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.

Die Kernaufgabe der parameterisierten Komplexität ist zu verstehen, wie Eingabestruktur die Problemkomplexität beeinflusst. Wir untersuchen diese Fragestellung aus einer granularen Perspektive und betrachten Problem-Parameter-Kombinationen mit einfach exponentieller Laufzeit, d.h., Laufzeit a^k n^c, wobei n die Eingabegröße ist, k der Parameterwert, und a und c zwei positive Konstanten sind. Unser Ziel ist es, die optimale Laufzeitbasis a für eine gegebene Kombination zu bestimmen. Für viele Zusammenhangsprobleme, wie Connected Vertex Cover oder Connected Dominating Set, ist die optimale Basis bezüglich dem Parameter Baumweite bekannt. Die Baumweite gehört zu der Klasse der Weiteparameter, welche auf natürliche Weise zu Algorithmen mit dem Prinzip der dynamischen Programmierung führen. Im ersten Teil dieser Dissertation untersuchen wir, wie sich die optimale Laufzeitbasis für diverse Zusammenhangsprobleme verändert, wenn wir zu ausdrucksstärkeren Weiteparametern wechseln. Wir entwerfen neue parameterisierte Algorithmen und (bedingte) untere Schranken, um diese optimalen Basen zu bestimmen. Insbesondere zeigen wir für die Parametersequenz Baumweite, modulare Baumweite, und Cliquenweite, dass die optimale Basis von Connected Vertex Cover bei 3 startet, sich erst auf 5 erhöht und dann auf 6, wobei hingegen die optimale Basis von Connected Dominating Set bei 4 startet, erst bei 4 bleibt und sich dann auf 5 erhöht. Im zweiten Teil gehen wir über Weiteparameter hinaus und analysieren restriktivere Arten von Parametern. Für die Baumtiefe entwerfen wir platzsparende Verzweigungsalgorithmen. Die Beweistechniken für untere Schranken bezüglich Weiteparametern übertragen sich nicht zu den restriktiveren Parametern, weshalb nur wenige optimale Laufzeitbasen bekannt sind. Um dies zu beheben untersuchen wir Knotenlöschungsprobleme. Insbesondere zeigen wir, dass die optimale Basis von Odd Cycle Transversal parameterisiert mit einem Modulator zu Baumweite 2 den Wert 3 hat. / The question at the heart of parameterized complexity is how input structure governs the complexity of a problem. We investigate this question from a fine-grained perspective and study problem-parameter-combinations with single-exponential running time, i.e., time a^k n^c, where n is the input size, k the parameter value, and a and c are positive constants. Our goal is to determine the optimal base a for a given combination. For many connectivity problems such as Connected Vertex Cover or Connecting Dominating Set, the optimal base is known relative to treewidth. Treewidth belongs to the class of width parameters, which naturally admit dynamic programming algorithms. In the first part of this thesis, we study how the optimal base changes for these connectivity problems when going to more expressive width parameters. We provide new parameterized dynamic programming algorithms and (conditional) lower bounds to determine the optimal base, in particular, we obtain for the parameter sequence treewidth, modular-treewidth, clique-width that the optimal base for Connected Vertex Cover starts at 3, increases to 5, and then to 6, whereas the optimal base for Connected Dominating Set starts at 4, stays at 4, and then increases to 5. In the second part, we go beyond width parameters and study more restrictive parameterizations like depth parameters and modulators. For treedepth, we design space-efficient branching algorithms. The lower bound techniques for width parameterizations do not carry over to these more restrictive parameterizations and as a result, only a few optimal bases are known. To remedy this, we study standard vertex-deletion problems. In particular, we show that the optimal base of Odd Cycle Transversal parameterized by a modulator to treewidth 2 is 3. Additionally, we show that similar lower bounds can be obtained in the realm of dense graphs by considering modulators consisting of so-called twinclasses.

Dans cette thèse, nous étudions la complexité algorithmique de problèmes d'optimisation impliquant un processus de diffusion dans un graphe. Plus précisément, nous nous intéressons tout d'abord au problème de sélection d'un ensemble cible. Ce problème consiste à trouver le plus petit ensemble de sommets d'un graphe à “activer” au départ tel que tous les autres sommets soient activés après un nombre fini d'étapes de propagation. Si nous modifions ce processus en permettant de “protéger” un sommet à chaque étape, nous obtenons le problème du pompier dont le but est de minimiser le nombre total de sommets activés en protégeant certains sommets. Dans ce travail, nous introduisons et étudions une version généralisée de ce problème dans laquelle plus d'un sommet peut être protégé à chaque étape. Nous proposons plusieurs résultats de complexité pour ces problèmes à la fois du point de vue de l'approximation mais également de la complexité paramétrée selon des paramètres standards ainsi que des paramètres liés à la structure du graphe. / In this thesis, we investigate the computational complexity of optimization problems involving a “diffusion process” in a graph. More specifically, we are first interested to the target set selection problem. This problem consists of finding the smallest set of initially “activated” vertices of a graph such that all the other vertices become activated after a finite number of propagation steps. If we modify this process by allowing the possibility of ``protecting'' a vertex at each step, we end up with the firefighter problem that asks for minimizing the total number of activated vertices by protecting some particular vertices. In fact, we introduce and study a generalized version of this problem where more than one vertex can be protected at each step. We propose several complexity results for these problems from an approximation point of view and a parameterized complexity perspective according to standard parameterizations as well as parameters related to the graph structure.

The formalized and algorithmic study of human language within the field of Natural Language Processing (NLP) has motivated much theoretical work in the related field of formal languages, in particular the subfields of grammar and automata theory. Motivated and informed by NLP, the papers in this thesis explore the connections between expressibility – that is, the ability for a formal system to define complex sets of objects – and algorithmic complexity – that is, the varying amount of effort required to analyse and utilise such systems. Our research studies formal systems working not just on strings, but on more complex structures such as trees and graphs, in particular syntax trees and semantic graphs. The field of mildly context-sensitive languages concerns attempts to find a useful class of formal languages between the context-free and context-sensitive. We study formalisms defining two candidates for this class; tree-adjoining languages and the languages defined by linear context-free rewriting systems. For the former, we specifically investigate the tree languages, and define a subclass and tree automaton with linear parsing complexity. For the latter, we use the framework of parameterized complexity theory to investigate more deeply the related parsing problems, as well as the connections between various formalisms defining the class. The field of semantic modelling aims towards formally and accurately modelling not only the syntax of natural language statements, but also the meaning. In particular, recent work in semantic graphs motivates our study of graph grammars and graph parsing. To the best of our knowledge, the formalism presented in Paper III of this thesis is the first graph grammar where the uniform parsing problem has polynomial parsing complexity, even for input graphs of unbounded node degree.

Le formalisme des états graphes consiste à modéliser des états quantiques par des graphes. Ce formalisme permet l'utilisation des notions et des outils de théorie des graphes (e.g. flot, domination, méthodes probabilistes) dans le domaine du traitement de l'information quantique. Ces dernières années, cette modélisation combinatoire a permis plusieurs avancées décisives, notamment (i) dans la compréhension des propriétés de l'intrication quantique (ii) dans l'étude des modèles de calcul particulièrement prometteurs en terme d'implémentation physique, et (iii) dans l'analyse et la construction de protocoles de cryptographie quantique. L'objectif de cette thèse est d'étudier les propriétés graphiques émergeant des problématiques d'informatique quantique, notamment pour la simulation quantique. En particulier, l'étude des propriétés de causalité et de localité des états graphes, en étendant par exemple la notion existante de flot de causalité à une notion intégrant des contraintes de localité, permettrait d'ouvrir de nouvelles perspectives pour la simulation de systèmes quantiques à l'aide d'états graphes. Des connections formelles avec les automates cellulaires quantiques bruités pourront également émerger de cette étude. / Graph States formalism consist in using graphs to model quantum states. This formalism allows us to use notion and tools of graph theory (e.g. flow, domination, probabilistic methods) in quantum information processing. Last years, this combinatorial modelisation had lead to many decisiv breakthroughs, in particular (i) in the comprehension of the quantum entranglement properties (ii) in very promising in term of physical implementation quantum calculus model, and (iii) in the analysis and construction of quantum cryptography protocols. The goal of this thesis is to study the graphic properties emerging of those quantum information processing problematics, especially for quantum simulation. In particular, the properties of causality and locality in graph states, by extanding for exemple the existing notion of causality flows to a notion integring the locality constraints, would allow new perspectives for the quantum system simulation using graphs states. Formal connections with noisy quantum cellular automata would emerge from this study.

Nos recherches traitent de coloration de graphes avec des contraintes de distance (coloration de packing) ou des contraintes sur le voisinage (coloration de Grundy). Soit S={si| i in N*} une série croissante d’entiers. Une S -coloration de packing est une coloration propre de sommets telle que tout ensemble coloré i est un si-packing (un ensemble où tous les sommets sont à distance mutuelle supérieure à si). Un graphe G est (s1,... ,sk)-colorable si il existe une S -coloration de packing de G avec les couleurs 1, ...,,k. Une coloration de Grundy est une coloration propre de sommets telle que pour tout sommet u coloré i, u est adjacent à un sommet coloré j, pour chaque j<i.Dans cette exposé, nous présentons des résultats connus à propos de la S-coloration de packing. Nous apportons de nouveaux résultats à propos de la S-coloration de packing, pour des classes de graphes telles que les chemins, les cycles et les arbres. Nous étudions en détail la complexité du problème de complexité associé à la S-coloration de packing, noté S -COL. Pour certaines instances de S -COL, nous caractérisons des dichotomies entre problèmes NP-complets et problèmes résolubles en tempspolynomial. Nous nous intéressons aux différentes grilles infinies, les grilles hexagonale, carrée, triangulaire et du roi et nous déterminons des propriétés de subdivisions d’un i-packing en plusieurs j-packings, avec j>i. Ces résultats nous permettent de déterminer des S-colorations de packings de ces grilles pour plusieurs séries d’entiers. Nous examinons une classe de graphe jamais étudiée en ce qui concerne la S -coloration de packing: les graphes subcubiques. Nous déterminons que tous les graphes subcubiques sont (1,2,2,2,2,2,2)-colorables et (1,1,2,2,3)-colorables. Un certain nombre de résultats sont prouvés pour certaines sous-classes des graphes subcubiques. Pour finir, nous nous intéressons au nombre de Grundy des graphes réguliers. Nous déterminons une caractérisation des graphes cubiques avec un nombre de Grundy de 4. De plus, nous prouvons que tous les graphes r-réguliers sans carré induit ont pour nombre de Grundy de r+1, pour r<5. / Our research are about graph coloring with distance constraints (packing coloring) or neighborhood constraints (Grundy coloring). Let S={si| i in N*} be a non decreasing sequence of integers. An S-packing coloring is a proper coloring such that every set of color i is an si-packing (a set of vertices at pairwise distance greater than si). A graph G is (s1,... ,sk)-colorable if there exists a packing coloring of G with colors 1,... ,k. A Grundy coloring is a proper vertex coloring such that for every vertex of color i, u is adjacent to a vertex of color j, for each j<i.In this presentation, we present results about S-packing coloring. We prove new results about the S-coloring of graphs including paths, cycles and trees. We study the complexity problem associated to the S-packing coloring, this problem is denoted S-COL. For some instances of S-COL, we characterize dichotomy between NP-complete problems and problems solved by a polynomial time algorithm. We study also different lattices, the hexagonal, square, triangular and king lattices. We determine properties on the subdivision of an i-packing in several j-packings, for j>i. These results allow us to determine S-packing coloring of these lattices for several sequences of integers. We examine a class of graph that has never been studied for S-packing coloring: the subcubic graphs. We determine that every subcubic graph is (1,2,2,2,2,2,2)-colorable and (1,1,2,2,3)-colorable. Few results are proven about some subclasses. Finally, we study the Grundy number of regular graphs. We determine a characterization of the cubic graphs with Grundy number 4. Moreover, we prove that every r-regular graph without induced square has Grundy number r+1, for r<5.

In many real world situations, especially involving multiagent systems and artificial intelligence, participating agents often need to agree upon a common alternative even if they have differing preferences over the available alternatives. Voting is one of the tools of choice in these situations. Common and classic applications of voting in modern applications include collaborative filtering and recommender systems, metasearch engines, coordination and planning among multiple automated agents etc. Agents in these applications usually have computational power at their disposal. This makes the study of computational aspects of voting crucial. This thesis is devoted to a study of computational complexity of several fundamental algorithmic and complexity-theoretic problems arising in the context of voting theory. The typical setting for our work is an “election”; an election consists of a set of voters or agents, a set of alternatives, and a voting rule. The vote of any agent can be thought of as a ranking (more precisely, a complete order) of the set of alternatives. A voting profile comprises a collection of votes of all the agents. Finally, a voting rule is a mapping that takes as input a voting profile and outputs an alternative, which is called the “winner” or “outcome” of the election. Our contributions in this thesis can be categorized into three parts and are described below. Part I: Preference Elicitation. In the first part of the thesis, we study the problem of eliciting the preferences of a set of voters by asking a small number of comparison queries (such as who a voter prefers between two given alternatives) for various interesting domains of preferences. We commence with considering the domain of single peaked preferences on trees in Chapter 3. This domain is a significant generalization of the classical well studied domain of single peaked preferences. The domain of single peaked preferences and its generalizations are hugely popular among political and social scientists. We show tight dependencies between query complexity of preference elicitation and various parameters of the single peaked tree, for example, number of leaves, diameter, path width, maximum degree of a node etc. We next consider preference elicitation for the domain of single crossing preference profiles in Chapter 4. This domain has also been studied extensively by political scientists, social choice theorists, and computer scientists. We establish that the query complexity of preference elicitation in this domain crucially depends on how the votes are accessed and on whether or not any single crossing ordering is a priori known. Part II: Winner Determination. In the second part of the thesis, we undertake a study of the computational complexity of several important problems related to determining winner of an election. We begin with a study of the following problem: Given an election, predict the winners of the election under some fixed voting rule by sampling as few votes as possible. We establish optimal or almost optimal bounds on the number of votes that one needs to sample for many commonly used voting rules when the margin of victory is at least n (n is the number of voters and is a parameter). We next study efficient sampling based algorithms for estimating the margin of victory of a given election for many common voting rules. The margin of victory of an election is a useful measure that captures the robustness of an election outcome. The above two works are presented in Chapter 5. In Chapter 6, we design an optimal algorithm for determining the plurality winner of an election when the votes are arriving one-by-one in a streaming fashion. This resolves an intriguing question on finding heavy hitters in a stream of items, that has remained open for more than 35 years in the data stream literature. We also provide near optimal algorithms for determining the winner of a stream of votes for other popular voting rules, for example, veto, Borda, maximin etc. Voters’ preferences are often partial orders instead of complete orders. This is known as the incomplete information setting in computational social choice theory. In an incomplete information setting, an extension of the winner determination problem which has been studied extensively is the problem of determining possible winners. We study the kernelization complexity (under the complexity-theoretic framework of parameterized complexity) of the possible winner problem in Chapter 7. We show that there do not exist kernels of size that is polynomial in the number of alternatives for this problem for commonly used voting rules under a plausible complexity theoretic assumption. However, we also show that the problem of coalitional manipulation which is an important special case of the possible winner problem admits a kernel whose size is polynomial bounded in the number of alternatives for common voting rules. \Part III: Election Control. In the final part of the thesis, we study the computational complexity of various interesting aspects of strategic behaviour in voting. First, we consider the impact of partial information in the context of strategic manipulation in Chapter 8. We show that lack of complete information makes the computational problem of manipulation intractable for many commonly used voting rules. In Chapter 9, we initiate the study of the computational problem of detecting possible instances of election manipulation. We show that detecting manipulation may be computationally easy under certain scenarios even when manipulation is intractable. The computational problem of bribery is an extensively studied problem in computational social choice theory. We study computational complexity of bribery when the briber is “frugal” in nature. We show for many common voting rules that the bribery problem remains intractable even when the briber’s behaviour is restricted to be frugal, thereby strengthening the intractability results from the literature. This forms the subject of Chapter 10.