23,093 research outputs found
Protecting a Graph with Mobile Guards
Mobile guards on the vertices of a graph are used to defend it against
attacks on either its vertices or its edges. Various models for this problem
have been proposed. In this survey we describe a number of these models with
particular attention to the case when the attack sequence is infinitely long
and the guards must induce some particular configuration before each attack,
such as a dominating set or a vertex cover. Results from the literature
concerning the number of guards needed to successfully defend a graph in each
of these problems are surveyed.Comment: 29 pages, two figures, surve
Holographic probabilities in eternal inflation
In the global description of eternal inflation, probabilities for vacua are
notoriously ambiguous. The local point of view is preferred by holography and
naturally picks out a simple probability measure. It is insensitive to large
expansion factors or lifetimes, and so resolves a recently noted paradox. Any
cosmological measure must be complemented with the probability for observers to
emerge in a given vacuum. In lieu of anthropic criteria, I propose to estimate
this by the entropy that can be produced in a local patch. This allows for
prior-free predictions.Comment: 5 pages, 3 figures. v4: published version, misprints corrected (mu ->
eta
Um problema de dominação eterna : classes de grafos, métodos de resolução e perspectiva prática
Orientadores: Cid Carvalho de Souza, Orlando LeeTese (doutorado) - Universidade Estadual de Campinas, Instituto de ComputaçãoResumo: O problema do conjunto dominante m-eterno é um problema de otimização em grafos que tem sido muito estudado nos últimos anos e para o qual se têm listado aplicações em vários domínios. O objetivo é determinar o número mínimo de guardas que consigam defender eternamente ataques nos vértices de um grafo; denominamos este número o índice de dominação m-eterna do grafo. Nesta tese, estudamos o problema do conjunto dominante
m-eterno: lidamos com aspectos de natureza teórica e prática e abordamos o problema
restrito a classes especícas de grafos e no caso geral. Examinamos o problema do conjunto dominante m-eterno com respeito a duas classes de grafos: os grafos de Cayley e os conhecidos grafos de intervalo próprios. Primeiramente, mostramos ser inválido um resultado sobre os grafos de Cayley presente na literatura, provamos que o resultado é válido para uma subclasse destes grafos e apresentamos outros achados. Em segundo lugar, fazemos descobertas em relação aos grafos de intervalo próprios, incluindo que, para estes grafos, o índice de dominação m-eterna é igual à cardinalidade máxima de um conjunto independente e, por consequência, o índice de dominação m-eterna pode ser computado em tempo linear.
Tratamos de uma questão que é fundamental para aplicações práticas do problema do
conjunto dominante m-eterno, mas que tem recebido relativamente pouca atenção. Para
tanto, introduzimos dois métodos heurísticos, nos quais formulamos e resolvemos modelos
de programação inteira e por restrições para computar limitantes ao índice de dominação
m-eterna. Realizamos um vasto experimento para analisar o desempenho destes métodos.
Neste processo, geramos um benchmark contendo 750 instâncias e efetuamos uma
avaliação prática de limitantes ao índice de dominação m-eterna disponíveis na literatura.
Por m, propomos e implementamos um algoritmo exato para o problema do conjunto
dominante m-eterno e contribuímos para o entendimento da sua complexidade: provamos
que a versão de decisão do problema é NP-difícil. Pelo que temos conhecimento, o algoritmo
proposto foi o primeiro método exato a ser desenvolvido e implementado para o
problema do conjunto dominante m-eternoAbstract: The m-eternal dominating set problem is a graph-protection optimization problem that has been an active research topic in the recent years and reported to have applications in various domains. It asks for the minimum number of guards that can eternally defend attacks on the vertices of a graph; this number is called the m-eternal domination number of the graph. In this thesis, we study the m-eternal dominating set problem by dealing with aspects of theoretical and practical nature and tackling the problem restricted to specic classes of graphs and in the general case. We examine the m-eternal dominating set problem for two classes of graphs: Cayley graphs and the well-known proper interval graphs. First, we disprove a published result on the m-eternal domination number of Cayley graphs, show that the result is valid for a subclass of these graphs, and report further ndings. Secondly, we present several discoveries regarding proper interval graphs, including that, for these graphs, the m- eternal domination number equals the maximum size of an independent set and, as a consequence, the m-eternal domination number can be computed in linear time. We address an issue that is fundamental to practical applications of the m-eternal dominating set problem but that has received relatively little attention. To this end, we introduce two heuristic methods, in which we propose and solve integer and constraint programming models to compute bounds on the m-eternal domination number. By performing an extensive experiment to validate the features of these methods, we generate a 750-instance benchmark and carry out a practical evaluation of bounds for the m-eternal domination number available in the literature. Finally, we propose and implement an exact algorithm for the m-eternal dominating set problem and contribute to the knowledge on its complexity: we prove that the decision version of the problem is NP-hard. As far as we know, the proposed algorithm was the first developed and implemented exact method for the m-eternal dominating set problemDoutoradoCiência da ComputaçãoDoutor em Ciência da Computação141964/2013-8CAPESCNP
The Greening of Faith: God, the Environment, and the Good Life (20th Anniversary Edition)
The recent release of Pope Francis’s much-discussed encyclical on the environment, Laudato Si’: On Care for Our Common Home, has reinforced environmental issues as also moral and spiritual issues. This anthology, twenty years ahead of the encyclical but very much in line with its agenda, offers essays by fifteen philosophers, theologians, and environmentalists who argue for a response to ecology that recognizes the tools of science but includes a more spiritual approach—one with a more humanistic, holistic view based on inherent reverence toward the natural world. Writers whose orientations range from Buddhism to evangelical Christianity to Catholicism to Native American beliefs explore ways to achieve this paradigm shift and suggest that “the environment is not only a spiritual issue, but the spiritual issue of our time.”https://scholars.unh.edu/unh_press/1003/thumbnail.jp
Eternal Independent Sets in Graphs
The use of mobile guards to protect a graph has received much attention in the literature of late in the form of eternal dominating sets, eternal vertex covers and other models of graph protection. In this paper, eternal independent sets are introduced. These are independent sets such that the following can be iterated forever: a vertex in the independent set can be replaced with a neighboring vertex and the resulting set is independent
Perpetually Dominating Large Grids
In the m-\emph{Eternal Domination} game, a team of guard tokens initially
occupies a dominating set on a graph . An attacker then picks a vertex
without a guard on it and attacks it. The guards defend against the attack: one
of them has to move to the attacked vertex, while each remaining one can choose
to move to one of his neighboring vertices. The new guards' placement must
again be dominating. This attack-defend procedure continues eternally. The
guards win if they can eternally maintain a dominating set against any sequence
of attacks, otherwise, the attacker wins.
The m-\emph{eternal domination number} for a graph is the minimum amount
of guards such that they win against any attacker strategy in (all guards
move model). We study rectangular grids and provide the first known general
upper bound on the m-eternal domination number for these graphs. Our novel
strategy implements a square rotation principle and eternally dominates grids by using approximately guards, which is
asymptotically optimal even for ordinary domination.Comment: latest full draft versio
Bounds for the -Eternal Domination Number of a Graph
Mobile guards on the vertices of a graph are used to defend the graph against an infinite sequence of attacks on vertices. A guard must move from a neighboring vertex to an attacked vertex (we assume attacks happen only at vertices containing no guard and that each vertex contains at most one guard). More than one guard is allowed to move in response to an attack. The -eternaldomination number, \edom(G), of a graph is the minimum number of guards needed to defend against any such sequence. We show that if is a connected graph with minimum degree at least~ and of order~, then \edom(G) \le \left\lfloor \frac{n-1}{2} \right\rfloor, and this bound is tight. We also prove that if is a cubic bipartite graph of order~, then \edom(G) \le \frac{7n}{16}
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