The present thesis is dedicated to two topics in Dynamics of Holomorphic maps. The first topic is dynamics of simple parabolic germs at the origin. The second topic is Polynomial-time Computability of Julia sets.\\ Dynamics of simple parabolic germs. Let $F$ be a germ with a simple parabolic fixed point at the origin: $F(w)=w+w^2+O(w^3).$ It is convenient to apply the change of coordinates $z=-1/w$ and consider the germ at infinity $$f(z)=-1/F(-1/z)=z+1+O(z^{-1}).$$ The dynamics of a germ $f$ can be described using Fatou coordinates. Fatou coordinates are analytic solutions of the equation $\phi(f(z))=\phi(z)+1.$ This equation has a formal solution \[\tilde\phi(z)=\text{const}+z+A\log z+\sum_{j=1}^\infty b_jz^{-j},\] where $\sum b_jz^{-j}$ is a divergent power series. Using \'Ecalle's Resurgence Theory we show that $\tilde$ can be interpreted as the asymptotic expansion of the Fatou coordinates at infinity. Moreover, the Fatou coordinates can be obtained from $\tilde \phi$ using Borel-Laplace summation. J.~\'Ecalle and S.~Voronin independently constructed a complete set of invariants of analytic conjugacy classes of germs with a parabolic fixed point. We give a new proof of validity of \'Ecalle's construction. \\ Computability of Julia sets. Informally, a compact subset of the complex plane is called \emph if it can be visualized on a computer screen with an arbitrarily high precision. One of the natural open questions of computational complexity of Julia sets is how large is the class of rational functions (in a sense of Lebesgue measure on the parameter space) whose Julia set can be computed in a polynomial time. The main result of Chapter II is the following: Theorem. Let $f$ be a rational function of degree $d\ge 2$. Assume that for each critical point $c\in J_f$ the $\omega$-limit set $\omega(c)$ does not contain either a critical point or a parabolic periodic point of $f$. Then the Julia set $J_f$ is computable in a polynomial time.

In this thesis we explore a number of ways in which combinatorial games can be used to help prove results in the area of propositional proof complexity. The results in this thesis can be divided into two sets, the first being dedicated to the study of Resolution space (memory) requirements, whereas the second is centered on formalizing the notion of `dangerous' reductions. The first group of results investigate Resolution space measures by asking questions of the form, `Given a formula F and integer k, does F have a [Type of Resolution] proof with [Type of Resource] at most k?'. We refer to this as a proof complexity resource problem, and provide comprehensive results for several forms of Resolution as well as various resources. These results include the PSPACE-Completeness of Tree Resolution clause space (and the Prover/Delayer game), the PSPACE-Completeness of Input Resolution derivation total space, and the PSPACE-Hardness of Resolution variable space. This research has theoretical as well as practical motivations: Proof complexity research has focused on the size of proofs, and Resolution space requirements are an interesting new theoretical area of study. In more practical terms, the Resolution proof system forms the underpinnings of all modern SAT-solving algorithms, including clause learning. In practice, the limiting factor on these algorithms is memory space, so there is a strong motivation for better understanding it as a resource. With the second group of results in this thesis we investigate and formalize what it means for a reduction to be `dangerous'. The area of SAT-solving necessarily employs reductions in order to translate from other domains to SAT, where the power of highly-optimized algorithms can be brought to bear. Researchers have empirically observed that it is unfortunately possible for reductions to map easy instances from the input domain to hard SAT instances. We develop a non-Hamiltonicity proof system and combine it with additional results concerning the Prover/Delayer game from the first part of this thesis as well as proof complexity results for intuitionistic logic in order to provide the first formal examples of harmful and beneficial reductions, ultimately leading to the development of a framework for studying and comparing translations from one language to another.

En el artículo "NP-completeness in Hedonic Games", identificamos algunas limitaciones significativas de los modelos estándar de juegos cooperativos: A menudo, es imposible alcanzar una organización estable de una sociedad en una cantidad de tiempo razonable. Las implicaciones básicas de estos resultados son las siguientes, Primero, desde un punto de vista positivo, las sociedades están "condenadas" a evolucionar constantemente, más que a alcanzar un estadio de equilibrio en el corto plazo. Segundo, desde una perspectiva normativa, un hipotético organizador de la sociedad debería tomar en consideración las limitaciones prácticas de tiempo a la hora de implementar un orden social estable.Para obtener nuestros resultados, utilizamos el concepto de NP-completitud, que es un modelo bien establecido de complejidad temporal en Ciencias de la Computación. En concreto, nos concentramos en estabilidad grupal y estabilidad individual en juegos hedónicos. Los juegos hedónicos son una clase simple de juegos cooperativos en los que la utilidad de cada individuo viene totalmente determinada por el grupo laboral al que pertenece. Nuestros resultados referentes a la complejidad, expresados en términos de NP-completitud, cubren un amplio espectro de dominios de las preferencias individuales, incluyendo preferencias estrictas, indiferencias en las preferencias o preferencias libres sobre el tamaño de los grupos. Dichos resultados también se cumplen si nos restringimos al caso en el que el tamaño máximo de los grupos es pequeño (dos o tres jugadores)En el artículo "Who is Who in Networks. Wanted: The Key Player" (junto con Antoni Calvó Armengol e Yves Zenou), analizamos un modelo de efectos de grupo en el que los agentes interactúan en un juego de influencias bilaterales. Los juegos no cooperativos con población finita y utilidades linales-cuadráticas, en los cuales cada jugador decide cuánto esfuerzo ejercer, pueden ser interpretados como juegos en red con complementariedades en los pagos, junto con un componente de susitucion global y uniforme, y un efecto de concavidad propia.Para dichos juegos, la acción de cada jugador en un equilibrio de Nash es proporcional a su centralidad de Bonacich en la red de complementariedades, estableciendo así un puente con la literatura de redes sociales. Dicho vínculo entre Bonacich y Nash implica que el equilibrio agregado aumenta con el tamaño y la densidad de la red. También analizamos una política que consiste en seleccionar al jugador clave, ésto es, el jugador que, una vez eliminado del juego, induce un cambio óptimo en la actividad agregada. Proveemos una caracterización geométrica del jugador clave, identificada con una medida de inter-centralidad, la cual toma en cuenta tanto la centralidad de cada jugador como su contribución a la centralidad de los otros.En el artículo "Optimal Targets in Peer Networks" (junto con Antoni Calvó Armengol e Yves Zenou), nos centramos en las consecuencias y limitaciones prácticas que se derivan del modelo de decisiones sobre delincuencia. Las principales metas que aborda el trabajo son las siguientes. Primero, la elección se extiende el concepto de delincuente clave en una red al de grupo clave. En dicha situación se trata de seleccionar de modo óptimo al conjunto de delincuentes a eliminar/neutralizar, dadas las restricciones presupuestarias para aplicar medidas. Dicho problema presenta una inherente complejidad computacional que solo puede salvarse mediante el uso de procedimientos aproximados, "voraces" o probabilísticos. Por otro lado, tratamos el problema del delincuente clave en el contexto de redes dinámicas, en las que, inicialmente, los individuos deciden acerca de su futuro como delincuentes o como ciudadanos que obtienen un salario fijo en el mercado. En dicha situación, la elección del delincuente clave es más compleja, ya que el objetivo de disminuir la delincuencia debe tener en cuenta los efectos en cadena que pueda traer consigo la desaparición de uno o varios delincuentes. Por último, estudiamos la complejidad computacional del problema de elección óptima y explotamos la propiedad de submodularidad de la intercentralidad de grupo, lo cual nos permite acotar el error relativo de la aproximación basada en un algoritmo voraz. / The aim of this thesis work is to contribute to the analysis of the interaction of agents in social networks and groups.In the chapter "NP-completeness in Hedonic Games", we identify some significant limitations in standard models of cooperation in games: It is often impossible to achieve a stable organization of a society in a reasonable amount of time. The main implications of these results are the following. First, from a positive point of view, societies are bound to evolve permanently, rather than reach a steady state configuration rapidly. Second, from a normative perspective, a planner should take into account practical time limitations in order to implement a stable social order.In order to obtain our results, we use the notion of NP-completeness, a well-established model of time complexity in Computer Science. In particular, we concentrate on group stability and individual stability in hedonic games. Hedonic games are a simple class of cooperative games in which each individual's utility is entirely determined by her group. Our complexity results, phrased in terms of NP-completeness, cover a wide spectrum of preference domains, including strict preferences, indifference in preferences or undemanding preferences over sizes of groups. They also hold if we restrict the maximum size of groups to be very small (two or three players).The last two chapters deal with the interaction of agents in the social setting. It focuses on games played by agents who interact among them. The actions of each player generate consequences that spread to all other players throughout a complex pattern of bilateral influences. In "Who is Who in Networks. Wanted: The Key Player" (joint with Antoni Calvó-Armengol and Yves Zenou), we analyze a model peer effects where agents interact in a game of bilateral influences. Finite population non-cooperative games with linear-quadratic utilities, where each player decides how much action she exerts, can be interpreted as a network game with local payoff complementarities, together with a globally uniform payoff substitutability component and an own-concavity effect.For these games, the Nash equilibrium action of each player is proportional to her Bonacich centrality in the network of local complementarities, thus establishing a bridge with the sociology literature on social networks. This Bonacich-Nash linkage implies that aggregate equilibrium increases with network size and density. We then analyze a policy that consists in targeting the key player, that is, the player who, once removed, leads to the optimal change in aggregate activity. We provide a geometric characterization of the key player identified with an inter-centrality measure, which takes into account both a player's centrality and her contribution to the centrality of the others.Finally, in the last chapter, "Optimal Targets in Peer Networks" (joint with Antoni Calvó-Armengol and Yves Zenou), we analyze the previous model in depth and study the properties and the applicability of network design policies.In particular, the key group is the optimal choice for a planner who wishes to maximally reduce aggregate activity. We show that this problem is computationally hard and that a simple greedy algorithm used for maximizing submodular set functions can be used to find an approximation. We also endogeneize the participation in the game and describe some of the properties of the key group. The use of greedy heuristics can be extended to other related problems, like the removal or addition of new links in the network.

This thesis comprises of two separate game theoretic models that fall under the general umbrella of network formation games. The first is a coalitional model of interaction in social networks that is based on the idea of social distance, in which players seek interactions with similar others. Our model captures some of the phenomena observed on such networks, such as homophily driven interactions and the formation of small worlds for groups of players. Using social distance games, we analyze the interactions between players on the network, study the properties of efficient and stable networks, relate them to the underlying graphical structure of the game, and give an approximation algorithm for finding optimal social welfare. We then show that efficient networks are not necessarily stable, and stable networks do not necessarily maximise welfare. We use the stability gap to investigate the welfare of stable coalition structures, and propose two new solution concepts with improved welfare guarantees. The second model is a compact formulation of matchings with externalities. Our formulation achieves tractability of the representation at the expense of full expressivity. We formulate a template of solution concept that applies to games where externalities are involved, and instantiate it in the context of optimistic, neutral, and pessimistic reasoning. Then we investigate the complexity of the representation in the context of many-to-many and one-to-one matchings, and provide both computational hardness results and polynomial time algorithms where applicable.

We consider several related problems arising in geometric graphs. In particular, we investigate the computational complexity and approximability properties of several optimization problems in unit ball graphs and develop algorithms to find exact and approximate solutions. In addition, we establish complexity-based theoretical justifications for several greedy heuristics. Unit ball graphs, which are defined in the three dimensional Euclidian space, have several application areas such as computational geometry, facility location and, particularly, wireless communication networks. Efficient operation of wireless networks involves several decision problems that can be reduced to well known optimization problems in graph theory. For instance, the notion of a \virtual backbone" in a wire- less network is strongly related to a minimum connected dominating set in its graph theoretic representation. Motivated by the vastness of application areas, we study several problems including maximum independent set, minimum vertex coloring, minimum clique partition, max-cut and min-bisection. Although these problems have been widely studied in the context of unit disk graphs, which are the two dimensional version of unit ball graphs, there is no established result on the complexity and approximation status for some of them in unit ball graphs. Furthermore, unit ball graphs can provide a better representation of real networks since the nodes are deployed in the three dimensional space. We prove complexity results and propose solution procedures for several problems using geometrical properties of these graphs. We outline a matching-based branch and bound solution procedure for the maximum k-clique problem in unit disk graphs and demonstrate its effectiveness through computational tests. We propose using minimum bottleneck connected dominating set problem in order to determine the optimal transmission range of a wireless network that will ensure a certain size of "virtual backbone". We prove that this problem is NP-hard in general graphs but solvable in polynomial time in unit disk and unit ball graphs. We also demonstrate work on theoretical foundations for simple greedy heuristics. Particularly, similar to the notion of "best" approximation algorithms with respect to their approximation ratios, we prove that several simple greedy heuristics are "best" in the sense that it is NP-hard to recognize the gap between the greedy solution and the optimal solution. We show results for several well known problems such as maximum clique, maximum independent set, minimum vertex coloring and discuss extensions of these results to a more general class of problems. In addition, we propose a "worst-out" heuristic based on edge contractions for the max-cut problem and provide analytical and experimental comparisons with a well known "best-in" approach and its modified versions.

Order-picking operations constitute the costliest activities in a warehouse. The order-picking problem (OPP) aims to determine the route of the picker(s) in such a way that the total order-picking time, hence the order-picking costs are minimized. In this study, a warehouse that consists of parallel pick aisles is assumed, and various versions of the OPP are considered. Although the single-picker version of the problem has been well studied in the literature, the multiple-picker version has not received much attention in terms of algorithmic approaches. The literature also does not take into account the time taken by the number of turns during the picking route. In this thesis, a detailed discussion is made regarding the computational complexity of the OPP with a single picker. A heuristic procedure, which makes use of the exact algorithm for the OPP with no middle aisles, is proposed for the single-picker OPP with middle aisles, and computational results on randomly generated problems are given. Additionally, an evolutionary algorithm that makes use of the cluster-first, route-second and route-first, cluster-second heuristics for the VRP is provided. The parameters of the algorithm are determined based on preliminary runs and the algorithm is also tested on randomly generated problems, with different weights given to the cluster-first, route-second and route-first, cluster-second approaches. Lastly, a polynomial time algorithm is proposed for the problem of minimizing the number of turns in a parallel-aisle warehouse.

Description Logics (DLs) are a family of logic-based knowledge representation (KR) formalisms designed to represent and reason about static conceptual knowledge in a semantically well-understood way. On the other hand, standard action formalisms are KR formalisms based on classical logic designed to model and reason about dynamic systems. The largest part of the present work is dedicated to integrating DLs with action formalisms, with the main goal of obtaining decidable action formalisms with an expressiveness significantly beyond propositional. To this end, we offer DL-tailored solutions to the frame and ramification problem. One of the main technical results is that standard reasoning problems about actions (executability and projection), as well as the plan existence problem are decidable if one restricts the logic for describing action pre- and post-conditions and the state of the world to decidable Description Logics. A smaller part of the work is related to decidable extensions of Description Logics with concrete datatypes, most importantly with those allowing to refer to the notions of space and time.

Texte remanié de : Ph. D : Mathématiques : University of Chicago : 1999. / Notes bibliogr.

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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.