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NP-Hardness is a well-known theory to identify the hardness of computational problems. It is believed that NP-Hard problems are unlikely to admit polynomial-time algorithms. However since many NP-Hard problems are of practical significance, different approaches are proposed to solve them: Approximation algorithms, randomized algorithms and heuristic algorithms. None of the approaches meet the practical needs. Recently parameterized computation and complexity has attracted a lot of attention and been a fruitful branch of the study of efficient algorithms. By taking advantage of the moderate value of parameters in many practical instances, we can design efficient algorithms for the NP-Hard problems in practice. In this dissertation, we discuss a new approach to design efficient parameterized algorithms, kernelization. The motivation is that instances of small size are easier to solve. Roughly speaking, kernelization is a preprocess on the input instances and is able to significantly reduce their sizes. We present a 2k kernel for the cluster editing problem, which improves the previous best kernel of size 4k; We also present a linear kernel of size 7k 2d for the d-cluster editing problem, which is the first linear kernel for the problem. The kernelization algorithm is simple and easy to implement. We propose a quadratic kernel for the pseudo-achromatic number problem. This implies that the problem is tractable in term of parameterized complexity. We also study the general problem, the vertex grouping problem and prove it is intractable in term of parameterized complexity. In practice, many problems seek a set of good solutions instead of a good solution. Motivated by this, we present the framework to study enumerability in term of parameterized complexity. We study three popular techniques for the design of parameterized algorithms, and show that combining with effective enumeration techniques, they could be transferred to design efficient enumeration algorithms.

Clustering is the unsupervised classification of patterns into groups, which is easy provided the data of patterns are consistent. However, real data are almost always tempered with inconsistencies, which make it a hard problem, and actually, the most widely studied formulations, correlation clustering and hierarchical clustering, are both NP-hard. In the graph representation of data, inconsistencies also frequently present themselves as cycles, also called deadlocks, and to break cycles by removing vertices is the objective of the classical feedback vertex set (FVS) problem. This dissertation studies the three problems, correlation clustering, hierarchical clustering, and disjoint-FVS (a variation of FVS), from a kernelization approach. A kernelization algorithm in polynomial time reduces a problem instance provably to speed up the further processing with other approaches. For each of the problems studied, an efficient kernelization algorithm of linear or sub-quadratic running time is presented. All the kernels obtained in this dissertation have linear size with very small constants. Better parameterized algorithms are also designed based on the kernels for the last two problems. Finally, some concluding remarks on possible directions for future research are briefly mentioned.

The StringToStringCorrection problem asks, given mutable string M, target string T, and positive integer k, can M be mutated into T using at most k operations (single symbol deletions or swaps of adjacent symbols) applied to M? The problem is known to be NP-complete. Abu-Khzam et al. give the first fixed-parameter algorithm for the problem when the parameter is the number of operations permitted. Their technique, charge and reduce, gives a O^∗(1.612k) bounded search tree algorithm, but leaves open whether a poly-size kernel exists. We show, using two polynomial time reduction rules on large regions of the input strings, that the problem has a problem kernel of size O(k^4). Our algorithm achieves this in a polynomial running time. Additionally, we introduce the problem k-MultiStringToStringCorrection (k-MS2SC), a generalized version of StringToStringCorrection, and prove it to be fixed-parameter tractable. / Graduate / 0984 / nwatt@uvic.ca

The Constraint Satisfaction Problem (CSP) is a fundamental NP-complete problem with many applications in artificial intelligence. This problem has enjoyed considerable scientific attention in the past decades due to its practical usefulness and the deep theoretical questions it relates to. However, there is a wide gap between practitioners, who develop solving techniques that are efficient for industrial instances but exponential in the worst case, and theorists who design sophisticated polynomial-time algorithms for restrictions of CSP defined by certain algebraic properties. In this thesis we attempt to bridge this gap by providing polynomial-time algorithms to test for membership in a selection of major tractable classes. Even if the instance does not belong to one of these classes, we investigate the possibility of decomposing efficiently a CSP instance into tractable subproblems through the lens of parameterized complexity. Finally, we propose a general framework to adapt the concept of kernelization, central to parameterized complexity but hitherto rarely used in practice, to the context of constraint reasoning. Preliminary experiments on this last contribution show promising results.

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Le problème de satisfaction de contraintes (CSP) est un problème NP-complet classique en intelligence artificielle qui a suscité un engouement important de la communauté scientifique grâce à la richesse de ses aspects pratiques et théoriques. Cependant, au fil des années un gouffre s'est creusé entre les praticiens, qui développent des méthodes exponentielles mais efficaces pour résoudre des instances industrielles, et les théoriciens qui conçoivent des algorithmes sophistiqués pour résoudre en temps polynomial certaines restrictions de CSP dont l'intérêt pratique n'est pas avéré. Dans cette thèse nous tentons de réconcilier les deux communautés en fournissant des méthodes polynomiales pour tester automatiquement l'appartenance d'une instance de CSP à une sélection de classes traitables majeures. Anticipant la possibilité que les instances réelles ne tombent que rarement dans ces classes traitables, nous analysons également de manière systématique la possibilité de décomposer efficacement une instance en sous-problèmes traitables en utilisant des méthodes de complexité paramétrée. Finalement, nous introduisons un cadre général pour exploiter dans les CSP les idées développées pour la kernelization, un concept fondamental de complexité paramétrée jusqu'ici peu utilisé en pratique. Ce dernier point est appuyé par des expérimentations prometteuses. / The Constraint Satisfaction Problem (CSP) is a fundamental NP-complete problem with many applications in artificial intelligence. This problem has enjoyed considerable scientific attention in the past decades due to its practical usefulness and the deep theoretical questions it relates to. However, there is a wide gap between practitioners, who develop solving techniques that are efficient for industrial instances but exponential in the worst case, and theorists who design sophisticated polynomial-time algorithms for restrictions of CSP defined by certain algebraic properties. In this thesis we attempt to bridge this gap by providing polynomial-time algorithms to test for membership in a selection of major tractable classes. Even if the instance does not belong to one of these classes, we investigate the possibility of decomposing efficiently a CSP instance into tractable subproblems through the lens of parameterized complexity. Finally, we propose a general framework to adapt the concept of kernelization, central to parameterized complexity but hitherto rarely used in practice, to the context of constraint reasoning. Preliminary experiments on this last contribution show promising results.

The classical graph edge coloring problem deals in coloring the edges of a given graph with minimum number of colors such that no two adjacent edges in the graph, get the same color in the proposed coloring. In the following work, we look at the other end of the spectrum where in our goal is to maximize the number of colors used for coloring the edges of the graph under some vertex specific constraints. We deal with the MAXIMUM EDGE COLORING problem which is defined as the following –For an integer q ≥2 and a graph G, the goal is to find a coloring of the edges of G with the maximum number of colors such that every vertex of the graph sees at most q colors. The question is very well motivated by the problem of channel assignment in wireless networks. This problem is NP-hard for q ≥ 2, and has been well-studied from the point of view of approximation. This problem has not been studied in the parameterized context before. Hence as a next step, this thesis investigates the parameterized complexity of this problem where the standard parameter is the solution size. The main focus of the work is the special case of q=2 ,i.e. MAXIMUM EDGE 2-COLORING which is theoretically intricate and practically relevant in the wireless networks setting. We first show an exponential kernel for the MAXIMUM EDGE q-COLORING problem where q is a fixed constant and q ≥ 2.We do a more specific analysis for the kernel of the MAXIMUM EDGE 2-COLORING problem. The kernel obtained here is still exponential in size but is better than the kernel obtained for MAXIMUM EDGE q-COLORING problem in case of q=2. We then show a fixed parameter tractable algorithm for the MAXIMUM EDGE 2-COLORING problem with a running time of O*∗(kO(k)).We also show a fixed parameter tractable algorithm for the MAXIMUM EDGE q-COLORING problem with a running time of O∗(kO(qk) qO(k)). The fixed parameter tractability of the dual parametrization of the MAXIMUM EDGE 2-COLORING problem is established by arguing a linear vertex kernel for the problem. We also show that the MAXIMUM EDGE 2-COLORING problem remains hard on graphs where the maximum degree is a constant and also on graphs without cycles of length four. In both these cases, we obtain quadratic kernels. A closely related variant of the problem is the question of MAX EDGE{1,2-}COLORING. For this problem, the vertices in the input graph may have different qε,{1.2} values and the goal is to use at least k colors for the edge coloring of the graph such that every vertex sees at most q colors, where q is either one or two. We show that the MAX EDGE{1,2}-COLORING problem is W[1]-hard on graphs that have no cycles of length four.

Dans le cadre de cette thèse, nous considérons la complexité paramétrée de problèmes NP-complets. Plus précisément, nous nous intéressons à l'existence d'algorithmes de noyau polynomiaux pour des problèmes d'édition de graphes et de contraintes. Nous introduisons en particulier la notion de branches, qui permet d'obtenir des algorithmes polynomiaux pour des problèmes d'édition de graphes lorsque la classe de graphes cible respecte une décomposition d'adjacence. Cette technique nous permet ainsi d'élaborer les premiers algorithmes de noyaux polynomiaux pour les problèmes Closest 3-Leaf Power, Cograph Edition et Proper Interval Completion. Ces résultats constituent les premiers noyaux polynomiaux pour ces problèmes. Concernant les problèmes d'édition de contraintes, nous étendons la notion de Conflict Packing, qui a déjà été utilisée dans quelques problèmes paramétrés et permet d'élaborer des algorithmes de noyau linéaires pour différents problèmes. Nous présentons un noyau linéaire pour le problème Feedback Arc Set in Tournaments, et adaptons les techniques utilisées pour obtenir un noyau linéaire pour le problème Dense Rooted Triplet Inconsistency. Dans les deux cas, nos résultats améliorent la meilleure borne connue, à savoir un noyau quadratique. Finalement, nous appliquons cette technique sur les problèmes Betweenness in Tournaments et Dense Circular Ordering, obtenant à nouveau des noyaux linéaires, qui constituent les premiers algorithmes de noyau polynomiaux connus pour ces problèmes. / In this thesis, we study the parameterized complexity of several NP-complete problems. More precisely, we study the existence of polynomial kernels for graph and constraints modification problems. In particular, we introduce the concept of branches, which provides polynomial kernels for some graph modification problems when the target graph class admits a so-called adjacency decomposition. This technique allows us to obtain the first known polynomial kernels for the Closest 3-Leaf Power, Cograph Edition and Proper Interval Completion problems. Regarding constraint modification problems, we develop and push further the concept of Conflict Packing, a technique that has already been used in a few parameterized problems and that provides polynomial kernels for several problems. We thus present a linear vertex-kernel for the Feedback Arc Set in Tournaments problem, and adapt these techniques to obtain a linear vertex-kernel for the Dense Rooted Triplet Inconsistency problem as well. In both cases, our results improve the best known bound of $O(k^2)$ vertices. Finally, we apply the Conflict Packing technique on the Betweenness in Tournaments and Dense Circular Ordering problems, obtaining once again linear vertex-kernels. To the best of our knowledge, these results constitute the first known polynomial kernels for these problems.

In der klassischen Komplexitätstheorie unterscheiden wir zwischen der Klasse P von in Polynomialzeit lösbaren Problemen, und der Klasse NP-schwer von Problemen bei denen die allgemeine Annahme ist, dass diese nicht in Polynomialzeit lösbar sind. Allerdings sind viele Probleme, die wir lösen möchten, NP-schwer. Gleichzeitig besteht eine große Diskrepanz zwischen den empirisch beobachteten und den festgestellten worst-case Laufzeiten. Es ist bekannt, dass Vorverarbeitung oder Datenreduktion auf realen Instanzen zu Laufzeitverbesserungen führt. Hier stoßen wir an die Grenze der klassischen Komplexitätstheorie. Der Fokus dieser Arbeit liegt auf Vorverarbeitungsalgorithmen für NP-schwere Probleme. Unser Ziel ist es, bestimmte Instanzen eines NP-schweren Problems vorverarbeiten zu können, indem wir die Struktur betrachten. Genauer gesagt, für eine gegebene Instanz und einen zusätzlichen Parameter l, möchten wir in Polynomialzeit eine äquivalente Instanz berechnen, deren Größe und Parameterwert nur durch eine Funktion im Parameterwert l beschränkt ist. In der parametrisierten Komplexitätstheorie heißen diese Algorithmen Kernelisierung. Wir werden drei NP-schwere Graphenprobleme betrachten, nämlich Vertex Cover, Edge Dominating Set und Subset Feedback Vertex Set. Für Vertex Cover werden wir bekannte Ergebnisse für Kernelisierungen vereinheitlichen, wenn der Parameter die Größe einer Entfernungsmenge zu einer gegebenen Graphklasse ist. Anschließend untersuchen wir die Kernelisierbarkeit von Edge Dominating Set. Es stellt sich heraus, dass die Kernelisierbarkeit deutlich komplexer ist. Dennoch klassifizieren wir die Existenz einer polynomiellen Kernelisierung, wenn jeder Graph in der Graphklasse eine disjunkte Vereinigung von konstant großen Komponenten ist. Schließlich betrachten wir das Subset Feedback Vertex Set Problem und zeigen, dass es eine randomisierte polynomielle Kernelisierung hat, wenn der Parameter die Lösungsgröße ist. / In classical complexity theory, we distinguish between the class P, of polynomial-time solvable problems, and the class NP-hard, of problems where the widely-held belief is that we cannot solve these problems in polynomial time. Unfortunately, many of the problems we want to solve are NP-hard. At the same time, there is a large discrepancy between the empirically observed running times and the established worst-case bounds. Using preprocessing or data reductions on real-world instances is known to lead to huge improvements in the running time. Here we come to the limits of classical complexity theory. In this thesis, we focus on preprocessing algorithms for NP-hard problems. Our goal is to find ways to preprocess certain instances of an NP-hard problem by considering the structure of the input instance. More precisely, given an instance and an additional parameter l, we want to compute in polynomial time an equivalent instance whose size and parameter value is bounded by a function in the parameter l only. In the field of parameterized complexity, these algorithms are called kernelizations. We will consider three NP-hard graph problems, namely Vertex Cover, Edge Dominating Set, and Subset Feedback Vertex Set. For Vertex Cover, we will unify known results for kernelizations when parameterized by the size of a deletion set to a specified graph class. Afterwards, we study the existence of polynomial kernelizations for Edge Dominating Set when parameterized by the size of a deletion set to a graph class. We point out that the existence of polynomial kernelizations is much more complicated than for Vertex Cover. Nevertheless, we fully classify the existence of polynomial kernelizations when every graph in the graph class is a disjoint union of constant size components. Finally, we consider graph cut problems, especially the Subset Feedback Vertex Set problem. We show that this problem has a randomized polynomial kernelization when the parameter is the solution size.