In this manuscript we study hitting sets both from a combinatorial and from an algorithmic point of view. A hitting set is a subset of vertices of a hypergraph which intersects all the hyperedges. A packing is a subset of pairwise disjoint hyperedges. In the general case, there is no function linking the minimum size of a hitting set and a maximum size of a packing.The first part of this thesis is devoted to present upper bounds on the size of hitting sets, in particular this upper bounds are expressed in the size of the maximum packing. Most of them are satisfied when the dimension of Vapnik-Chervonenkis of the hypergraph is bounded. The originality of this thesis consists in using these hypergraph tools in order to obtain several results on graph problems. First we prove that a conjecture of Scott holds for maximal triangle-free graphs. Then we generalize a result of Chepoi, Estellon and Vaxès on dominating sets at large distance. We finally study a conjecture of Yannakakis and prove that it holds for several graph subclasses using VC-dimension.The second part of this thesis explores algorithmic aspects of hitting sets. More precisely we focus on parameterized complexity of graph separation problems where we are looking for hitting sets of a set of paths. Combining connectivity tools, important separator technique and Dilworth's theorem, we design an FPT algorithm for the Multicut problem parameterized by the size of the solution.
| Identifer | oai:union.ndltd.org:CCSD/oai:tel.archives-ouvertes.fr:tel-01012106 |
| Date | 09 December 2013 |
| Creators | Bousquet, Nicolas |
| Publisher | Université Montpellier II - Sciences et Techniques du Languedoc |
| Source Sets | CCSD theses-EN-ligne, France |
| Language | English |
| Detected Language | English |
| Type | PhD thesis |
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