Exact Algorithms for the Graph Coloring Problem

The graph coloring problem is the problem of partitioning the vertices of a graph into the smallest possible set of independent sets. Since it is a well-known NP-Hard problem, it is of great interest of the computer science finding results over exact algorithms that solve it. The main algorithms of this kind, though, are scattered through the literature. In this paper, we group and contextualize some of these algorithms, which are based in Dynamic Programming, Branch-and-Bound and Integer Linear Programming. The algorithms for the first group are based in the work of Lawler, which searches maximal independent sets on each subset of vertices of a graph as the base of his algorithm. In the second group, the algorithms are based in the work of Brelaz, which adapted the DSATUR procedure to an exact version, and in the work of Zykov, which introduced the definition of Zykov trees. The third group contains the algorithms based in the work of Mehrotra and Trick, which uses the Column Generation method

Approximation algorithms in graphs via sample complexity

Orientador: André L. VignattiCoorientador: Murilo V. G. da SilvaTese (doutorado) - Universidade Federal do Paraná, Setor de Ciências Exatas, Programa de Pós-Graduação em Informática. Defesa : Curitiba, 28/10/2022Inclui referênciasÁrea de concentração: Ciência da ComputaçãoResumo: Grafos de grande porte advém de diversos contextos em fenômenos naturais e sociais. Contudo, algoritmos que escalam em complexidade de tempo cúbica e até mesmo quadrática, quando executados nesses grafos, podem ser computacionalmente custosos na prática. Neste trabalho, apresentamos esquemas de aproximação aleatorizados de tempo próximo de linear para dois problemas em grafos onde não se tem algoritmo conhecido de tempo estritamente subcúbico no número de vértices. Adicionalmente, para um terceiro problema, também apresentamos um esquema de aproximação aleatorizado de tempo próximo de linear, mas para o caso em que o grafo de entrada tem diâmetro logarítmico. Os algoritmos propostos foram projetados utilizando análise de complexidade de amostra, teoria de dimensão Vapnik–Chervonenkis e médias de Rademacher. Seja G = (V ,E) um grafo com n = |V| e m = |E|. O primeiro problema tratado neste trabalho é o de centralidade de percolação em um grafo G direcionado com pesos reais não-negativos. Tal medida quantifica a importância de um vértice em um grafo que está passando por um processo de contágio. Neste trabalho, apresentamos um algorimo de aproximação de tempo O(mlog nlog diamV (G)) para estimar a centralidade de percolação de cada vértice de V, onde diamV (G) é número máximo de vértices em um caminho mínimo em G. O segundo problema é uma versão relaxada do problema de computar todos os caminhos mínimos entre todos os pares de vértices (APSP). Nesta versão, iremos computar todos os caminhos de G que tenham "centralidade" pelo menos e, para 0 < epsilon < 1 constante, medida esta que está relacionada a uma generalização da centralidade de intermediação. O algoritmo proposto executa em tempo O(m log n + (diamV (G))2). O terceiro problema é o de coeficiente de agrupamento local de cada vértice de um grafo G. Neste trabalho propomos um algoritmo para este problema que roda em tempo O(delta lg delta +m), onde delta é o grau máximo de um vértice de um grafo. Finalmente, neste trabalho também apresentamos algoritmos de aproximação para os problemas do conjunto dominante mínimo (MDS) e da cobertura por vértices mínima (MVC). Em ambos aplicamos técnicas probabilísticas, contudo não utilizamos análise de complexidade de amostra nestes casos. Para estes problemas, obtivemos limitantes mais justos lidando com o problema diretamente em um modelo específico de grafo aleatório lei de potência. Em particular, mostramos que o fator de aproximação esperado para o problema MDS é assintoticamente no máximo 9.14, e para o problema MVC, o fator é assintoticamente menor do que 2. Tais valores superam os limitantes conhecidos da literatura.Abstract: Very large graphs arise in many contexts in natural and social phenomena. Algorithms with time complexity that scales in cubic or even in quadratic time in such graphs, although polynomial, can be computationally expensive in practice. In this thesis we present near-linear time randomized approximation schemes for two problems in graphs that are not known to admit algorithms that are truly subcubic in the number of vertices. Additionally, for a third problem, we also present a near-linear time randomized approximation scheme, but under the assumption that the input graph has logarithmic diameter, which is a very common scenario for large graphs in real-world applications. Our algorithms have been built using tools from sample complexity analysis, theory of Vapnik–Chervonenkis dimension, and Rademacher Averages. Let G = (V ,E) be a graph with n = |V| and m = |E|. The first problem we deal with is the percolation centrality in a directed weighted graph G, a measure that quantifies the importance of a vertex in a graph that is going through a contagious process. In this work we present an expected O(mlog nlog DiamV (G)) time approximation algorithm for the estimation of the percolation centrality for all vertices of G, wherein DiamV (G) is the maximum number of vertices in a shortest path in G. The second problem is a relaxed version of the all-pairs shortest paths (APSP) where we are interested in computing all shortest paths with "centrality" at least e, where 0 < epsilon < 1 is a constant. This centrality measure is a certain generalization of the betweenness centrality. We propose an algorithm for this relaxed problem that runs in O(m log n + (DiamV (G))2) time. The third problem corresponds to the computation of the local clustering coefficient of each vertex in a graph G, a measure that quantifies the degree in which a vertex is a part of a cluster in G. The algorithm that estimates the local clustering of each vertex in G runs in O(delta lg delta+m), wherein delta is the maximum degree of a vertex in a graph. Finally, we also propose approximation algorithms for the minimum dominating set (MDS) and the minimum vertex cover (MVC). However, for both problems we apply probabilistic techniques that are not related to sample complexity analysis, since we found tighter bounds for the approximation factor for these problems by attacking them directly using a particular random graph model for power-law graphs. In particular, we show that the expected approximation factor for the MDS problem is assymtotically at most 9.14, and for the MVC problem, the factor is assymptotically smaller than two. Such values outperforms the known bounds of the literature

Algoritmos exatos para o problema da coloração de grafos

Orientador : Renato CarmoDissertação (mestrado) - Universidade Federal do Paraná, Setor de Ciências Exatas, Programa de Pós-Graduação em Informática. Defesa: Curitiba, 21/08/2017Inclui referências : p. 81-83Resumo: O problema de coloração de grafos consiste em particionar os vértices de um grafo na menor quantidade possível de conjuntos independentes. Este trabalho tem como objetivo agrupar e contextualizar alguns dos principais algoritmos para o problema de coloração de grafos, reunindo num mesmo texto informações que se encontram espalhadas pela literatura técnica sob a forma de artigos científicos. Discutimos os principais métodos apresentados na literatura para o problema de coloração de grafos, a saber, soluções baseadas em Programação Linear Inteira, Branch-and-Bound e Programação Dinâmica. Palavras-chave: Teoria dos Grafos, Coloração de Grafos, Algoritmos Exatos.Abstract: The graph coloring problem is the problem of partitioning the vertices of a graph into the smallest possible set of independent sets. The goal of this work is to group and contextualize some of the main algorithms for the graph coloring problem, bringing into a single text information which is scattered in the technical literature in scientific papers. We discuss the main methods in the literature for the graph coloring problem, namely, solutions based on Integer Linear Programming, Branch-and-Bound and Dynamic Programming. Keywords: Graph Theory, Graph Coloring, Exact Algorithms

Exact Algorithms for the Graph Coloring Problem

The graph coloring problem is the problem of partitioning the vertices of a graph into the smallest possible set of independent sets. Since it is a well-known NP-Hard problem, it is of great interest of the computer science finding results over exact algorithms that solve it. The main algorithms of this kind, though, are scattered through the literature. In this paper, we group and contextualize some of these algorithms, which are based in Dynamic Programming, Branch-and-Bound and Integer Linear Programming. The algorithms for the first group are based in the work of Lawler, which searches maximal independent sets on each subset of vertices of a graph as the base of his algorithm. In the second group, the algorithms are based in the work of Brelaz, which adapted the DSATUR procedure to an exact version, and in the work of Zykov, which introduced the definition of Zykov trees. The third group contains the algorithms based in the work of Mehrotra and Trick, which uses the Column Generation method

Universal Dependencies 2.8.1

Universal Dependencies is a project that seeks to develop cross-linguistically consistent treebank annotation for many languages, with the goal of facilitating multilingual parser development, cross-lingual learning, and parsing research from a language typology perspective. The annotation scheme is based on (universal) Stanford dependencies (de Marneffe et al., 2006, 2008, 2014), Google universal part-of-speech tags (Petrov et al., 2012), and the Interset interlingua for morphosyntactic tagsets (Zeman, 2008). Version 2.8.1 fixes a bug in 2.8 where a portion of the Dutch Alpino treebank was accidentally omitted

Universal Dependencies 2.10

Universal Dependencies is a project that seeks to develop cross-linguistically consistent treebank annotation for many languages, with the goal of facilitating multilingual parser development, cross-lingual learning, and parsing research from a language typology perspective. The annotation scheme is based on (universal) Stanford dependencies (de Marneffe et al., 2006, 2008, 2014), Google universal part-of-speech tags (Petrov et al., 2012), and the Interset interlingua for morphosyntactic tagsets (Zeman, 2008)

Universal Dependencies 2.7

Universal Dependencies is a project that seeks to develop cross-linguistically consistent treebank annotation for many languages, with the goal of facilitating multilingual parser development, cross-lingual learning, and parsing research from a language typology perspective. The annotation scheme is based on (universal) Stanford dependencies (de Marneffe et al., 2006, 2008, 2014), Google universal part-of-speech tags (Petrov et al., 2012), and the Interset interlingua for morphosyntactic tagsets (Zeman, 2008)

Universal Dependencies 2.3

Universal Dependencies is a project that seeks to develop cross-linguistically consistent treebank annotation for many languages, with the goal of facilitating multilingual parser development, cross-lingual learning, and parsing research from a language typology perspective. The annotation scheme is based on (universal) Stanford dependencies (de Marneffe et al., 2006, 2008, 2014), Google universal part-of-speech tags (Petrov et al., 2012), and the Interset interlingua for morphosyntactic tagsets (Zeman, 2008)

Universal Dependencies 2.11

Universal Dependencies is a project that seeks to develop cross-linguistically consistent treebank annotation for many languages, with the goal of facilitating multilingual parser development, cross-lingual learning, and parsing research from a language typology perspective. The annotation scheme is based on (universal) Stanford dependencies (de Marneffe et al., 2006, 2008, 2014), Google universal part-of-speech tags (Petrov et al., 2012), and the Interset interlingua for morphosyntactic tagsets (Zeman, 2008)