Dans cette thèse, nous étudions des problèmes de transversaux d'un point de vue tant algorithmique que combinatoire. Etant donné un hypergraphe, un transversal est un ensemble de sommets qui touche toutes les hyperarêtes. Un packing est un ensemble d'hyperarêtes deux à deux disjointes. Alors que la taille minimale d'un transversal est au moins égale à la taille maximale d'un packing on ne peut pas dans le cas général borner la taille minimale d'un transversal par une fonction du packing maximal. Dans un premier temps, un état de l'art rappelle les différentes conditions qui assurent l'existence de bornes supérieures sur la taille des transversaux, en particulier en fonction de la taille d'un packing. La plupart d'entre elles sont valables lorsque la VC-dimension de Vapnik-Chervonenkis de l'hypergraphe, est bornée. L'originalité de la thèse consiste à utiliser ces outils d'hypergraphes pour obtenir des résultats sur des problèmes de graphes. Nous prouvons notamment une conjecture de coloration de Scott dans le cas des graphes sans-triangle maximaux; ensuite, nous généralisons un résultat de Chepoi, Estellon et Vaxès traitant de domination à grande distance; enfin nous nous attaquons à une conjecture de Yannakakis sur la séparation des cliques et des stables d'un graphe.Dans un second temps, nous étudions les transversaux d'un point de vue algorithmique. On se concentre plus particulièrement sur les problèmes de séparation de graphe où on cherche des transversaux à un ensemble de chemin. En combinant des outils de connexité, les séparateurs importants et le théorème de Dilworth, nous obtenons un algorithme FPT pour le problème Multicut paramétré par la taille de la solution. / 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.

Both the maximum agreement forest problem and the multicut on trees problem are NP-hard, thus cannot be solved efficiently if P /=NP. The maximum agreement forest problem was motivated in the study of evolution trees in bioinformatics, in which we are given two leaf-labeled trees and are asked to find a maximum forest that is a subgraph of both trees. The multicuton trees problem has applications in networks, in which we are given a forest and a set of pairs of termianls and are asked to find a cut that separates all pairs of terminals. We develop combinatorial and algorithmic techniques that lead to improved parameterized algorithms, approximation algorithms, and kernelization algorithms for these problems. For the maximum agreement forest problem, we proceed from the bottommost level of trees and extend solutions to whole trees. With this technique, we show that the maxi- mum agreement forest problem is fixed-parameterized tractable in general trees, resolving an open problem in this area. We also provide the first constant ratio approximation algorithm for the problem in general trees. For the multicut on trees problem, we take a new look at the problem through the eyes of vertex cover problem. This connection allows us to develop an kernelization algorithm for the problem, which gives an upper bound of O(k3) on the kernel size, significantly improving the previous best upper bound O(k6). We further exploit this connection to give a parameterized algorithm for the problem that runs in time O∗ (1.62k), thus improving the previous best algorithm of running time O∗ (2k). In the protein complex prediction problem, which comes directly from the study of bioinformatics, we are given a protein-protein interaction network, and are asked to find dense regions in this graph. We formulate this problem as a graph clustering problem and develop an algorithm to refine the results for identifying protein complexes. We test our algorithm on yeast protein- protein interaction networks, and we show that our algorithm is able to identify complexes more accurately than other existing algorithms.

Cut problems on graphs are a well-known and intensively studied class of optimization problems. In this thesis, we study the maximum cut problem, the maximum bond problem, and the minimum multicut problem through their associated polyhedra, i.e., the cut polytope, the bond polytope, and the multicut dominant, respectively. Continuing the research on the maximum cut problem and the cut polytope, we present a linear description for cut polytopes of K_{3,3}-minor-free graphs as well as an algorithm solving the maximum cut problem on these graphs in the same running time as planar maximum cut. Moreover, we give a complete characterization of simple and simplicial cut polytopes. Considering the maximum cut problem from an algorithmic point of view, we propose an FPT-algorithm for the maximum cut problem parameterized by the crossing number. We start the structural study of the bond polytope by investigating its basic properties and the effect of graph operations on the bond polytope and its facet-defining inequalities. After presenting a linear-time reduction of the maximum bond problem to the maximum bond problem on 3-connected graphs, we discuss valid and facet defining inequalities arising from edges and cycles. These inequalities yield a complete linear description for bond polytopes of 3-connected planar (K_5-e)-minor-free graphs. This polytopal result is mirrored algorithmically by proposing a linear-time algorithm for the maximum bond problem on (K_5-e)-minor-free graphs. Finally, we start the structural study of the multicut dominant. We investigate basic properties, which gives rise to lifting and projection results for the multicut dominant. Then, we study the effect of graph operations on the multicut dominant and its facet-defining inequalities. Moreover, we present facet-defining inequalities supported on stars, trees, and cycles as well as separation algorithms for the former two classes of inequalities.

O Problema do Multicorte Dirigido Mínimo é um problema clássico em otimização combinatória. Ele é NP-difícil mesmo para instâncias muito simples. Este trabalho faz uma análise dos algoritmos exatos e de aproximação para resolver o problema. Também implementa alguns desses algoritmos e compara seus desempenhos. / The directed multicut problem is a classical problem in combinatorial optimization. It is NP-hard even for very simple families of instances. This work makes an analysis of the exact and approximation algorithms for the problem. It also implements some of these algorithms and compares their performances.

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.

O Problema do Multicorte Dirigido Mínimo é um problema clássico em otimização combinatória. Ele é NP-difícil mesmo para instâncias muito simples. Este trabalho faz uma análise dos algoritmos exatos e de aproximação para resolver o problema. Também implementa alguns desses algoritmos e compara seus desempenhos. / The directed multicut problem is a classical problem in combinatorial optimization. It is NP-hard even for very simple families of instances. This work makes an analysis of the exact and approximation algorithms for the problem. It also implements some of these algorithms and compares their performances.

Dans cette thèse, nous abordons des problèmes NP-difficiles à l'aide de techniques combinatoires, en se focalisant sur le domaine de la complexité paramétrée. Les principaux problèmes que nous considérons sont les problèmes de Multicoupe et d'Arbre Orienté Couvrant avec Beaucoup de Feuilles. La Multicoupe est une généralisation naturelle du très classique problème de coupe, et consiste à séparer un ensemble donné de paires de sommets en supprimant le moins d'arêtes possible dans un graphe. Le problème d'Arbre Orienté Couvrant avec Beaucoup de Feuilles consiste à trouver un arbre couvrant avec le plus de feuilles possible dans un graphe dirigé. Les résultats principaux de cette thèse sont les suivants. Nous montrons que le problème de Multicoupe paramétré par la taille de la solution est FPT (soluble à paramètre fixé), c'est-à-dire que l'existence d'une multicoupe de taille $k$ dans un graphe à $n$ sommets peut être décidée en temps $f(k)*poly(n)$. Nous montrons que Multicoupe dans les arbres admet un noyau polynomial, c'est-à-dire est réductible aux instances de taille polynomiale en $k$. Nous donnons un algorithme en temps $O^*(3.72^k)$ pour le problème d'Arbre Orienté Couvrant avec Beaucoup de Feuilles et le premier algorithme exponentiel exact non trivial (c'est-à-dire meilleur que $2^n$). Nous fournissons aussi un noyau quadratique et une approximation à facteur constant. Ces résultats algorithmiques sont basés sur des résultats combinatoires et des propriétés structurelles qui concernent, entre autres, les décompositions arborescentes, les mineurs, des règles de réduction et les $s-t$ numberings. Nous présentons des résultats combinatoires hors du domaine de la complexité paramétrée: une caractérisation des graphes de cercle Helly comme les graphes de cercle sans diamant induit, et une caractérisation partielle des classes de graphes 2-bel-ordonnées. / This thesis tackles NP-hard problems with combinatorial techniques, focusing on the framework of Fixed-Parameter Tractability. The main problems considered here are Multicut and Maximum Leaf Out-branching. Multicut is a natural generalisation of the cut problem, and consists in simultaneously separating prescribed pairs of vertices by removing as few edges as possible in a graph. Maximum Leaf Out-branching consists in finding a spanning directed tree with as many leaves as possible in a directed graph. The main results of this thesis are the following. We show that Multicut is FPT when parameterized by the solution size, i.e. deciding the existence of a multicut of size $k$ in a graph with $n$ vertices can be done in time $f(k)*poly(n)$. We show that Multicut In Trees admits a polynomial kernel, i.e. can be reduced to instances of size polynomial in $k$. We give an $O^*(3.72^k)$ algorithm for Maximum Leaf Out-branching and the first non-trivial (better than $2^n$) exact algorithm. We also provide a quadratic kernel and a constant factor approximation algorithm. These algorithmic results are based on combinatorial results and structural properties, involving tree decompositions, minors, reduction rules and $s-t$ numberings, among others. We present results obtained with combinatorial techniques outside the scope of parameterized complexity: a characterization of Helly circle graphs as the diamond-free circle graphs, and a partial characterisation of 2-well-quasi-ordered classes of graphs.

Cette thèse est composée de deux parties, chacune portant sur un sous-domaine de l'optimisation combinatoire a priori distant de l'autre. Le premier thème de recherche abordé est la programmation biniveau stochastique. Se cachent derrière ce terme deux sujets de recherche relativement peu étudiés conjointement, à savoir d'un côté la programmation stochastique, et de l'autre la programmation biniveau. La programmation mathématique (PM) regroupe un ensemble de méthodes de modélisation et de résolution, pouvant être utilisées pour traiter des problèmes pratiques que se posent des décideurs. La programmation stochastique et la programmation biniveau sont deux sous-domaines de la PM, permettant chacun de modéliser un aspect particulier de ces problèmes pratiques. Nous élaborons un modèle mathématique issu d'un problème appliqué, où les aspects biniveau et stochastique sont tous deux sollicités, puis procédons à une série de transformations du modèle. Une méthode de résolution est proposée pour le PM résultant. Nous démontrons alors théoriquement et vérifions expérimentalement la convergence de cette méthode. Cet algorithme peut être utilisé pour résoudre d'autres programmes biniveaux que celui qui est proposé.Le second thème de recherche de cette thèse s'intitule "problèmes de coupe et de couverture partielles dans les graphes". Les problèmes de coupe et de couverture sont parmi les problèmes de graphe les plus étudiés du point de vue complexité et algorithmique. Nous considérons certains de ces problèmes dans une variante partielle, c'est-à-dire que la propriété de coupe ou de couverture dont il est question doit être vérifiée partiellement, selon un paramètre donné, et non plus complètement comme c'est le cas pour les problèmes originels. Précisément, les problèmes étudiés sont le problème de multicoupe partielle, de coupe multiterminale partielle, et de l'ensemble dominant partiel. Les versions sommets des ces problèmes sont également considérés. Notons que les problèmes en variante partielle généralisent les problèmes non partiels. Nous donnons des algorithmes exacts lorsque cela est possible, prouvons la NP-difficulté de certaines variantes, et fournissons des algorithmes approchés dans des cas assez généraux. / This thesis is composed of two parts, each part belonging to a sub-domain of combinatorial optimization a priori distant from the other. The first research subject is stochastic bilevel programming. This term regroups two research subject rarely studied together, namely stochastic programming on the one hand, and bilevel programming on the other hand. Mathematical Programming (MP) is a set of modelisation and resolution methods, that can be used to tackle practical problems and help take decisions. Stochastic programming and bilevel programming are two sub-domains of MP, each one of them being able to model a specific aspect of these practical problems. Starting from a practical problem, we design a mathematical model where the bilevel and stochastic aspects are used together, then apply a series of transformations to this model. A resolution method is proposed for the resulting MP. We then theoretically prove and numerically verify that this method converges. This algorithm can be used to solve other bilevel programs than the ones we study.The second research subject in this thesis is called "partial cut and cover problems in graphs". Cut and cover problems are among the most studied from the complexity and algorithmical point of view. We consider some of these problems in a partial variant, which means that the cut or cover property that is looked into must be verified partially, according to a given parameter, and not completely, as it was the case with the original problems. More precisely, the problems that we study are the partial multicut, the partial multiterminal cut, and the partial dominating set. Versions of these problems were vertices are