Dans cette thèse, nous considérons plusieurs paramètres des hypergraphes et nous étudions si les restrictions aux sous-classes des hypergraphes permettent d’obtenir des propriétés combinatoires et algorithmiques souhaitables. La plupart des paramètres que nous prenons en compte sont des instances spéciales des packings et transversals des hypergraphes.Dans la première partie, nous allons nous concentrer sur les line graphs des graphes subcubiques sans triangle et nous allons démontrer que pour tous ces graphes il y a un independent set de taille au moins 3|V(G)|/10 et cette borne est optimale. Conséquence immédiate: nous obtenons une borne inférieure optimale pour la taille d’un couplage maximum dans les graphes subcubiques sans triangle. De plus, nous montrons plusieurs résultats algorithmiques liés au FEEDBACK VERTEX SET, HAMILTONIAN CYCLE et HAMILTONIAN PATH quand restreints aux line graphs des graphes subcubiques sans triangle.Puis nous examinons trois hypergraphes ayant la propriété d’Erdős-Pósa et nous cherchons à déterminer les fonctions limites optimales. Tout d’abord, nous apportons une fonction theta-bounding pour la classe des graphes subcubiques et nous étudions CLIQUE COVER: en répondant à une question de Cerioli et al., nous montrons qu’il admet un PTAS pour les graphes planaires. Par la suite, nous nous intéressons à la Conjecture de Tuza et nous montrons que la constante 2 peut être améliorée pour les graphes avec arêtes contenues dans au maximum quatre triangles et pour les graphes sans certains odd-wheels. Enfin, nous nous concentrons sur la Conjecture de Jones: nous la démontrons dans le cas des graphes sans griffes avec degré maximal 4 et nous faisons quelques observations dans le cas des graphes subcubiques.Nous étudions ensuite la VC-dimension de certains hypergraphes résultants des graphes. En particulier, nous considérons l’hypergraphe sur l’ensemble des sommets d’un certain graphe qui est induit par la famille de ses sous-graphes k-connexes. En généralisant les résultats de Kranakis et al., nous fournissons des bornes supérieures et inférieures optimales pour la VC-dimension et nous montrons que son calcul est NP-complet, pour chacun k > 0. Enfin, nous démontrons que ce problème (dans le cas k = 1) et le problème étroitement lié CONNECTED DOMINATING SET sont soit solvables en temps polynomial ou NP-complet, quand restreints aux classes de graphes obtenues en interdisant un seul sous-graphe induit.Dans la partie finale de cette thèse, nous nous attaquons aux meta-questions suivantes: Quand est-ce qu’un certain problème “difficile” de graphe devient “facile”?; Existe-t-il des frontières séparant des instances “faciles” et “difficiles”? Afin de répondre à ces questions, dans le cas des classes héréditaires, Alekseev a introduit la notion de boundary class pour un problème NP-difficile et a montré qu’un problème Pi est NP-difficile pour une classe héréditaire X finiment défini si et seulement si X contient un boundary class pour Pi. Nouscontinuons la recherche des boundary classes pour les problèmes suivants: HAMILTONIAN CYCLE THROUGH SPECIFIED EDGE, HAMILTONIAN PATH, FEEDBACK VERTEX SET, CONNECTED DOMINATING SET and CONNECTED VERTEX COVER. / In this thesis, we consider several hypergraph parameters and study whether restrictions to subclasses of hypergraphs allow to obtain desirable combinatorial or algorithmic properties. Most of the parameters we consider are special instances of packings and transversals of hypergraphs.In the first part, we focus on line graphs of subcubic triangle-free graphs and show that any such graph G has an independent set of size at least 3|V(G)|/10, the bound being sharp. As an immediate consequence, we obtain a tight lower bound for the matching number of subcubic triangle-free graphs. Moreover, we prove several algorithmic results related to FEEDBACK VERTEX SET, HAMILTONIAN CYCLE and HAMILTONIAN PATH when restricted to line graphs of subcubic triangle-free graphs.Then we consider three hypergraphs having the Erdős-Pósa Property and we seek to determine the optimal bounding functions. First, we provide an optimal theta-bounding function for the class of subcubic graphs and we study CLIQUE COVER: answering a question by Cerioli et al., we show it admits a PTAS for planar graphs. Then we focus on Tuza’s Conjecture and show that the constant 2 in the statement can be improved for graphs whose edges are contained in at most four triangles and graphs obtained by forbidding certain odd-wheels. Finally, we concentrate on Jones’ Conjecture: we prove it in the case of claw-free graphs with maximum degree at most 4 and we make some observations in the case of subcubic graphs.Then we study the VC-dimension of certain set systems arising from graphs. In particular, we consider the set system on the vertex set of some graph which is induced by the family of its k-connected subgraphs. Generalizing results by Kranakis et al., we provide tight upper and lower bounds for the VC-dimension and we show that its computation is NP-complete, for each k > 0. Finally, we show that this problem (in the case k = 1) and the closely related CONNECTED DOMINATING SET are either NP-complete or polynomial-time solvable when restricted to classes of graphs obtained by forbidding a single induced subgraph.In the final part of the thesis, we consider the following meta-questions: When does a certain “hard” graph problem become “easy”?; Is there any “boundary” separating “easy” and “hard” instances? In order to answer these questions in the case of hereditary classes, Alekseev introduced the notion of a boundary class for an NP-hard problem and showed that a problem Pi is NP-hard for a finitely defined (hereditary) class X if and only if X contains a boundary class for Pi. We continue the search of boundary classes for the following problems: HAMILTONIAN CYCLE THROUGH SPECIFIED EDGE, HAMILTONIAN PATH, FEEDBACK VERTEX SET, CONNECTED DOMINATING SET and CONNECTED VERTEX COVER.

The Support Vector (SV) machine is a novel type of learning machine, based on statistical learning theory, which contains polynomial classifiers, neural networks, and radial basis function (RBF) networks as special cases. In the RBF case, the SV algorithm automatically determines centers, weights and threshold such as to minimize an upper bound on the expected test error. The present study is devoted to an experimental comparison of these machines with a classical approach, where the centers are determined by $k$--means clustering and the weights are found using error backpropagation. We consider three machines, namely a classical RBF machine, an SV machine with Gaussian kernel, and a hybrid system with the centers determined by the SV method and the weights trained by error backpropagation. Our results show that on the US postal service database of handwritten digits, the SV machine achieves the highest test accuracy, followed by the hybrid approach. The SV approach is thus not only theoretically well--founded, but also superior in a practical application.

<p dir="ltr">The generalization error of a classifier is related to the complexity of the set of functions among which the classifier is chosen. We study a family of low-complexity classifiers consisting of thresholding a random one-dimensional feature. The feature is obtained by projecting the data on a random line after embedding it into a higher-dimensional space parametrized by monomials of order up to k. More specifically, the extended data is projected n-times and the best classifier among those n, based on its performance on training data, is chosen. </p><p dir="ltr">We show that this type of classifier is extremely flexible, as it is likely to approximate, to an arbitrary precision, any continuous function on a compact set as well as any Boolean function on a compact set that splits the support into measurable subsets. In particular, given full knowledge of the class conditional densities, the error of these low-complexity classifiers would converge to the optimal (Bayes) error as k and n go to infinity. On the other hand, if only a training dataset is given, we show that the classifiers will perfectly classify all the training points as k and n go to infinity. </p><p dir="ltr">We also bound the generalization error of our random classifiers. In general, our bounds are better than those for any classifier with VC dimension greater than O(ln(n)). In particular, our bounds imply that, unless the number of projections n is extremely large, there is a significant advantageous gap between the generalization error of the random projection approach and that of a linear classifier in the extended space. Asymptotically, as the number of samples approaches infinity, the gap persists for any such n. Thus, there is a potentially large gain in generalization properties by selecting parameters at random, rather than optimization. </p><p dir="ltr">Given a classification problem and a family of classifiers, the Rashomon ratio measures the proportion of classifiers that yield less than a given loss. Previous work has explored the advantage of a large Rashomon ratio in the case of a finite family of classifiers. Here we consider the more general case of an infinite family. We show that a large Rashomon ratio guarantees that choosing the classifier with the best empirical accuracy among a random subset of the family, which is likely to improve generalizability, will not increase the empirical loss too much. </p><p dir="ltr">We quantify the Rashomon ratio in two examples involving infinite classifier families in order to illustrate situations in which it is large. In the first example, we estimate the Rashomon ratio of the classification of normally distributed classes using an affine classifier. In the second, we obtain a lower bound for the Rashomon ratio of a classification problem with a modified Gram matrix when the classifier family consists of two-layer ReLU neural networks. In general, we show that the Rashomon ratio can be estimated using a training dataset along with random samples from the classifier family and we provide guarantees that such an estimation is close to the true value of the Rashomon ratio.</p>