abstract: A central concept of combinatorics is partitioning structures with given constraints. Partitions of on-line posets and on-line graphs, which are dynamic versions of the more familiar static structures posets and graphs, are examined. In the on-line setting, vertices are continually added to a poset or graph while a chain partition or coloring (respectively) is maintained. %The optima of the static cases cannot be achieved in the on-line setting. Both upper and lower bounds for the optimum of the number of chains needed to partition a width $w$ on-line poset exist. Kierstead's upper bound of $\frac{5^w-1}{4}$ was improved to $w^{14 \lg w}$ by Bosek and Krawczyk. This is improved to $w^{3+6.5 \lg w}$ by employing the First-Fit algorithm on a family of restricted posets (expanding on the work of Bosek and Krawczyk) . Namely, the family of ladder-free posets where the $m$-ladder is the transitive closure of the union of two incomparable chains $x_1\le\dots\le x_m$, $y_1\le\dots\le y_m$ and the set of comparabilities $\{x_1\le y_1,\dots, x_m\le y_m\}$. No upper bound on the number of colors needed to color a general on-line graph exists. To lay this fact plain, the performance of on-line coloring of trees is shown to be particularly problematic. There are trees that require $n$ colors to color on-line for any positive integer $n$. Furthermore, there are trees that usually require many colors to color on-line even if they are presented without any particular strategy. For restricted families of graphs, upper and lower bounds for the optimum number of colors needed to maintain an on-line coloring exist. In particular, circular arc graphs can be colored on-line using less than 8 times the optimum number from the static case. This follows from the work of Pemmaraju, Raman, and Varadarajan in on-line coloring of interval graphs. / Dissertation/Thesis / Ph.D. Mathematics 2012

Orientadores: Célia Picinin de Mello, Christiane Neme Campos / Texto em português e inglês / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Computação / Made available in DSpace on 2018-08-25T12:30:28Z (GMT). No. of bitstreams: 1 Luiz_AtilioGomes_M.pdf: 1439406 bytes, checksum: e2dc07f910b1876087a8f61428919e30 (MD5) Previous issue date: 2014 / Resumo: O problema da coloração total semiforte foi introduzido por Zhang et al. por volta de 2005. Este problema consiste em associar cores às arestas e aos vértices de um grafo G=(V(G),E(G)), utilizando o menor número de cores possível, de forma que: (i) quaisquer dois vértices ou duas arestas adjacentes possuam cores distintas; (ii) cada vértice tenha cor diferente das cores das arestas que nele incidem; e, além disso, (iii) para quaisquer dois vértices adjacentes u,v pertencentes a V(G), o conjunto das cores que colorem u e suas arestas incidentes é distinto do conjunto das cores que colorem v e suas arestas incidentes. Denominamos esse menor número de cores para o qual um grafo admite uma coloração total semiforte como número cromático total semiforte. Zhang et al. também determinaram o número cromático total semiforte de algumas famílias clássicas de grafos e observaram que todas elas possuem uma coloração total semiforte com no máximo Delta(G)+3 cores. Com base nesta observação, eles conjeturaram que Delta(G)+3 cores seriam suficientes para construir uma coloração total semiforte para qualquer grafo simples G. Essa conjetura é denominada Conjetura da Coloração Total Semiforte e permanece aberta para grafos arbitrários, tendo sido verificada apenas para algumas famílias de grafos. Nesta dissertação, apresentamos uma resenha dos principais resultados existentes envolvendo a coloração total semiforte. Além disso, determinamos o número cromático total semiforte para as seguintes famílias: os grafos simples com Delta(G)=3 e sem vértices adjacentes de grau máximo; os snarks-flor; os snarks de Goldberg; os snarks de Blanusa generalizados; os snarks de Loupekine LP1; e os grafos equipartidos completos de ordem par. Verificamos que os grafos destas famílias possuem número cromático total semiforte menor ou igual a Delta(G)+2. Investigamos também a coloração total semiforte dos grafos tripartidos e dos grafos equipartidos completos de ordem ímpar e verificamos que os grafos destas famílias possuem número cromático total semiforte menor ou igual a Delta(G)+3. Os resultados obtidos confirmam a validade da Conjetura da Coloração Total Semiforte para todas as famílias consideradas nesta dissertação / Abstract: The adjacent-vertex-distinguishing-total-colouring (AVD-total-colouring) problem was introduced and studied by Zhang et al. around 2005. This problem consists in associating colours to the vertices and edges of a graph G=(V(G),E(G)) using the least number of colours, such that: (i) any two adjacent vertices or adjacent edges receive distinct colours; (ii) each vertex receive a colour different from the colours of its incident edges; and (iii) for any two adjacent vertices u,v of G, the set of colours that color u and its incident edges is distinct from the set of colours that color v and its incident edges. The smallest number of colours for which a graph G admits an AVD-total-colouring is named its AVD-total chromatic number. Zhang et al. determined the AVD-total chromatic number for some classical families of graphs and noted that all of them admit an AVD-total-colouring with no more than Delta(G)+3 colours. Based on this observation, the authors conjectured that Delta(G)+3 colours would be sufficient to construct an AVD-total-colouring for any simple graph G. This conjecture is called the AVD-Total-Colouring Conjecture and remains open for arbitrary graphs, having been verified for a few families of graphs. In this dissertation, we present an overview of the main existing results related to the AVD-total-colouring of graphs. Furthermore, we determine the AVD-total-chromatic number for the following families of graphs: simple graphs with Delta(G)=3 and without adjacent vertices of maximum degree; flower-snarks; Goldberg snarks; generalized Blanusa snarks; Loupekine snarks; and complete equipartite graphs of even order. We verify that the graphs of these families have AVD-total-chromatic number at most Delta(G)+2. Additionally, we verify that the AVD-Total-Colouring Conjecture is true for tripartite graphs and complete equipartite graphs of odd order. These results confirm the validity of the AVD-Total-Colouring Conjecture for all the families considered in this dissertation / Mestrado / Ciência da Computação / Mestre em Ciência da Computação

This dissertation is in two parts, each of three chapters. In Part 1, I shall prove some results concerning variants of the 'secretary problem'. In Part 2, I shall bound several generalizations of the acyclic chromatic number of a graph as functions of its maximum degree. I shall begin Chapter 1 by describing the classical secretary problem, in which the aim is to select the best candidate for the post of a secretary, and its solution. I shall then summarize some of its many generalizations that have been studied up to now, provide some basic theory, and briefly outline the results that I shall prove. In Chapter 2, I shall suppose that the candidates come as ‘m’ pairs of equally qualified identical twins. I shall describe an optimal strategy, a formula for its probability of success and the asymptotic behaviour of this strategy and its probability of success as m → ∞. I shall also find an optimal strategy and its probability of success for the analagous version with ‘c’-tuplets. I shall move away from known posets in Chapter 3, assuming instead that the candidates come from a poset about which the only information known is its size and number of maximal elements. I shall show that, given this information, there is an algorithm that is successful with probability at least ¹/e . For posets with ‘k ≥ 2’ maximal elements, I shall prove that if their width is also ‘k’ then this can be improved to ‘k-1√1/k’ and show that no better bound of this type is possible. In Chapter 4, I shall describe the history of acyclic colourings, in which a graph must be properly coloured with no two-coloured cycle, and state some results known about them and their variants. In particular, I shall highlight a result of Alon, McDiarmid and Reed, which bounds the acyclic chromatic number of a graph by a function of its maximum degree. My results in the next two chapters are of this form. I shall consider two natural generalizations in Chapter 5. In the first, only cycles of length at least ’l’ must receive at least three colours. In the second, every cycle must receive at least ‘c’ colours, except those of length less than ‘c’, which must be multicoloured. My results in Chapter 6 generalize the concept of a cycle; it is now subgraphs with minimum degree ‘r’ that must receive at least three colours, rather than subgraphs with minimum degree two (which contain cycles). I shall also consider a natural version of this problem for hypergraphs.

Orientadores: Pedro Jussieu de Rezende, Cid Carvalho de Souza / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Computação / Made available in DSpace on 2018-08-26T19:09:30Z (GMT). No. of bitstreams: 1 Zambon_MauricioJosedeOliveira_M.pdf: 2774684 bytes, checksum: 1d0ed1f5c1ae01b7646e4bffea6a3736 (MD5) Previous issue date: 2014 / Resumo: Nesta dissertação, apresentamos a primeira abordagem algorítmica e os primeiros resultados experimentais da literatura para tratamento do Problema Cromático Discreto da Galeria de Arte (DCAGP). Trata-se de um problema de natureza geométrica que consiste de uma variante do clássico Problema da Galeria de Arte. Neste, deseja-se encontrar um conjunto de guardas com cardinalidade mínima que consiga vigiar toda uma dada galeria. Já no DCAGP temos por objetivo obter um conjunto de observadores que cubra a galeria e que admita uma coloração válida com o menor número de cores. Uma coloração é válida se dois observadores que veem um mesmo ponto recebem cores distintas. Abordamos a resolução deste problema através de duas abordagens: uma exata e uma heurística. Inicialmente, apresentamos uma heurística primal que fornece limitantes superiores de boa qualidade e, em seguida, um modelo de programação linear inteira para resolução exata do DCAGP. Este método foi capaz de resolver todas as instâncias de um extenso conjunto de galerias, representadas por polígonos simples aleatoriamente gerados, de até 2500 vértices, em menos de um minuto. Já num outro conjunto de instâncias onde a representação inclui polígonos com buracos e polígonos fractais de von Koch com até 800 vértices, o método encontrou soluções comprovadamente ótimas para 80% das instâncias em menos de 30 minutos. No contexto dessas soluções, discutimos o uso de lazy-constraints e de técnicas de fortalecimento do modelo, assim como uma breve análise da dificuldade das instâncias. Reportamos ainda resultados da utilização de relaxação Lagrangiana, para obtenção de bons limitantes, principalmente superiores, e também resultados obtidos por meio de uma variação da técnica relax-and-fix. Finalmente, discutimos um processo de branch-and-price para resolução exata do DCAGP / Abstract: In this dissertation, we present the first algorithmic approach and the first experimental results in the literature for solving the Discrete Chromatic Art Gallery Problem (DCAGP). This problem is geometric in nature and consists of a variation of the classic Art Gallery Problem. In the latter, we want to find a minimum cardinality guard set that is able to watch over a given gallery. On the other hand, in the DCAGP, the objective is to find a set of watchers that covers the gallery and admits a valid coloring with a minimum number of colors. A coloring is valid if two watchers that observe a same point are assigned different colors. To solve this problem we apply two approaches: an exact and a heuristic one. Firstly, we present a primal heuristic able to provide good quality upper bounds, and subsequently an integer programming model that yields exact solutions for the DCAGP. This method was able to solve all instances from an extensive set of galleries, represented by randomly generated simple polygons, of up to 2500 vertices, in less than one minute. On another set of instances, where the representation includes polygons with holes and fractal von Koch polygons, with up to 800 vertices, this method found proven optimal solutions for 80% of the instances in less than 30 minutes. In the context of these solutions, we discuss the use of lazy constraints and techniques for strengthening the model, besides a brief analysis of the hardness of the instances. Moreover, we report on results obtained through a Lagrangian relaxation, mainly as a means to obtain good upper bounds, as well as from a variation of the relax-and-fix technique. Lastly, we discuss a branch-and-price process for solving the DCAGP to exactness / Mestrado / Ciência da Computação / Mestre em Ciência da Computação

<div>Spectral Methods have had huge influence of modern algorithm design. For algorithmic problems on graphs, this is done by using a deep connection between random walks and the powers of various natural matrices associated with the graph. The major contribution</div><div>of this thesis initiates attempts to recover algorithmic results in Graph Minor Theory via spectral methods.</div><div><br></div><div>We make progress towards this goal by exploring these questions in the Property Testing Model for bounded degree graphs. Our main contributions are</div><div>1. The first result gives an almost query optimal one-sided tester for the property of H-minor-freeness. Benjamini-Schramm-Shapira (STOC 2008) conjectured that for fixed H, this can be done in time O(sqrt n). Our algorithm solves this in time n^{1/2+o(1)} which nearly resolves this upto n^{o(1)} factors.</div><div><br></div><div>2. BSS also conjectured that in the two-sided model, H-minor-freeness can be tested in time poly(1/eps). We resolve this conjecture in the affirmative.</div><div><br></div><div>3.Lastly, in a previous work on the two-sided-question above, Hassidim-Kelner-Nguyen-Onak (FOCS 2009) introduced a tool they call partition oracle. They conjectured that partition oracles could be implemented in time poly(1/eps) and gave an implementation which took exp(poly(1/eps)) time. In this work, we resolve this conjecture and produce such an oracle.</div><div><br></div><div><br></div><div>Additionally, this work also presents an algorithm which can recover a planted 3-coloring in a graph with some random like properties and suggests some future research directions alongside.</div>

Cette thèse explore des problématiques liées aux jeux. Les jeux qui nous intéressent sont ceux pour lesquels il n'y a pas d'information cachée: tout les joueurs ont accès à toute l'information relative au jeu; il n'y a pas non plus d'aléa. Certains jeux de société (comme les échecs ou le go) satisfont ces propriétés et sont représentatifs des jeux que nous considérons ici. La notion de stratégie est au centre de l'étude de ces jeux. En termes simples, une stratégie est une façon de jouer qui permet de s'assurer un certain résultat. La question centrale, à la fois quand on joue à un jeu et quand on l'étudie, est le problème de trouver la 'meilleure' stratégie, qui assure la victoire du joueur qui l'applique. Dans cette thèse, nous considérons à la fois des jeux à un joueurs, appelés puzzles combinatoires, et des jeux à deux joueurs. Le Rubik's cube, Rush-Hour, et le taquin sont des exemples biens connus de puzzles combinatoires. Récemment, un certain nombre de jeux -- des jeux à un joueur notamment -- ont connu un regain d'intérêt en tant qu'objets d'étude dans un domaine plus grand appelé reconfiguration. Les puzzles que l'on vient de mentionner peuvent tous être décrit de la façon suivante: il y a un ensemble de configurations, qui représente tous les états possibles du jeu; et le joueur est autorisé à transformer une configuration en une autre à via un certain nombre de règles. Le but est d'atteindre une certaine configuration cible à partir d'une configuration initiale. Le domaine de la reconfiguration étend cette définition à des problèmes de recherche: l'ensemble des configurations devient l'ensemble des solutions à une instance d'un problème donné, et l'on se demande si l'on peut transformer une solution donnée en une autre en utilisant uniquement un ensemble de règles de transformation précises. La recherche sur ce type de problèmes a été guidée par des motivations algorithmiques: ce processus peut être vu comme un moyen d'adapter une solution déjà en place en une nouvelle solution à l'aide de petits changements locaux; et des connections avec d'autres problèmes comme la génération aléatoire, ainsi que des problèmes de physique statistique. Les jeux à deux joueurs, qui sont aussi appelés jeux combinatoires, ont été étudiés depuis le début du XXème siècle avec les travaux de Bouton, continués par Berlekamp, Conway et Guy avec le développement d'une théorie unifiant un certain nombre de jeux classiques. Ce travail se focalise sur des joueurs parfaits, c'est à dire qui choisissent toujours le coup optimal. Notre but est de caractériser lequel des deux joueurs possède une stratégie qui lui assure la victoire, quelque soient les coups de son adversaire. Cette approche est au coeur de ce qui est appelé la Théorie des Jeux Combinatoires. Une grande parties des recherche est focalisée sur des 'jeux mathématiques', qui sont des jeux inventés par des mathématiciens, souvent avec des règles très simples, et rarement connus en dehors de la recherche. La motivation principale pour étudier ces jeux est la présence de liens parfois surprenant entre ces jeux et d'autres domaines des mathématiques comme entre autre la théorie des nombres, les automates ou les systèmes dynamiques. Dans cette thèse, nous étudions ainsi des jeux à un et deux joueurs. Nous nous intéressons tout particulièrement à la complexité de ces jeux, c'est à dire évaluer à quel point il est difficile (d'un point de vue algorithmique) de calculer une stratégie gagnante. Nous nous intéressons aussi à leur structure et à certaines propriétés qu'ils peuve satisfaire. Finalement, un des outils principaux dans cette étude est la notion de graphe, et nous utilisons notamment des méthodes et des techniques provenant de théorie des graphes pour résoudre ces problèmes / This thesis explores problems related to games. The games that we consider in this study are games for which there is no hidden information: all the players have access to all the information related to the game; there is also no randomness and everything is deterministic. A few well-known board games (such as chess or go) fall in this category and are representative of the kinds of games that we consider here. Central to the study of these games is the notion of strategy, which roughly speaking, is a way of playing which ensures a certain objective. The main question of interest, when both playing and studying a game, is the problem of finding the 'best' strategy, which secures the victory for the player following it. In this thesis, we consider both one-player games, also called combinatorial puzzles, and two-player games. Examples of combinatorial puzzles include Rubik's cube, Rush-Hour, Sokoban, the 15 puzzle, or peg solitaire. Recently, some types of one-player games in particular have received a strong regain of interest as part of the larger area of reconfiguration problems. The puzzles we described above can all be described in the following way: there is a set of configurations, which represents all the possible states of the game; and the player is allowed to transform a configuration using a prescribed set of moves. Starting from an initial configuration, the goal is to reach a target configuration by a succession of valid moves. Reconfiguration extends this definition to any search problem: the set of configuration becomes the set of solutions to an instance of a given problem, and we ask whether we can transform one given solution to another using only a prescribed set of moves. Hence, in addition to the combinatorial puzzles, reconfiguration problems also include many `games' which are not played by humans anymore but are instead mathematical problems sharing a lot of similarities with combinatorial puzzles. The study of reconfiguration problems has been driven by many different motivations. It has algorithmic applications: it can be seen as a way to adapt a current solution already in place to reach a new one by only making small local changes. It is also connected to other problems such as random sampling, approximate counting or problems coming from statistical physics. It can also be used as a tool for understanding the performance of some heuristic algorithms based on local modifications of solutions such as local search. Two-player games, which are also called combinatorial games, have been studied since the beginning of the twentieth century, with the works of Bouton which were continued with the development of a nice theory by Berlekamp, Conway, and Guy, unifying a certain number of classical games. We focus in this study on perfect strategies (i.e., players always choosing the best possible move), and try to characterize who wins under perfect play for various games. This approach is at the heart of what is called Combinatorial Game Theory. Most of the research in this area is focused on `math games' which are games invented by mathematicians, often with simple rules and almost never played by humans. The main motivation for studying these games comes from the nice, and sometimes unexpected connections these games have with other areas of mathematics, such as for example number theory, automatons, or dynamical systems. In this thesis, we study one- and two-player games. The questions we consider are often related to complexity. Complexity theory consists in trying to classify problems depending on their hardness. By hardness we mean to quantify how much time it would take for a computer to solve the problem. An other aspect of this research consists in investigating structural properties that these games can satisfy. Finally, one of our main tools is the notion of graph, and we use in particular methods and techniques from graph theory to answer the different questions we just mentioned

La prise de conscience générale de l'importance de vérifier plus scrupuleusement les programmes a engendré une croissance considérable des efforts de vérification formelle de programme durant cette dernière décennie. Néanmoins, le code qu'exécute l'ordinateur, ou code exécutable, n'est pas le code écrit par le développeur, ou code source. La vérification formelle de compilateurs est donc un complément indispensable à la vérification de code source.L'une des tâches les plus complexes de compilation est l'allocation de registres. C'est lors de celle-ci que le compilateur décide de la façon dont les variables du programme sont stockées en mémoire durant son exécution. La mémoire comporte deux types de conteneurs : les registres, zones d'accès rapide, présents en nombre limité, et la pile, de capacité supposée suffisamment importante pour héberger toutes les variables d'un programme, mais à laquelle l'accès est bien plus lent. Le but de l'allocation de registres est de tirer au mieux parti de la rapidité des registres, car une allocation de registres de bonne qualité peut conduire à une amélioration significative du temps d'exécution du programme.Le modèle le plus connu de l'allocation de registres repose sur la coloration de graphe d'interférence-affinité. Dans cette thèse, l'objectif est double : d'une part vérifier formellement des algorithmes connus d'allocation de registres par coloration de graphe, et d'autre part définir de nouveaux algorithmes optimisants pour cette étape de compilation. Nous montrons tout d'abord que l'assistant à la preuve Coq est adéquat à la formalisation d'algorithmes d'allocation de registres par coloration de graphes. Nous procédons ainsi à la vérification formelle en Coq d'un des algorithmes les plus classiques d'allocation de registres par coloration de graphes, l'Iterated Register Coalescing (IRC), et d'une généralisation de celui-ci permettant à un utilisateur peu familier du système Coq d'implanter facilement sa propre variante de cet algorithme au seul prix d'une éventuelle perte d'efficacité algorithmique. Ces formalisations nécessitent des réflexions autour de la formalisation des graphes d'interférence-affinité, de la traduction sous forme purement fonctionnelle d'algorithmes impératifs et de l'efficacité algorithmique, la terminaison et la correction de cette version fonctionnelle. Notre implantation formellement vérifiée de l'IRC a été intégrée à un prototype du compilateur CompCert.Nous avons ensuite étudié deux représentations intermédiaires de programmes, dont la forme SSA, et exploité leurs propriétés pour proposer de nouvelles approches de résolution optimale de la fusion, l'une des optimisations opéréeslors de l'allocation de registres dont l'impact est le plus fort sur la qualité du code compilé. Ces approches montrent que des critères de fusion tenant compte de paramètres globaux du graphe d'interférence-affinité, tels que sa largeur d'arbre, ouvrent la voie vers de nouvelles méthodes de résolution potentiellement plus performantes. / The need for trustful programs led to an increasing use of formal verication techniques the last decade, and especially of program proof. However, the code running on the computer is not the source code, i.e. the one written by the developper, since it has to betranslated by the compiler. As a result, the formal verication of compilers is required to complete the source code verication. One of the hardest phases of compilation is register allocation. Register allocation is the phase within which the compiler decides where the variables of the program are stored in the memory during its execution. The are two kinds of memory locations : a limited number of fast-access zones, called registers, and a very large but slow-access stack. The aim of register allocation is then to make a great use of registers, leading to a faster runnable code.The most used model for register allocation is the interference graph coloring one. In this thesis, our objective is twofold : first, formally verifying some well-known interference graph coloring algorithms for register allocation and, second, designing new graph-coloring register allocation algorithms. More precisely, we provide a fully formally veri ed implementation of the Iterated Register Coalescing, a very classical graph-coloring register allocation heuristics, that has been integrated into the CompCert compiler. We also studied two intermediate representations of programs used in compilers, and in particular the SSA form to design new algorithms, using global properties of the graph rather than local criteria currently used in the litterature.

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Orientador: Maria do Socorro Nogueira Rangel / Banca: Samuel Jurkiewicz / Banca: Cleonice Fátima Bracciali / Resumo: O objetivo desse trabalho é apresentar o problema de coloração em grafos sob diferentes perspectivas. Caracterizamos o polinômio cromático de um grafo e enunciamos algumas de suas propriedades. Apresentamos duas formulações matemáticas para o problema de coloração de vértices e um método de solução para cada formulação. Apresentamos e discutimos propostas de atividades para o desenvolvimento de uma Oficina de Coloração para alunos do Ensino Médio e Fundamental. / Abstract: In this work the graph coloring problem was presented under di erent perspectives. We define the chromatic polynomials of a graph and describe some of its properties. Furthermore, two solution methods for the vertex coloring problem, through integer programming formulation, has been presented. We propose and discuss some activities for the development of a Workshop for students of secondary school. / Mestre