a graph with some random like properties and suggests some future research directions alongside.</div>

Savoir comment transmettre une information est fondamental dans un réseau. Il est essentiel que chaque entité du réseau soit capable de décider localement, avec sa vue du réseau, du chemin par lequel l'information doit passer. Ainsi, il est souvent utile d'étudier la topologie du réseau, modélisée par un graphe, pour répondre à ces exigences. Nous nous intéressons dans un premier temps, à la décomposition arborescente des graphes planaires. En effet, comme dans beaucoup de problèmes de graphes, l'étude de la topologie des graphes nous conduit à procéder à une décomposition du graphe afin d'exploiter les propriétés structurelles qui en découlent. En suite, nous nous sommes aussi intéressés à la structure des graphes qui excluent un mineur H, en particulier le graphe K_{2,r}. Ces travaux nous ont permis d'améliorer les bornes actuelles connues sur la largeur arborescente de ces graphes. Dans la dernière partie, nous abordons le problème du routage compact. Nous nous sommes intéressés aux schémas de routage de plus courts chemins utilisant des adresses, des tables de routage de tailles optimales de O(log n) bits, où n est le nombre de sommets du graphe. Nous proposons un tel schéma de routage pour une famille de graphes valués contenant les arbres et les graphes planaire-extérieurs. / In a network, it is crucial to know how to construct an efficent routing scheme. It is fundamental for each entity with its local knowledge of the network, to be able to decide on which link to forward messages. Thus, it is important to sutdy the underlying network topology in order to design routing schemes. In the first part of this thesis, we construct a new tree-decomposition for planar graphs. In fact, as in many graph problems, the study of the graph structure leads to do a tree-decomposition for exploiting structural propertys of the graphs. In second part, we studied the structure of H-minor free graphs, in particular whenever H = K_{2,r}. Our results improve upon previous known bounds about the tree-width of K_{2,r}-minor free graphs. At last, we treat the problème of compact routing scheme. More precisely, we are interested in shortest-path routing schemes that use O(\log n) bits for addresses, headers and routing tables, where n is the number of vertices in the graph. We propose such a routing scheme for a large family of weighted graphs including outerplanar graphs.