Dissertations / Theses on the topic 'Parameterized complexity algorithms'

Many problems of practical significance are known to be NP-hard, and hence, are unlikely to be solved by polynomial-time algorithms. There are several ways to cope with the NP-hardness of a certain problem. The most popular approaches include heuristic algorithms, approximation algorithms, and randomized algorithms. Recently, parameterized computation and complexity have been receiving a lot of attention. By taking advantage of small or moderate parameter values, parameterized algorithms provide new venues for practically solving problems that are theoretically intractable. In this dissertation, we design efficient parameterized algorithms for several wellknown NP-hard problems and prove strong lower bounds for some others. In doing so, we place emphasis on the development of new techniques that take advantage of the structural properties of the problems. We present a simple parameterized algorithm for Vertex Cover that uses polynomial space and runs in time O(1.2738k + kn). It improves both the previous O(1.286k + kn)-time polynomial-space algorithm by Chen, Kanj, and Jia, and the very recent O(1.2745kk4 + kn)-time exponential-space algorithm, by Chandran and Grandoni. This algorithm stands out for both its performance and its simplicity. Essential to the design of this algorithm are several new techniques that use structural information of the underlying graph to bound the search space. For Vertex Cover on graphs with degree bounded by three, we present a still better algorithm that runs in time O(1.194k + kn), based on an “almost-global” analysis of the search tree. We also show that an important structural property of the underlying graphs – the graph genus – largely dictates the computational complexity of some important graph problems including Vertex Cover, Independent Set and Dominating Set. We present a set of new techniques that allows us to prove almost tight computational lower bounds for some NP-hard problems, such as Clique, Dominating Set, Hitting Set, Set Cover, and Independent Set. The techniques are further extended to derive computational lower bounds on polynomial time approximation schemes for certain NP-hard problems. Our results illustrate a new approach to proving strong computational lower bounds for some NP-hard problems under reasonable conditions.

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.

Alliances are used to denote agreements between members of a group with similar interests. Alliances can occur between nations, biological sequences, business cartels, and other entities. The notion of alliances in graphs was first introduced by Kristiansen, Hedetniemi, and Hedetniemi in . A defensive alliance in a graph G=(V,E) is a non empty set S⊆V S where, for all x ∈S, |N∩S|≥|N-S|. Consequently, every vertex that is a member of a defensive alliance has at least as many vertices defending it as there are vertices attacking it. Alliances can be used to model a variety of applications such as classification problems, communities in the web distributed protocols, etc . In , Gerber and Kobler introduced the problem of partitioning a graph into strong defensive alliances for the first time as the "Satisfactory Graph Partitioning (SGP)" problem. In his dissertation , Shafique used the problem of partitioning a graph into alliances to model problems in data clustering. Decision problems for several types of alliances and alliance partitions have been shown to be NP-complete. However, because of their applicability, it is of interest to study methods to overcome the complexity of these problems. In this thesis, we will present a variety of algorithms for finding alliances in different families of graphs with a running time that is polynomial in terms of the size of the input, and allowing exponential running time as a function of a chosen parameter. This study is guided by the theory of parameterized complexity introduced by Rod Downey and Michael Fellows in . In addition to parameterized algorithms for alliance related problems, we study the partition of series-parallel graphs into alliances. The class of series-parallel graphs is a special class in graph theory since many problems known to be NP-complete on general graphs have been shown to have polynomial time algorithms on series-parallel graphs. For example, the problem of finding a minimum defensive alliance has been shown to have a linear time algorithm when restricted to series-parallel graphs . Series-parallel graphs have also been to focus of study in a wide range of applications including CMOS layout and scheduling problems [ML86, Oud97]. Our motivation is driven by clustering properties that can be modeled with alliances. We observe that partitioning series-parallel graphs into alliances of roughly the same size can be used to partition task graphs to minimize the communication between processors and balance the workload of each processor. We present a characterization of series-parallel graphs that allow a partition into defensive alliances and a subclass of series-parallel graphs with a satisfactory partitions.
Ph.D.
School of Electrical Engineering and Computer Science
Engineering and Computer Science
Computer Science PhD

Much research has been done on wireless sensor networks. However, most protocols and algorithms for such networks are based on the ideal model Unit Disk Graph (UDG) model or do not assume any model. Furthermore, many results assume the knowledge of location information of the network. In practice, sensor networks often deviate from the UDG model significantly. It is not uncommon to observe stable long links that are more than five times longer than unstable short links in real wireless networks. A more general network model, the quasi unit-disk graph (quasi-UDG) model, captures much better the characteristics of wireless networks. However, the understanding of the properties of general quasi-UDGs has been very limited, which is impeding the design of key network protocols and algorithms. In this dissertation we study the properties for general wireless sensor networks and develop new topological/geometrical techniques for wireless sensor networking. We assume neither the ideal UDG model nor the location information of the nodes. Instead we work on the more general quasi-UDG model and focus on figuring out the relationship between the geometrical properties and the topological properties of wireless sensor networks. Based on such relationships we develop algorithms that can compute useful substructures (planar subnetworks, boundaries, etc.). We also present direct applications of the properties and substructures we constructed including routing, data storage, topology discovery, etc. We prove that wireless networks based on quasi-UDG model exhibit nice properties like separabilities, existences of constant stretch backbones, etc. We develop efficient algorithms that can obtain relatively dense planar subnetworks for wireless sensor networks. We also present efficient routing protocols and balanced data storage scheme that supports ranged queries. We present algorithmic results that can also be applied to other fields (e.g., information management). Based on divide and conquer and improved color coding technique, we develop algorithms for path, matching and packing problem that significantly improve previous best algorithms. We prove that it is unlikely for certain problems in operation science and information management to have any relatively effective algorithm or approximation algorithm for them.

We can visualize a graph by producing a geometric representation of the graph in which each node is represented by a single point on the plane, and each edge is represented by a curve that connects its two endpoints. Directed graphs are often used to model hierarchical structures; in order to visualize the hierarchy represented by such a graph, it is desirable that a drawing of the graph reflects this hierarchy. This can be achieved by drawing all the edges in the graph such that they all point in an upwards direction. A graph that has a drawing in which all edges point in an upwards direction and in which no edges cross is known as an upward planar graph. Unfortunately, testing if a graph is upward planar is NP-complete. Parameterized complexity is a technique used to find efficient algorithms for hard problems, and in particular, NP-complete problems. The main idea is that the complexity of an algorithm can be constrained, for the most part, to a parameter that describes some aspect of the problem. If the parameter is fixed, the algorithm will run in polynomial time. In this thesis, we investigate contracting an edge in an upward planar graph that has a specified embedding, and show that we can determine whether or not the resulting embedding is upward planar given the orientation of the clockwise and counterclockwise neighbours of the given edge. Using this result, we then show that under certain conditions, we can join two upward planar graphs at a vertex and obtain a new upward planar graph. These two results expand on work done by Hutton and Lubiw. Finally, we show that a biconnected graph has at most k!8^k-1 planar embeddings, where k is the number of triconnected components. By using an algorithm by Bertolazzi et al. that tests whether a given embedding is upward planar, we obtain a parameterized algorithm, where the parameter is the number of triconnected components, for testing the upward planarity of a biconnected graph. This algorithm runs in O(k!8^kn³) time.

Networks are a mathematical abstraction of the interactions between a set of entities, with extensive applications in social science, epidemiology, bioinformatics, and cybersecurity, among others. There are many fundamental problems when analyzing network data, such as anomaly detection, dense subgraph mining, motif finding, information diffusion, and epidemic spread. A common underlying task in all these problems is finding an "interesting subgraph"; that is, finding a part of the graph---usually small relative to the whole---that optimizes a score function and has some property of interest, such as connectivity or a minimum density. Finding subgraphs that satisfy common constraints of interest, such as the ones above, is computationally hard in general, and state-of-the-art algorithms for many problems in network analysis are heuristic in nature. These methods are fast and usually easy to implement. However, they come with no theoretical guarantees on the quality of the solution, which makes it difficult to assess how the discovered subgraphs compare to an optimal solution, which in turn affects the data mining task at hand. For instance, in anomaly detection, solutions with low anomaly score lead to sub-optimal detection power. On the other end of the spectrum, there have been significant advances on approximation algorithms for these challenging graph problems in the theoretical computer science community. However, these algorithms tend to be slow, difficult to implement, and they do not scale to the large datasets that are common nowadays. The goal of this dissertation is developing scalable algorithms with theoretical guarantees for various network analysis problems, where the underlying task is to find subgraphs with constraints. We find interesting subgraphs with guarantees by adapting techniques from parameterized complexity, convex optimization, and submodularity optimization. These techniques are well-known in the algorithm design literature, but they lead to slow and impractical algorithms. One unifying theme in the problems that we study is that our methods are scalable without sacrificing the theoretical guarantees of these algorithm design techniques. We accomplish this combination of scalability and rigorous bounds by exploiting properties of the problems we are trying to optimize, decomposing or compressing the input graph to a manageable size, and parallelization. We consider problems on network analysis for both static and dynamic network models. And we illustrate the power of our methods in applications, such as public health, sensor data analysis, and event detection using social media data.
Ph. D.

La première partie de cette thèse s'intéresse au groupage de trafic dans les réseaux de télécommunications. La notion de groupage de trafic correspond à l'agrégation de flux de faible débit dans des conduits de plus gros débit. Cependant, à chaque insertion ou extraction de trafic sur une longueur d'onde il faut placer dans le noeud du réseau un multiplexeur à insertion/extraction (ADM). De plus il faut un ADM pour chaque longueur d'onde utilisée dans le noeud, ce qui représente un coût d'équipements important. Les objectifs du groupage de trafic sont d'une part le partage efficace de la bande passante et d'autre part la réduction du coût des équipements de routage. Nous présentons des résultats d'inapproximabilité, des algorithmes d'approximation, un nouveau modèle qui permet au réseau de pouvoir router n'importe quel graphe de requêtes de degré borné, ainsi que des solutions optimales pour deux scénarios avec trafic all-to-all: l'anneau bidirectionnel et l'anneau unidirectionnel avec un facteur de groupage qui change de manière dynamique. La deuxième partie de la thèse s'intéresse aux problèmes consistant à trouver des sous-graphes avec contraintes sur le degré. Cette classe de problèmes est plus générale que le groupage de trafic, qui est un cas particulier. Il s'agit de trouver des sous-graphes d'un graphe donné avec contraintes sur le degré, tout en optimisant un paramètre du graphe (très souvent, le nombre de sommets ou d'arêtes). Nous présentons des algorithmes d'approximation, des résultats d'inapproximabilité, des études sur la complexité paramétrique, des algorithmes exacts pour les graphes planaires, ainsi qu'une méthodologie générale qui permet de résoudre efficacement cette classe de problèmes (et de manière plus générale, la classe de problèmes tels qu'une solution peut être codé avec une partition d'un sous-ensemble des sommets) pour les graphes plongés dans une surface. Finalement, plusieurs annexes présentent des résultats sur des problèmes connexes.

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∗(3.72k) 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 2n). 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.

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.

The modern mathematical treatment of the study of decisions taken by participants whose interests are in conflict is now generally labeled as “game theory”. To understand these interactions the theory provides some solution concepts. An important such a concept is the notion of Nash equilibrium, which provides a way of predicting the behavior of strategic participants in situations of conflicts. However, many decision problems regarding to the computation of Nash equilibrium are computationally hard. Motivated by these hardness results, we study the parameterized complexity of the Nash equilibrium. In parameterized complexity one considers computational problems in a twodimensional setting: the first dimension is the usual input size n, the second dimension is a positive integer k, the parameter. A problem is fixed-parameter tractable (FPT) if it can be solved in time f(k)nO(1) where f denotes a computable, possibly exponential, function. We show that some decision problems regarding to the computation of Nash equilibrium are hard even in parameterized complexity theory. However, we provide FPT algorithms for some other problems relevant to the computation of Nash equilibrium.
Thesis (PhD Doctorate)
Doctor of Philosophy (PhD)
Institute for Integrated and Intelligent Systems
Science, Environment, Engineering and Technology
Full Text

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 relations. 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. Concernant les problèmes d'édition de relations, 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 tech- nique sur les problèmes DENSE BETWEENNESS et DENSE CIRCULAR ORDERING, obtenant à nouveau des noyaux linéaires, qui constituent les premiers algorithmes de noyau polynomiaux connus pour ces problèmes.

Les décompositions de graphes, lorsqu'elles sont de petite largeur, sont souvent utilisées pour résoudre plus efficacement des problèmes étant difficiles dans le cas de graphes quelconques. Dans ce travail de thèse, nous nous intéressons aux limites liées à ces décompositions, et à la construction d'obstructions certifiant leur grande largeur. Dans une première partie, nous donnons un algorithme généralisant et unifiant la construction d'obstructions pour différentes largeurs de graphes, en temps XP lorsque paramétré par la largeur considérée. Nous obtenons en particulier le premier algorithme permettant de construire efficacement une obstruction à la largeur arborescente en temps O^{tw+4}. La seconde partie de notre travail porte sur l'étude du problème Ensemble [Sigma,Rho]-Dominant, une généralisation des problèmes de domination sur les graphes et caractérisée par deux ensembles d'entiers Sigma et Rho. Les diverses études de ce problème apparaissant dans la littérature concernent uniquement les cas où le problème est FPT, lorsque paramétré par la largeur arborescente. Nous montrons que ce problème ne l'est pas toujours, et que pour certains cas d'ensembles Sigma et Rho, il devient W[1]-difficile lorsque paramétré par la largeur arborescente. Dans la dernière partie, nous étudions la complexité d'un nouveau problème de coloration appelé k-Coloration Additive, combinant théorie des graphes et théorie des nombres. Nous montrons que ce nouveau problème est NP-complet pour tout k >= 4 fixé, tandis qu'il peut être résolu en temps polynomial sur les arbres pour k quelconque et non fixé.

Le problème de la détermination de la qualité d’une solution partielle se pose dans la majeure partie des approches algorithmiques cherchant à calculer progressivement une solution globale. L’élagage des arbres de recherche, la preuve de garanties d’approximation et l’efficacité des stratégies d’énumération sont des approches algorithmiques qui exigent souvent un moyen approprié de décider si une solution partielle donnée est un bon candidat pour l’étendre à une solution globale de bonne qualité. Dans cette thèse, nous étudions un type particulier de problèmes d’optimisation, appelés problèmes d’extension pour un grand nombre de problèmes basés sur des graphes. Contredisant peut-être l’intuition, ces problèmes ont tendance à être NP-difficile, même quand le problème d’optimisation sous-jacent peut être résolu en temps polynomial. Nous présentons de nombreux résultats positifs et négatifs de NP-difficulté et d’approximation pour différents scénarios d’entrée. De plus, nous étudions la complexité paramétrée des problèmes d’extension par rapport à la taille des pré-solutions, ainsi que l’optimalité de certains algorithmes exacts sous l’hypothèse du temps exponentielle
The problem of determining the quality of a partial solution occurs in almost every algorithmic approach that gradually computes a global solution. Pruning search trees, proving approximation guarantees, or the efficiency of enumeration strategies usually require a suitable way to decide if a given partial solution is a reasonable candidate to pursue for extension to a global one, of assured quality. In this thesis, we study a special type of optimization problems, called extension problems for a large number of optimization problems on graphs. Possibly contradicting intuition, these problems tend to be NP-hard, even for problems where the underlying optimization problem can be solved in polynomial time. We present many positive/negative hardness and approximation results for different input scenarios. Moreover, the parameterized complexity of extension problems with respect to size of partial solutions, as well as the optimality of some exact algorithms under the Exponential-Time Hypothesis (ETH) are studied

De nombreux problèmes de la vie réelle sont NP-Difficiles et ne peuvent pas être résolus en temps polynomial. Deux paradigmes notables pour les résoudre quand même sont: l'approximation et la complexité paramétrée. Dans cette thèse, on présente une nouvelle technique appelée "gloutonnerie-Pour-La-Paramétrisation". On l'utilise pour établir ou améliorer la complexité paramétrée de nombreux problèmes et également pour obtenir des algorithmes paramétrés pour des problèmes à cardinalité contrainte sur les graphes bipartis. En vue d'établir des résultats négatifs sur l'approximabilité en temps sous-Exponentiel et en temps paramétré, on introduit différentes méthodes de sparsification d'instances préservant l'approximation. On combine ces "sparsifieurs" à des réductions nouvelles ou déjà connues pour parvenir à nos fins. En guise de digestif, on présente des résultats de complexité de jeux comme le Bridge et Havannah
Several real-Life problems are NP-Hard and cannot be solved in polynomial time.The two main options to overcome this issue are: approximation and parameterized complexity. In this thesis, we present a new technique called greediness-For-Parameterization and we use it to improve the parameterized complexity of many problems. We also use this notion to obtain parameterized algorithms for some problems in bipartite graphs. Aiming at establishing negative results on the approximability in subexponential time and in parameterized time, we introduce new methods of sparsification that preserves approximation. We combine those "sparsifiers" with known or new reductions to achieve our goal. Finally, we present some hardness results of games such as Bridge and Havannah

De nouvelles stratégies thérapeutiques ont émergé grâce aux aptamères qui sont des ligands à haute affinité générés par une méthode expérimentale appelée SELEX. Toutefois, leur utilisation demeure limitée en raison de leur manque de sélectivité et de leur liaison à d'autres cibles. Ces effets hors-cibles ("off-target") sont fréquemment observés dans toutes les stratégies thérapeutiques basées sur l'ARN. Pour accroître leur spécificité et sélectivité, des aptamères modifiés chimiquement in situ ont été générés par SELEX (SOMAmers). Néanmoins, des limitations techniques de SELEX persistent notamment dans le nombre et le type de modifications chimiques utilisables. Nous proposons donc une stratégie de conception in silico d'oligonucléotides modifiés pour s'affranchir de certaines de ces limitations. Pour ce faire, nous avons développé une nouvelle méthode de reconstruction d'ARN simple brin (ARNsb) basée sur l'approche par fragments, sur la connaissance 3D de la structure cible et sur la technique du Color-Coding développé par Alon, Yuster et Zwick. Celle-ci repose sur une modélisation de la connectivité des poses sous la forme d'un graphe permettant la mise en œuvre d'un algorithme combinatoire efficace par programmation dynamique. Cette approche est implémentée pour le « Docking » (recherche du mode de liaison natif) à partir d'une distribution donnée de fragments, pour le « Design » thérapeutique de-novo d'un ARNsb et pour l'analyse statistique de caractéristiques afin d'obtenir des informations entre les poses et la protéine (par exemple, l'analyse de profils de séquence ou la probabilité d'une pose d'appartenir à une chaine ARNsb). Cette nouvelle approche a fait l'objet d'une preuve de concept sur sept complexes ARNsb/protéines et a démontré des résultats pertinents, robustes et exploitables dans une perspective de design.Pour mettre à l'épreuve cette nouvelle méthodologie, une étude de cas a été faite sur l'enzyme Beta-Sécrétase 1 (BACE1, impliquée dans la maladie d'Alzheimer) et son homologue Beta-Sécrétase 2 (BACE2) afin de concevoir un ARNsb sélectif de BACE1. Pour ce faire, une étude a été réalisée afin de déterminer les caractéristiques clés de la spécificité et sélectionner des nucléotides modifiés pouvant se lier aux sites et sous-sites fonctionnels de BACE-1. Ces résultats pourront servir de base pour le design d'ARNsb sélectif et pour l'évaluation de la méthodologie basée sur le Color-Coding
New therapeutic strategies have emerged using aptamers that are high-affinity ligands generated using a procedure called SELEX. Nevertheless, their use are limited due to the lack of selectivity and off-target effects frequently observed as in all RNA-based therapeutic strategies. In order to increase their specificity and selectivity, a class of chemically modified aptamers has been designed in-situ by SELEX (“SOMAmers”). But, various limitations remain due to technical constraints in SELEX, in particular the number and the type of chemical modifications that can be used. We propose a new fragment-based strategy to design in-silico modified aptamers that will overcome some of these limitations, based on the knowledge of 3D structure and the color-coding technique introduced by Alon, Yuster and Zwick. This approach is based on the modeling of pose connectivity in the form of a graph, enabling the implementation of an efficient combinatorial algorithm using dynamic programming. It is used for the “Docking” from a given fragment distribution, the “Design” and the equilibrium statistics for learning features between fragments and the 3D protein, and has been the subject of a proof of concept on 7 ssRNA/protein complexes.To test this new approach on a real therapeutic case study, an analysis has been carried out on the Beta-Secretase 1 (BACE1), an enzyme involved in Alzheimer's disease, and the Beta-Secretase 2 (a homologous protein to BACE1) to determine the key features of specificity and to select the modified nucleotides that bind sites and subsites of BACE1. These results can provide a basis for the design of selective ssRNA and for the evaluation of the Color-Coding based methodology

Nous étudions des problèmes NP-difficiles portant sur les graphes contenant des blocages.Nous traitons les problèmes de coupes du point de vue de la complexité paramétrée. La taille p de la coupe est le paramètre. Étant donné un ensemble de sources {s1,...,sk} et une cible t, nous proposons un algorithme qui construit une coupe de taille au plus p séparant au moins r sources de t. Nous nommons ce problème NP-complet Partial One-Target Cut. Notre algorithme est FPT. Nous prouvons également que la variante de Partial One-Target Cut, où la coupe est composée de noeuds, est W[1]-difficile. Notre seconde contribution est la construction d'un algorithme qui compte les coupes minimums entre deux ensembles S et T en temps $2^{O(plog p)}n^{O(1)}$.Nous présentons ensuite plusieurs résultats sur le ratio de compétitivité des stratégies déterministes et randomisées pour le problème du voyageur canadien.Nous prouvons que les stratégies randomisées n'utilisant pas de mémoire ne peuvent pas améliorer le ratio 2k+1. Nous apportons également des éléments concernant les bornes inférieures de compétitivité de l'ensemble des stratégies randomisées. Puis, nous étudions la compétitivité en distance d'un groupe de voyageurs avec et sans communication. Enfin, nous nous penchons sur la compétitivité des stratégies déterministes pour certaines familles de graphes. Deux stratégies, avec un ratio inférieur à 2k+1 sont proposées: une pour les graphes cordaux avec poids uniformes et l'autre pour les graphes où la taille de la plus grande coupe minimale séparant s et t est au plus k
We study NP-hard problems on graphs with blockages seen as models of networks which are exposed to risk of failures.We treat cut problems via the parameterized complexity framework. The cutset size p is taken as a parameter. Given a set of sources {s1,...,sk} and a target $t, we propose an algorithm which builds a small edge cut of size p separating at least r sources from t. This NP-complete problem is called Partial One-Target Cut. It belongs to the family of multiterminal cut problems. Our algorithm is fixed-parameter tractable (FPT) as its execution takes $2^{O(p^2)}n^{O(1)}$. We prove that the vertex version of this problem, which imposes cuts to contain vertices instead of edges, is W[1]-hard. Then, we design an FPT algorithm which counts the minimum vertex (S,T)-cuts of an undirected graph in time $2^{O(plog p)}n^{O(1)}$.We provide numerous results on the competitive ratio of both deterministic and randomized strategies for the Canadian Traveller Problem. The optimal ratio obtained for the deterministic strategies on general graphs is 2k+1, where k is a given upper bound on the number of blockages. We show that randomized strategies which do not use memory cannot improve the bound 2k+1. In addition, we discuss the tightness of lower bounds on the competitiveness of randomized strategies. The distance competitive ratio for a group of travellers possibly equipped with telecommunication devices is studied. Eventually, a strategy dedicated to equal-weight chordal graphs is proposed while another one is built for graphs with small maximum (s,t)-cuts. Both strategies outperform the ratio 2k+1

La théorie de la NP-complétude nous apprend que pour un certain nombre de problèmes d'optimisation, il est vain d'espérer un algorithme efficace calculant une solution optimale. Partant de ce constat, un moyen pour contourner cet obstacle est de réaliser un compromis sur chacun de ces critères, engendrant deux approches devenues classiques. La première, appelée approximation polynomiale, consiste à développer des algorithmes efficaces et retournant une solution proche d'une solution optimale. La seconde, appelée complexité paramétrée, consiste à développer des algorithmes retournant une solution optimale mais dont l'explosion combinatoire est capturée par un paramètre de l'entrée bien choisi. Cette thèse comporte deux objectifs. Dans un premier temps, nous proposons d'étudier et d'appliquer les méthodes classiques de ces deux domaines afin d'obtenir des résultats positifs et négatifs pour deux problèmes d'optimisation dans les graphes : un problème de partition appelé Sparsest k-Compaction, et un problème de recherche d'un sous-graphe avec une cardinalité fixée appelé Sparsest k-Subgraph. Dans un second temps, nous présentons comment les méthodes de ces deux domaines ont pu se combiner ces dernières années pour donner naissance au principe d'approximation paramétrée. En particulier, nous étudierons les liens entre approximation et algorithmes de noyaux
The theory of NP-completeness tells us that for many optimization problems, there is no hope for finding an efficient algorithm computing an optimal solution. Based on this, two classical approaches have been developped to deal with these problems. The first one, called polynomial- time approximation, consists in designing efficient algorithms computing a solution that is close to an optimal one. The second one, called param- eterized complexity, consists in designing exact algorithms which com- binatorial explosion is captured by a carefully chosen parameter of the instance. The goal of this thesis is twofold. First, we study and apply classical methods from these two domains in order to obtain positive and negative results for two optimization problems in graphs: a partitioning problem called Sparsest k-Compaction, and a cardinality constraint subgraph problem called Sparsest k-Subgraph. Then, we present how the different methods from these two domains have been combined in recent years in a concept called parameterized approximation. In particular, we study the links between approximation and kernelization algorithms

Dans un graphe orienté contenant un nœud appelé racine, un sous ensemble de nœuds appelés terminaux et une pondération sur les arcs, le problème de l’arborescence de Steiner (DST) consiste en la recherche d’une arborescence de poids minimum contenant pour chaque terminal un chemin de la racine vers ce terminal. Ce problème est NP-Complet. Cette thèse se penche sur l’étude de l’approximabilité de ce problème. Sauf si P=NP, il n’existe pas pour ce problème d’approximation de rapport constant ou logarithmique en k, oú k est le nombre de terminaux. Le plus petit rapport d’approximation connu est O (k") où " est un réel strictement positif. Dans la première partie, nous donnons trois algorithmes d’approximation : un algorithme glouton efficace qui associe deux techniques d’approximations connues pour DST, un algorithme dans le cas des graphes structurés en paliers qui étudie l’approximabilité du problème quand les terminaux sont éloignés de la racine, et un algorithme exponentiel qui combine un algorithme d’approximation et un algorithme exact, dont le rapport d’approximation et la complexité temporelle sont paramétrés par le nombre de terminaux couverts par l’algorithme exact. Dans la seconde partie, nous étudions deux problèmes issus de DST auquel est ajoutée une contrainte sur les nœuds de branchement. Cette contrainte réduit le nombre de solutions réalisables et peut faciliter la recherche d’une solution optimale parmi ce sous-ensemble de solutions. En fonction de la contrainte, nous étudions la possibilité de la trouver en temps polynomial et quel est le rapport d’approximation entre cette solution et la solution du problème non contraint
The directed Steiner tree problem (DST) asks, considering a directed weighted graph, a node r called root and a set of nodes X called terminals, for a minimum cost directed tree rooted in r spanning X. DST is an NP-complete problem. We are interested in the search for polynomial approximations for DST. Unless P = NP, DST can not be approximated neither within a constant ratio nor a logarithmic ratio with respected to k, where k is the number of terminals. The smallest known approximation ratio is O(kԑ)$ where ԑ is a positive real.In the first part, we provide three new approximation algorithms : a practical greedy algorithm merging two of the main approximation techniques for DST, an algorithm for the case where the graph is layered and where the distance between the terminals and the root is high, and an exponential approximation algorithm combining an approximation algorithm and an exact algorithm, parameterized with the number of terminals the exact algorithm must cover.In the last part we study DST with two branching constraints. With this technique, we are able to reduce the number of feasible solutions, and possibly facilitate the search for an optimal solution of the constraint problem. We study how it is possible to build such a solution in polynomial time and if this solution is a good approximation of an optimal solution of the non-constraint problem

Graph colourings and combinatorial games are two very widely studied topics in discrete mathematics. This thesis addresses the computational complexity of a range of problems falling within one or both of these subjects. Much of the thesis is concerned with the computational complexity of problems related to the combinatorial game (Free-)Flood-It, in which players aim to make a coloured graph monochromatic ("flood" the graph) with the minimum possible number of flooding operations; such problems are known to be computationally hard in many cases. We begin by proving some general structural results about the behaviour of the game, including a powerful characterisation of the number of moves required to flood a graph in terms of the number of moves required to flood its spanning trees; these structural results are then applied to prove tractability results about a number of flood-filling problems. We also consider the computational complexity of flood-filling problems when the game is played on a rectangular grid of fixed height (focussing in particular on 3xn and 2xn grids), answering an open question of Clifford, Jalsenius, Montanaro and Sach. The final chapter concerns the parameterised complexity of list problems on graphs of bounded treewidth. We prove structural results determining the list edge chromatic number and list total chromatic number of graphs with bounded treewidth and large maximum degree, which are special cases of the List (Edge) Colouring Conjecture and Total Colouring Conjecture respectively. Using these results, we show that the problem of determining either of these quantities is fixed parameter tractable, parameterised by the treewidth of the input graph. Finally, we analyse a list version of the Hamilton Path problem, and prove it to be W[1]-hard when parameterised by the pathwidth of the input graph. These results answer two open questions of Fellows, Fomin, Lokshtanov, Rosamond, Saurabh, Szeider and Thomassen.

En algorithmique et en complexité, la plus grande part de la recherche se base sur l’hypothèse que P ≠ NP (Polynomial time et Non deterministic Polynomial time), c'est-à-dire qu'il existe des problèmes dont la solution peut être vérifiée mais non construite en temps polynomial. Si cette hypothèse est admise, de nombreux problèmes naturels ne sont pas dans P (c'est-à-dire, n'admettent pas d'algorithme efficace), ce qui a conduit au développement de nombreuses branches de l'algorithmique. L'une d'elles est la complexité paramétrée. Elle propose des algorithmes exacts, dont l'analyse est faite en fonction de la taille de l'instance et d'un paramètre. Ce paramètre permet une granularité plus fine dans l'analyse de la complexité.Un algorithme sera alors considéré comme efficace s'il est à paramètre fixé, c'est-à-dire, lorsque sa complexité est exponentielle en fonction du paramètre et polynomiale en fonction de la taille de l'instance. Ces algorithmes résolvent les problèmes de la classe FPT (Fixed Parameter Tractable).L'extraction de noyaux est une technique qui permet, entre autre, d’élaborer des algorithmes à paramètre fixé. Elle peut être vue comme un pré-calcul de l'instance, avec une garantie sur la compression des données. Plus formellement, une extraction de noyau est une réduction polynomiale depuis un problème vers lui même, avec la contrainte supplémentaire que la taille du noyau (l'instance réduite) est bornée en fonction du paramètre. Pour obtenir l’algorithme à paramètre fixé, il suffit de résoudre le problème dans le noyau, par exemple par une recherche exhaustive (de complexité exponentielle, en fonction du paramètre). L’existence d'un noyau implique donc l'existence d'un algorithme à paramètre fixé, la réciproque est également vraie. Cependant, l’existence d'un algorithme à paramètre fixé efficace ne garantit pas un petit noyau, c'est a dire un noyau dont la taille est linéaire ou polynomiale. Sous certaines hypothèses, il existe des problèmes n’admettant pas de noyau (c'est-à-dire hors de FPT) et il existe des problèmes de FPT n’admettant pas de noyaux polynomiaux.Un résultat majeur dans le domaine des noyaux est la construction d'un noyau linéaire pour le problème Domination dans les graphes planaires, par Alber, Fellows et Niedermeier.Tout d'abord, la méthode de décomposition en régions proposée par Alber, Fellows et Niedermeier, a permis de construire de nombreux noyaux pour des variantes de Domination dans les graphes planaires. Cependant cette méthode comportait un certain nombre d’imprécisions, ce qui rendait les preuves invalides. Dans la première partie de notre thèse, nous présentons cette méthode sous une forme plus rigoureuse et nous l’illustrons par deux problèmes : Domination Rouge Bleue et Domination Totale.Ensuite, la méthode a été généralisée, d'une part, sur des classes de graphes plus larges (de genre borné, sans-mineur, sans-mineur-topologique), d'autre part, pour une plus grande variété de problèmes. Ces méta-résultats prouvent l’existence de noyaux linéaires ou polynomiaux pour tout problème vérifiant certaines conditions génériques, sur une classe de graphes peu denses. Cependant, pour atteindre une telle généralité, il a fallu sacrifier la constructivité des preuves : les preuves ne fournissent pas d'algorithme d'extraction constructif et la borne sur le noyau n'est pas explicite. Dans la seconde partie de notre thèse nous effectuons un premier pas vers des méta-résultats constructifs ; nous proposons un cadre général pour construire des noyaux linéaires en nous inspirant des principes de la programmation dynamique et d'un méta-résultat de Bodlaender, Fomin, Lokshtanov, Penninkx, Saurabh et Thilikos
In the fields of Algorithmic and Complexity, a large area of research is based on the assumption that P ≠ NP(Polynomial time and Non deterministic Polynomial time), which means that there are problems for which a solution can be verified but not constructed in polynomial time. Many natural problems are not in P, which means, that they have no efficient algorithm. In order to tackle such problems, many different branches of Algorithmic have been developed. One of them is called Parametric Complexity. It consists in developing exact algorithms whose complexity is measured as a function of the size of the instance and of a parameter. Such a parameter allows a more precise analysis of the complexity. In this context, an algorithm will be considered to be efficient if it is fixed parameter tractable (fpt), that is, if it has a complexity which is exponential in the parameter and polynomial in the size of the instance. Problems that can be solved by such an algorithm form the FPT class.Kernelisation is a technical that produces fpt algorithms, among others. It can be viewed as a preprocessing of the instance, with a guarantee on the compression of the data. More formally, a kernelisation is a polynomial reduction from a problem to itself, with the additional constraint that the size of the kernel, the reduced instance, is bounded by a function of the parameter. In order to obtain an fpt algorithm, it is sufficient to solve the problem in the reduced instance, by brute-force for example (which has exponential complexity, in the parameter). Hence, the existence of a kernelisiation implies the existence of an fpt algorithm. It holds that the converse is true also. Nevertheless, the existence of an efficient fpt algorithm does not imply a small kernel, meaning a kernel with a linear or polynomial size. Under certain hypotheses, it can be proved that some problems can not have a kernel (that is, are not in FPT) and that some problems in FPT do not have a polynomial kernel.One of the main results in the field of Kernelisation is the construction of a linear kernel for the Dominating Set problem on planar graphs, by Alber, Fellows and Niedermeier.To begin with, the region decomposition method proposed by Alber, Fellows and Niedermeier has been reused many times to develop kernels for variants of Dominating Set on planar graphs. Nevertheless, this method had quite a few inaccuracies, which has invalidated the proofs. In the first part of our thesis, we present a more thorough version of this method and we illustrate it with two examples: Red Blue Dominating Set and Total Dominating Set.Next, the method has been generalised to larger classes of graphs (bounded genus, minor-free, topological-minor-free), and to larger families of problems. These meta-results prove the existence of a linear or polynomial kernel for all problems verifying some generic conditions, on a class of sparse graphs. As a price of generality, the proofs do not provide constructive algorithms and the bound on the size of the kernel is not explicit. In the second part of our thesis, we make a first step to constructive meta-results. We propose a framework to build linear kernels based on principles of dynamic programming and a meta-result of Bodlaender, Fomin, Lokshtanov, Penninkx, Saurabh and Thilikos

Pour mieux saisir les liens complexes entre génotype et phénotype, une méthode utilisée consiste à étudier les relations entre différents éléments biologiques (entre les protéines, entre les métabolites...). Celles-ci forment ce qui est appelé un réseau biologique, que l'on représente algorithmiquement par un graphe. Nous nous intéressons principalement dans cette thèse au problème de la recherche d'un motif (multi-ensemble de couleurs) dans un graphe coloré, représentant un réseau biologique. De tels motifs correspondent généralement à un ensemble d'éléments conservés au cours de l'évolution et participant à une même fonction biologique. Nous continuons l'étude algorithmique de ce problème et de ses variantes (qui admettent plus de souplesse biologique), en distinguant les instances difficiles algorithmiquement et en étudiant différentes possibilités pour contourner cette difficulté (complexité paramétrée, réduction d'instance, approximation...). Nous proposons également un greffon intégré au logiciel Cytoscape pour résoudre efficacement ce problème, que nous testons sur des données réelles.Nous nous intéressons également à différents problèmes de génomique comparative. La démarche scientifique adoptée reste la même: depuis une formalisation d'un problème biologique, déterminer ses instances difficiles algorithmiquement et proposer des solutions pour contourner cette difficulté (ou prouver que de telles solutions sont impossibles à trouver sous des hypothèses fortes)

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.

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.

This doctoral dissertation is based on 3 articles, which I have written during my PhD studies. They consist of novel results in the theory of approximation algorithms. Two of them concern the subdomain of stochastic optimization, whereas the focus of the last one lies in the intersection of theories of approximation algorithms and parameterized complexity. The results include approximation algorithms for covering problems under data uncertainty. In this setting we consider problems such as Set Cover, where only a part of data is known in advance. The algorithm must construct an assignment encoding a solution for all potential inputs, basing on a probability distribution over input scenarios. Another area is designing mechanisms and auctions, where each player/item can be approached only once and we look for a strategy that would maximize the mechanism's revenue with respect to a given distribution. In the last part I focus on graph problems in which we want to delete a small set of vertices to impose some property of a graph. These results provide significant improvements of approximation coefficients or running time for several problems of this kind.
Niniejsza rozprawa doktorska oparta jest na 3 artykułach, które opublikowałem w trakcie studiów III stopnia. Zawierają one nowatorskie wyniki z dziedziny algorytmów aproksymacyjnych. Dwie prace dotyczą poddziedziny optymalizacji stochastycznej, zaś wyniki ostatniej pracy leżą w przecięciu teorii algorytmów aproksymacyjnych i złożoności parametryzowanej. Przedstawione wyniki zawierają m.in. analizę algorytmów dla problemów pokryciowych w obliczu niepewności danych. W tym modelu rozważamy problemy takie jak np. Set Cover, przy założeniu, że cześć danych wejściowych jest nieznana z góry, zaś algorytm ma dostęp do rozkładu prawdopodobieństwa opisującego dane. Celem jest zbudowanie takiej struktury danych, która pozwoli odtworzyć rozwiązanie, kiedy całość danych zostanie ujawniona. Innym zagadnieniem jest konstrukcja mechanizmów i aukcji, w których decyzję odnośnie każdego klienta/przedmiotu należy podjąć zanim poznamy stan pozostałych. Poszukujemy strategii maksymalizującej oczekiwany zysk mechanizmu w oparciu o rozkład prawdopodobieństwa modelujący stan klientów/przedmiotów. W ostatniej części skupiam się na problemach grafowych, w których celem jest usunięcie możliwie małego podzbioru wierzchołków, tak aby graf należał do zadanej klasy. W prezentowanej pracy udało się osiągnąć znaczącą poprawę współczynników aproksymacji lub czasu działania algorytmów dla szeregu problemów tego typu.

Evolutionary algorithms are general problem solvers that have been successfully used in solving combinatorial optimization problems. However, due to the great amount of randomness in these algorithms, theoretical understanding of them is quite challenging. In this thesis we analyse the parameterized complexity of evolutionary algorithms on combinatorial optimization problems. Studying the parameterized complexity of these algorithms can help us understand how different parameters of problems influence the runtime behaviour of the algorithm and consequently lead us in finding better performing algorithms. We focus on two NP-hard combinatorial optimization problems; the generalized travelling salesman problem (GTSP) and the vertex cover problem (VCP). For solving the GTSP, two hierarchical approaches with different neighbourhood structures have been proposed in the literature. In this thesis, local search algorithms and simple evolutionary algorithms based on these approaches are investigated from a theoretical perspective and complementary abilities of the two approaches are pointed out by presenting instances where they mutually outperform each other. After investigating the runtime behaviour of the mentioned randomised algorithms on GTSP, we turn our attention to the VCP. Evolutionary multi-objective optimization for the classical vertex cover problem has been previously analysed in the context of parameterized complexity analysis. We extend the analysis to the weighted version of the problem. We also examine a dynamic version of the classical problem and analyse evolutionary algorithms with respect to their ability to maintain a 2-approximation. Inspired by the concept of duality, an edge-based evolutionary algorithm for solving the VCP has been introduced in the literature. Here we show that this edge-based EA is able to maintain a 2-approximation solution in the dynamic setting. Moreover, using the dual form of the problem, we extend the edge-based approach to the weighted vertex cover problem.
Thesis (Ph.D.) -- University of Adelaide, School of Computer Science, 2017.

Structural Properties of Graphs and Eficient Algorithms: Problems Between Parameters Dušan Knop Parameterized complexity became over last two decades one of the most impor- tant subfield of computational complexity. Structural graph parameters (widths) play important role both in graph theory and (parameterized) algoritmh design. By studying some concrete problems we exhibit the connection between struc- tural graph parameters and parameterized tractability. We do this by examining tractability and hardness results for the Target Set Selection, Minimum Length Bounded Cut, and other problems. In the Minimum Length Bounded Cut problem we are given a graph, source, sink, and a positive integer L and the task is to remove edges from the graph such that the distance between the source and the sink exceeds L in the resulting graph. We show that an optimal solution to the Minimum Length Bounded Cut problem can be computed in time f(k)n, where f is a computable function and k denotes the tree-depth of the input graph. On the other hand we prove that (under assumption that FPT ̸= W[1]) no such algorithm can exist if the parameter k is the tree-width of the input graph. Currently only few such problems are known. The Target Set Selection problem exibits the same phenomenon for the vertex cover number and...

A combinatorial game is a game in which all players have perfect information and there is no element of chance; some well-known examples include othello, checkers, and chess. When people play combinatorial games they develop strategies, which can be viewed as a function which takes as input a game position and returns a move to make from that position. A strategy is winning if it guarantees the player victory despite whatever legal moves any opponent may make in response. The classical complexity of deciding whether a winning strategy exists for a given position in some combinatorial game has been well-studied both in general and for many specific combinatorial games. The vast majority of these problems are, depending on the specific properties of the game or class of games being studied, complete for either PSPACE or EXP. In the parameterized complexity setting, Downey and Fellows initiated a study of "short" (or k-move) winning strategy problems. This can be seen as a generalization of "mate-in-k" chess problems, in which the goal is to find a strategy which checkmates your opponent within k moves regardless of how he responds. In their monograph on parameterized complexity, Downey and Fellows suggested that AW[*] was the "natural home" of short winning strategy problems, but there has been little work in this field since then. In this thesis, we study the parameterized complexity of finding short winning strategies in combinatorial games. We consider both the general and several specific cases. In the general case we show that many short games are as hard classically as their original variants, and that finding a short winning strategy is hard for AW[P] when the rules are implemented as succinct circuits. For specific short games, we show that endgame problems for checkers and othello are in FPT, that alternating hitting set, hex, and the non-endgame problem for othello are in AW[*], and that short chess is AW[*]-complete. We also consider pursuit-evasion parameterized by the number of cops. We show that two variants of pursuit-evasion are AW[*]-hard, and that the short versions of these problems are AW[*]-complete.