Geometric covering problems have always been of focus in computer scientific research. The generic geometric covering problem asks to cover a set S of n objects with another set of objects whose cardinality is minimum, in a geometric setting. Many versions of geometric cover have been studied in detail, one of which is line cover: Given a set of points in the plane, find the minimum number of lines to cover them. In Euclidean space Rm, this problem is known as Hyperplane Cover, where lines are replaced by affine hyperplanes bounded by dimension d. Line cover is NP-hard, so is its hyperplane analogue. Our thesis focuses on few extensions of hyperplane cover and line cover. One of the techniques used to study NP-hard problems is Fixed Parameter Tractability (FPT), where, in addition to input size, a parameter k is provided for input instance. We ask to solve the problem with respect to k, such that the running time is a function in both n and k, strictly polynomial in n, while the exponential component is limited to k. In this thesis, we study FPT and parameterized complexity theory, the theory of classifying hard problems involving a parameter k. We focus on two new geometric covering problems: covering a set of points in the plane with conics (conic cover) and covering a set of points with strips or fat lines of given width in the plane (fat line cover). A conic is a non-degenerate curve of degree two in the plane. A fat line is defined as a strip of finite width w. In this dissertation, we focus on the parameterized versions of these two problems, where, we are asked to cover the set of points with k conics or k fat lines. We use the existing techniques of FPT algorithms, kernelization and approximation algorithms to study these problems. We do a comprehensive study of these problems, starting with NP-hardness results to studying their parameterized hardness in terms of parameter k. We show that conic cover is fixed parameter tractable, and give an algorithm of running time O∗ ((k/1.38)^4k), where, O∗ implies that the running time is some polynomial in input size. Utilizing special properties of a parabola, we are able to achieve a faster algorithm and show a running time of O∗ ((k/1.15)^3k). For fat line cover, first we establish its NP-hardness, then we explore algorithmic possibilities with respect to parameterized complexity theory. We show W [1]-hardness of fat line cover with respect to the number of fat lines, by showing a parameterized reduction from the problem of stabbing axis-parallel squares in the plane. A parameterized reduction is an algorithm which transforms an instance of one parameterized problem into an instance of another parameterized problem using a FPT-algorithm. In addition, we show that some restricted versions of fat line cover are also W [1]-hard. Further, in this thesis, we explore a restricted version of fat line cover, where the set of points are integer coordinates and allow only axis-parallel lines to cover them. We show that the problem is still NP-hard. We also show that this version is fixed parameter tractable having a kernel size of O (k^2) and give a FPT-algorithm with a running time of O∗ (3^k). Finally, we conclude our study on this problem by giving an approximation algorithm for this version having a constant approximation ratio 2.

While polynomial-time approximation algorithms remain a dominant notion in tackling computationally hard problems, the framework of parameterized complexity has been emerging rapidly in recent years. Roughly speaking, the analytic framework of parameterized complexity attempts to grasp the difference between problems which admit O(c^k . poly(n))-time algorithms such as Vertex Cover, and problems like Dominating Set for which essentially brute-force O(n^k)-algorithms are best possible until now. Problems of the former type is said to be fixed-parameter tractable (FPT) and those of the latter type are regarded intractable. In this thesis, we investigate some problems on directed graphs and a number of constraint satisfaction problems (CSPs) from the parameterized perspective. We develop fixed-parameter algorithms for some digraph problems. In particular, we focus on the basic problem of finding a tree with certain property embedded in a given digraph. New or improved fpt-algorthms are presented for finding an out-branching with many or few leaves (Directed Maximum Leaf, Directed Minimum Leaf problems). For acyclic digraphs, Directed Maximum Leaf is shown to allow a kernel with linear number of vertices. We suggest a kernel for Directed Minimum Leaf with quadratic number of vertices. An improved fpt-algorithm for finding k-Out-Tree is presented and this algorithm is incorporated as a subroutine to obtain a better algorithm for Directed Minimum Leaf. In the second part of this thesis, we concentrate on several CSPs in which we want to maximize the number of satisfied constraints and consider parameterization "above tight lower bound" for these problems. To deal with this type of parameterization, we present a new method called SABEM using probabilistic approach and applying harmonic analysis on pseudo-boolean functions. Using SABEM we show that a number of CSPs admit polynomial kernels, thus being fixed-parameter tractable. Moreover, we suggest some problem-specific combinatorial approaches to Max-2-Sat and a wide special class of Max-Lin2, which lead to a kernel of smaller size than what can be obtained using SABEM for respective problems.

Rampant lateral gene transfer (LGT) among prokaryotes, hybridization in plants and other reticulate evolutionary processes invalidate typical phylogenetic tree models by violating the assumption that organisms only inherit genetic information from a single parent species. Comparing the different evolutionary histories of multiple genes is necessary to identify and assess these processes. In this work I develop efficient approximation and fixed-parameter algorithms for computing rooted maximum agreement forests (MAFs) and maximum acyclic agreement forests (MAAFs) of pairs of phylogenetic trees. Their sizes correspond to the subtree-prune-and-regraft (SPR) distance and the hybridization number of these pairs of trees, which are important measures of the dissimilarity of phylogenies used in studying reticulate evolution. Although these MAFs and MAAFs are NP-hard to compute, my fixed-parameter algorithms are practical because they scale exponentially with the computed distance rather than the size of the trees. I contribute efficient fixed-parameter algorithms for computing MAFs and MAAFs of two binary rooted trees and give the first efficient fixed-parameter and approximation algorithms for computing MAFs of two multifurcating rooted trees. My open-source implementation of the MAF algorithms is orders of magnitude faster than previous approaches, reducing the time required to compute SPR distances of 46 between trees of 144 species to fractions of a second whereas previous approaches required hours to compute SPR distances of 25. These fast MAF-based distance metrics enable the construction of supertrees to reconcile a collection of gene trees and rapid inference of LGT. Simulations demonstrate that supertrees minimizing the SPR distance are more accurate than other supertree methods under plausible rates of LGT. I constructed an SPR supertree from a phylogenomic dataset of 40,631 gene trees covering 244 genomes from several major bacterial phyla and inferred "highways" of gene transfer between these bacterial classes and genera; a small number of these highways connect distantly related genera and can highlight specific genes implicated in long-distance LGT. These fast MAF algorithms are thus practical and enable new analyses of reticulate evolution.

L'étude expérimentale d'algorithmes est un sujet crucial dans la conception de nouveaux algorithmes, puisque le contexte d'évaluation influence inévitablement la mesure de la qualité des algorithmes. Le sujet particulier qui nous intéresse dans l'étude expérimentale est la pertinence des instances choisies pour servir de base de test à l'expérimentation. Nous formalisons ce critère par la notion de "difficulté d'instance" qui dépend des performances pratiques de méthodes de résolution. Le coeur de la thèse porte sur un outil pour évaluer empiriquement la difficulté d'instance. L'approche proposée présente une méthode de benchmarking d'instances sur des jeux de test d'algorithmes. Nous illustrons cette méthode expérimentale pour évaluer des classes d'instances à travers plusieurs exemples d'applications sur le problème du voyageur de commerce. Nous présentons ensuite une approche pour générer des instances difficiles. Elle repose sur des opérations qui modifient les instances, mais qui permettent de retrouver facilement une solution optimale, d'une instance à l'autre. Nous étudions théoriquement et expérimentalement son impact sur les performances de méthodes de résolution. / The empirical study of algorithms is a crucial topic in the design of new algorithms because the context of evaluation inevitably influences the measure of the quality of algorithms. In this topic, we particularly focus on the relevance of instances forming testbeds. We formalize this criterion with the notion of 'instance hardness' that depends on practical performance of some resolution methods. The aim of the thesis is to introduce a tool to evaluate instance hardness. The approach uses benchmarking of instances against a testbed of algorithms. We illustrate our experimental methodology to evaluate instance classes through several applications to the traveling salesman problem. We also suggest possibilities to generate hard instances. They rely on operations that modify instances but that allow to easily find the optimal solution of one instance from the other. We theoretically and empirically study their impact on the performance of some resolution methods.

Dans cette thèse, nous étudions diverses largeurs de graphes autour de la largeur arborescente ainsi que de la largeur de clique. Nous commençons avec une étude comparative entre la largeur arborescente d’un graphe et la largeur de clique du graphe d’incidence associé, de laquelle nous extrayons des résultats algorithmiques encourageants. Puis nous présentons quelques propriétés structurelles liées à la largeur arborescente spéciale, largeur relativement récente qui est à mi-chemin entre les deux largeurs précédentes. Enfin nous nous intéressons à une notion plus générale connue sous le nom de fonction de partition sous-modulaire qui englobe, entre autres, les largeurs arborescentes « classique » et spéciale, la largeur de chemin ainsi que la largeur linéaire et les largeurs de branches de coupe et de découpe. Nous présentons alors un algorithme linéaire à paramètre fixé pour le calcul de ces différentes largeurs, lequel généralise un certain nombre de résultats propres à chacune de ces largeurs. / In this thesis, we study some width parameters on graphs, beyond tree-width and clique-width. Our first investigation is a comparative study between the tree-width of a graph and the clique-width of the associated incidence graph, from which we extract some strong algorithmic results. Then we present a few structural properties over a recently defined width called special tree-width and which takes its definition through both tree-width and clique-width. Finally, we end our journey with a more general notion named sub-modular partition fonction and which encompass both “classic” and special tree-widths, path-width, branch-width, linear-width, cut-width and carvingwidth among others. So, we introduce a fixed parameter tractable algorithm computing those widths parameters and thus we generalize a number of results specific to each of them.

L'étude expérimentale d'algorithmes est un sujet crucial dans la conception de nouveaux algorithmes, puisque le contexte d'évaluation influence inévitablement la mesure de la qualité des algorithmes. Le sujet particulier qui nous intéresse dans l'étude expérimentale est la pertinence des instances choisies pour servir de base de test à l'expérimentation. Nous formalisons ce critère par la notion de "difficulté d'instance" qui dépend des performances pratiques de méthodes de résolution. Le coeur de la thèse porte sur un outil pour évaluer empiriquement la difficulté d'instance. L'approche proposée présente une méthode de benchmarking d'instances sur des jeux de test d'algorithmes. Nous illustrons cette méthode expérimentale pour évaluer des classes d'instances à travers plusieurs exemples d'applications sur le problème du voyageur de commerce. Nous présentons ensuite une approche pour générer des instances difficiles. Elle repose sur des opérations qui modifient les instances, mais qui permettent de retrouver facilement une solution optimale, d'une instance à l'autre. Nous étudions théoriquement et expérimentalement son impact sur les performances de méthodes de résolution.

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