Etude topologique et algorithmique des structures tropicales dans des graphes sommet-coloriés

par Phong Pham

Projet de thèse en Informatique

Sous la direction de Yannis Manoussakis.

Thèses en préparation à Paris Saclay , dans le cadre de Sciences et Technologies de l'Information et de la Communication , en partenariat avec LRI - Laboratoire de Recherche en Informatique (laboratoire) , GALaC - Graphes, Algorithmes et Combinatoire (equipe de recherche) et de Université Paris-Sud (établissement de préparation de la thèse) depuis le 01-11-2015 .

  • Résumé

    Etude topologique et algorithmique des structures tropicales dans des graphes sommet-coloriés.

  • Titre traduit

    Topological and algorithmic study (NP-completeness, approximability, non-approximability) of tropical structures in vertex-colored graphs

  • Résumé

    Recent years there is a great interest on problems in vertex-colored graphs motivated by both their theoretical interest and applications in various field such molecular biology, Social Sciences, the Web graph etc. For instance the web graph may be considered as a vertex-colored graph where each color corresponds to the domain (mathematics, physics, chemistry etc) of the content of the page. Original problems here correspond to extract subgraphs (for ex. dominating sets, vertex covers, independent sets, connected components etc.) colored in a specified tropical patterns i.e. each color appears at least ones in the sub-graph under consideration. This is quite new subject, and there are no so much published results. Here we propose to start by studying some basic graph problems, namely, tropical vertex cover, tropical independent set, tropical matchings, and minimum tropical components for (classic and random) vertex-colored graphs. In a first research axis, we wish to see if some fundamental problems (maximum, vertex cover, independent set and k-connected component) are polynomial or remain NP-hard for very specific families of graphs (dense, planar, perfect or cubic graphs). In a second axis we wish to establish approximation results within a constant factor (if possible) or prove non-approximability results for general vertex-colored graphs. We are expecting to obtain some better approximation results for specific families of graphs, especially trees or dense graphs. In a third step we will establish some structural results involving sufficient conditions on degrees, number of edges etc and then prove that these results are the best possible. As a last step, we wish to study these problems in the case of random graphs, as they may be used to model the web graph. Step 1: Try to understand the difficulty of chosen fundamental graph problems. In particular, see for which families of graphs they remain NP-hard. Step 2 : Study approximation algorithms. Recall that, up to now, no approximation (or non approximation) results are unknown. Step 3: Prove that there are "good" approximations (i.e., within a constant factor) for trees and a non fixed number of colors. If not true, establish non-approximation results. In addition, determine specified families of graphs for which the problem is polynomial, as for example, dense graphs. Step 4 : Establish sufficient conditions involving several parameters of graphs (degrees, number of edges, connectivity etc.) guarantying the existence of tropical dominating sets, vertex covers, independent sets and connected components, with bounded number of vertices. Step 5: Study the above problems in the case of vertex-colored random graphs as these graphs may be used to model the web graph.