483 research outputs found
An Order-based Algorithm for Minimum Dominating Set with Application in Graph Mining
Dominating set is a set of vertices of a graph such that all other vertices
have a neighbour in the dominating set. We propose a new order-based randomised
local search (RLS) algorithm to solve minimum dominating set problem in
large graphs. Experimental evaluation is presented for multiple types of
problem instances. These instances include unit disk graphs, which represent a
model of wireless networks, random scale-free networks, as well as samples from
two social networks and real-world graphs studied in network science. Our
experiments indicate that RLS performs better than both a classical greedy
approximation algorithm and two metaheuristic algorithms based on ant colony
optimisation and local search. The order-based algorithm is able to find small
dominating sets for graphs with tens of thousands of vertices. In addition, we
propose a multi-start variant of RLS that is suitable for solving the
minimum weight dominating set problem. The application of RLS in graph
mining is also briefly demonstrated
Exact and Heuristic Algorithms for the Domination Problem
In a simple connected graph , a subset of vertices
is a dominating set if any vertex is adjacent to some
vertex from this subset. A number of real-life problems can be modeled
using this problem which is known to be among the difficult NP-hard problems in
its class. We formulate the problem as an integer liner program (ILP) and
compare the performance with the two earlier existing exact state-of-the-art
algorithms and exact implicit enumeration and heuristic algorithms that we
propose here. Our exact algorithm was able to find optimal solutions much
faster than ILP and the above two exact algorithms for middle-dense instances.
For graphs with a considerable size, our heuristic algorithm was much faster
than both, ILP and our exact algorithm. It found an optimal solution for more
than half of the tested instances, whereas it improved the earlier known
state-of-the-art solutions for almost all the tested benchmark instances. Among
the instances where the optimum was not found, it gave an average approximation
error of
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
Recommended from our members
The frequency assignment problem
This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.This thesis examines a wide collection of frequency assignment problems. One of the largest topics in this thesis is that of L(2,1)-labellings of outerplanar graphs. The main result in this topic is the fact that there exists a polynomial time algorithm to determine the minimum L(2,1)-span for an outerplanar graph. This result generalises the analogous result for trees, solves a stated open problem and complements the fact that the problem is NP-complete for planar graphs. We furthermore give best possible bounds on the minimum L(2,1)-span and the cyclic-L(2,1)-span in outerplanar graphs, when the maximum degree is at least eight.
We also give polynomial time algorithms for solving the standard constraint matrix problem for several classes of graphs, such as chains of triangles, the wheel and a larger class of graphs containing the wheel. We furthermore introduce the concept of one-close-neighbour problems, which have some practical applications. We prove optimal results for bipartite graphs, odd cycles and complete multipartite graphs. Finally we evaluate different algorithms for the frequency assignment problem, using domination analysis. We compute bounds for the domination number of some heuristics for both the fixed spectrum version of the frequency assignment problem and the minimum span frequency assignment problem. Our results show that the standard greedy algorithm does not perform well, compared to some slightly more advanced algorithms, which is what we would expect. In this thesis we furthermore give some background and motivation for the topics being investigated, as well as mentioning several open problems.EPSR
Development of hybrid metaheuristics based on instance reduction for combinatorial optimization problems
113 p.La tesis presentada describe el desarrollo de algoritmos metaheurísticos híbridos, basados en reducción de instancias de problema. Éstos son enfocados en la resolución de problemas de optimización combinatorial. La motivación original de la investigación radicó en lograr, a través de la reducción de instancias de problemas, el uso efectivo de modelos de programación lineal entera (ILP) sobre problemas que dado su tamaño no admiten el uso directo con esta técnica exacta. En este contexto se presenta entre otros desarrollos el framework Construct, Merge, Solve & Adapt (CMSA) para resolución de problemas de optimización combinatorial en general, el cual posteriormente fue adaptado para mejorar el desempeño de otras metaheurísticas sin el uso de modelos ILP. Los algoritmos presentados mostraron resultados que compiten o superan el estado del arte sobre los problemas Minimum Common String Partition (MCSP), Minimum Covering Arborescence (MCA) y Weighted Independent Domination (WID)
- …