6 research outputs found
On an application of convexity to discrete systems
AbstractWe prove the following result: Let A be a symmetric matrix, f be gradient (or certain subgradient) of a convex function, and {yi} be a sequence defined by yi + 1 = f(Ayi), y0 arbitrary. Then the only possible periods of {yi} are 1 or 2
On pre-periods of discrete influence systems
AbstractWe investigate mappings of the form g = ÆA where Æ is a cyclically monotonous mapping of finite range and A is a linear mapping given by a symmetric matrix. We give some upper bounds on the pre-period of g, i.e. the maximum q for which all g(x),g2(x),âŠ,gq(x) are distinct
On the Voting Time of the Deterministic Majority Process
In the deterministic binary majority process we are given a simple graph
where each node has one out of two initial opinions. In every round, every node
adopts the majority opinion among its neighbors. By using a potential argument
first discovered by Goles and Olivos (1980), it is known that this process
always converges in rounds to a two-periodic state in which every node
either keeps its opinion or changes it in every round.
It has been shown by Frischknecht, Keller, and Wattenhofer (2013) that the
bound on the convergence time of the deterministic binary majority
process is indeed tight even for dense graphs. However, in many graphs such as
the complete graph, from any initial opinion assignment, the process converges
in just a constant number of rounds.
By carefully exploiting the structure of the potential function by Goles and
Olivos (1980), we derive a new upper bound on the convergence time of the
deterministic binary majority process that accounts for such exceptional cases.
We show that it is possible to identify certain modules of a graph in order
to obtain a new graph with the property that the worst-case
convergence time of is an upper bound on that of . Moreover, even
though our upper bound can be computed in linear time, we show that, given an
integer , it is NP-hard to decide whether there exists an initial opinion
assignment for which it takes more than rounds to converge to the
two-periodic state.Comment: full version of brief announcement accepted at DISC'1
Local majorities, coalitions and monopolies in graphs: a review
AbstractThis paper provides an overview of recent developments concerning the process of local majority voting in graphs, and its basic properties, from graph theoretic and algorithmic standpoints
Complexity and algorithms related to two classes of graph problems
This thesis addresses the problems associated with conversions on graphs and editing by removing a matching. We study the f-reversible processes, which are those associated with a threshold value for each vertex, and whose dynamics depends on the number of neighbors with different state for each vertex. We set a tight upper bound for the period and transient lengths, characterize all trees that reach the maximum transient length for 2-reversible processes, and we show that determining the size of a minimum conversion set is NP-hard. We show that the AND-OR model defines a convexity on graphs. We show results of NP-completeness and efficient algorithms for certain convexity parameters for this new one, as well as approximate algorithms. We introduce the concept of generalized threshold processes, where the results are NP-completeness and efficient algorithms for both non relaxed and relaxed versions. We study the problem of deciding whether a given graph admits a removal of a matching in order to destroy all cycles. We show that this problem is NP-hard even for subcubic graphs, but admits efficient solution for several graph classes. We study the problem of deciding whether a given graph admits a removal of a matching in order to destroy all odd cycles. We show that this problem is NP-hard even for planar graphs with bounded degree, but admits efficient solution for some graph classes. We also show parameterized results.Esta tese aborda problemas associados a conversĂ”es em grafos e de edição pela remoção de um emparelhamento. Estudamos processos f-reversĂveis, que sĂŁo aqueles associados a um valor de limiar para cada vĂ©rtice e cuja dinĂąmica depende da quantidade de vizinhos com estado contrĂĄrio para cada vĂ©rtice. Estabelecemos um limite superior justo para o tamanho do perĂodo e transiente, caracterizamos todas as ĂĄrvores que alcançam o transiente mĂĄximo em processos 2-reversĂveis e mostramos que determinar o tamanho de um conjunto conversor mĂnimo Ă© NP-difĂcil. Mostramos que o modelo AND-OR define uma convexidade sobre grafos. Mostramos resultados de NP-completude e algoritmos eficientes para certos parĂąmetros de convexidade para esta nova, assim como algoritmos aproximativos. Introduzimos o conceito de processos de limiar generalizados, onde mostramos resultados de NP-completude e algoritmos eficientes para ambas as versĂ”es nĂŁo relaxada e relaxada. Estudamos o problema de decidir se um dado grafo admite uma remoção de um emparelhamento de modo a remover todos os ciclos. Mostramos que este problema Ă© NP-difĂcil mesmo para grafos subcĂșbicos, mas admite solução eficiente para vĂĄrias classes de grafos. Estudamos o problema de decidir se um dado grafo admite uma remoção de um emparelhamento de modo a remover todos os ciclos Ămpares. Mostramos que este problema Ă© NP-difĂcil mesmo para grafos planares com grau limitado, mas admite solução eficiente para algumas classes de grafos. Mostramos tambĂ©m resultados parametrizados