The maximum number of cliques in dense graphs

AbstractDenote the number of vertices of G by |G|. A clique of graph G is a maximal complete subgraph. The density ω(G) is the number of vertices in the largest clique of G. If ω(G)⩾12|G|, then G has at most 2|G|−ω(G) cliques. The extremal graphs are then examined as well

Lower bounds on dynamic programming for maximum weight independent set

Publisher Copyright: © 2021 Tuukka Korhonen.We prove lower bounds on pure dynamic programming algorithms for maximum weight independent set (MWIS). We model such algorithms as tropical circuits, i.e., circuits that compute with max and + operations. For a graph G, an MWIS-circuit of G is a tropical circuit whose inputs correspond to vertices of G and which computes the weight of a maximum weight independent set of G for any assignment of weights to the inputs. We show that if G has treewidth w and maximum degree d, then any MWIS-circuit of G has 2Ω(w/d) gates and that if G is planar, or more generally H-minor-free for any fixed graph H, then any MWIS-circuit of G has 2Ω(w) gates. An MWIS-formula is an MWIScircuit where each gate has fan-out at most one. We show that if G has treedepth t and maximum degree d, then any MWIS-formula of G has 2Ω(t/d) gates. It follows that treewidth characterizes optimal MWIS-circuits up to polynomials for all bounded degree graphs and H-minor-free graphs, and treedepth characterizes optimal MWIS-formulas up to polynomials for all bounded degree graphs.Peer reviewe

Analysis of an exhaustive search algorithm in random graphs and the n^{c\log n} -asymptotics

We analyze the cost used by a naive exhaustive search algorithm for finding a maximum independent set in random graphs under the usual G_{n,p} -model where each possible edge appears independently with the same probability p. The expected cost turns out to be of the less common asymptotic order n^{c\log n}, which we explore from several different perspectives. Also we collect many instances where such an order appears, from algorithmics to analysis, from probability to algebra. The limiting distribution of the cost required by the algorithm under a purely idealized random model is proved to be normal. The approach we develop is of some generality and is amenable for other graph algorithms.Comment: 35 page

Decomposição nula de grafos unicíclicos

Nesta tese, obtemos informações estruturais de um grafo unicíclico usando seu espaço nulo. Observando as entradas dos autovetores associados ao autovalor zero, obtemos a decomposição nula de um grafo. Utilizando a decomposição nula desse grafo unicíclico, obtemos fórmulas fechadas para os números de emparelhamento e independência do mesmo. Algumas dessas fórmulas permitem o cálculo desses dois parâmetros através da decomposição nula de subárvores do grafo unicíclico enquanto outras possibilitam o cálculo através da decomposição nula do próprio grafo unicíclico. Além disso, caracterizamos um grafo unicíclico singular através do suporte de suas árvores pendentes e obtemos uma base para o espaço nulo desse grafo usando uma base do espaço nulo de suas subárvores.In this dissertation, we obtain structural information of a unicyclic graph using its null space. Observing the entries of the eigenvectors corresponding to the eigenvalue zero we obtain the null decomposition of a graph. Using the null decomposition of a unicyclic graph we obtain closed formulas for the matching and independence numbers of unicyclic graphs. Some of these formulas allows one to compute these two parameters using the null decomposition of the subtrees of the unicyclic graph while others make it possible to compute these parameters by null decomposition of the unicyclic graph itself. Moreover, we obtain a characterization of a singular unicyclic graph using the support of their pendant trees and we obtain a basis for the null space of unicyclic graph using a basis of the null space of their subtrees

Algoritmos para o problema da clique máxima : análise e comparação experimental

Orientador : Prof. Dr. Renato CarmoTese (doutorado) - Universidade Federal do Paraná, Setor de Ciências Exatas, Programa de Pós-Graduação em Informática. Defesa: Curitiba, 28/09/2017Inclui referências : f. 107-113Resumo: O problema da Clique Máxima (CM) é um problema fundamental e há uma grande motivação pela busca de algoritmos tão eficientes quanto possível para resolvê-lo de forma exata. Como esperado para um problema NP-difícil, os melhores algoritmos com desempenho de pior caso conhecido tem custo de tempo exponencial. Por outro lado, resultados experimentais encontrados na literatura indicam que instâncias de tamanho considerável podem ser resolvidas usando algoritmos baseados na técnica de branch-and-bound. Com isso, observa-se uma distância entre os melhores resultados analíticos e os melhores resultados experimentais. Uma possível explicação para discrepância aparente entre teoria e prática foi encontrada pela análise de instâncias aleatórias. Diversos algoritmos de branch- and-bound para a solução exata do CM foram estudados, analisados e implementados. Com base nos resultados analíticos é proposta uma metodologia para comparação experimental de algoritmos, que tem como principal ponto positivo o fato de que algoritmos podem ser comparados independente de detalhes de implementação e execução. Vários algoritmos foram testados como prova de conceito. Também foram estudadas instâncias de pior caso para algoritmos de branch-and-bound que só utilizam coloração como limitante superior, resultando em um custo exponencial de tempo para estes algoritmos. Uma nova família de algoritmos foi desenvolvida, capaz de resolver tais instâncias em tempo polinomial. Recentemente, técnicas de resolvedores para problemas de satisfatibilidade têm sido aplicadas em algoritmos para CM. Tais técnicas dependem de uma redução entre os dois problemas, mas o significado em termos do grafo fica obscurecido nas descrições originais. Algumas técnicas foram estudadas e convertidas para uma descrição que não usa termos referentes aos problemas de satisfatibilidade. A implementação de vários algoritmos estudados foi disponibilizada em um repositório de acesso público. Palavras-chave: Solução exata. Branch-and-bound. Análise de algoritmos. Comparação experimental.Abstract: e Maximum Clique problem (CM) is a fundamental problem and there is a great motivation for the development of efficient exact algorithms to solve it. As expected for a NP-hard problem, the best algorithms where worst case analyses have been conducted present exponential running times. On the other hand, experimental results available in the literature show that instances of considerable size can be solved by branch and bound algorithms. Therefore, there is an apparent gap between the best theoretical results and the best experimental results. One possible explanation for this discrepancy between theory and practice was found through the analyses of random instances. Several exact branch and bound algorithm for CM were studied, analyzed and implemented. Based on these analytical results, a new methodology for the comparison of algorithms is proposed, where algorithms can be tested and compared regardless of implementation and execution details. Several algorithms were tested as a proof of concept. Worst case instances for some branch and bound algorithms were studied, namely algorithms that adopt only coloring-based bounding techniques to reduce the search space. These algorithms present exponential time cost for the studied instances. A new family of algorithms was developed, which is able to solve the mentioned instances in polinomial time. Recently, techniques from satisfiability solvers have been used in algorithms for CM. Such techniques depend on a reduction between the problems, and the original descriptions in terms of propositional calculus obscures their graph theoretic meaning. Some of these techniques were studied and converted to a description that uses only graph theory terminology. The implementation of several algorithms was made available in a public access repository. Keywords: Exact solution. Branch-and-bound. Analysis of algorithms. Experimental comparison

Graph transformations preserving the stability number

We analyse the relations between several graph transformations that were introduced to be used in procedures determining the stability number of a graph. We show that all these transformations can be decomposed into a sequence of edge deletions and twin deletions. We also show how some of these transformations are related to the notion of even pair introduced to color some classes of perfect graphs. Then, some properties of edge deletion and twin deletion are given and a conjecture is formulated about the class of graphs for which these transformations can be used to determine the stability number

Graph Transformations preserving the stability number

We analyze the relations between several graph transformations that were introduced to be used in procedures determining the stability number of a graph. We show that all these transformations can be decomposed into a sequence of edge deletions and twin deletions. We also show how some of these transformations are related to the notion of even pair introduced to color some classes of perfect graphs. Then, some properties of edge deletion and twin deletion are given and a conjecture is formulated about the class of graphs for which these transformations can be used to determine the stability number. (C) 2011 Elsevier B.V. All rights reserved