2 research outputs found
Integer programming models for the branchwidth problem
We consider the problem of computing the branchwidth and an optimal branch decomposition
of a graph. Branch decompositions and branchwidth were introduced in
1991 by Robertson and Seymour and were used in the proof of Graph Minors Theorem
(GMT), a well known conjecture (Wagner's conjecture) in graph theory. The
notions of branchwidth and branch decompositions have been proved to be useful for
solving many NP-hard problems that have applications in fields such as graph theory,
network design, sensor networks and biology. Branch decompositions have been
utilized for problems such as the traveling salesman problem by Cook and Seymour,
general minor containment and the branchwidth problem by Hicks by means of the
relevant branch decomposition-based algorithms.
Branch decomposition-based algorithms are fixed parameter tractable algorithms
obtained by combining dynamic programming techniques with branch decompositions.
The running time and space of these algorithms strongly depend on the width
of the utilized branch decomposition. Thus, finding optimal or close to optimal branch
decompositions is very important for the efficiency of the branch decomposition-based
algorithms. Motivated by the vastness of the fields of application, we aim to increase
the efficiency of the branch decomposition-based algorithms by investigating effective techniques to find optimal branch decompositions.
We present three integer programming models for the branchwidth problem.
Two similar formulations are based on the relationship of branchwidth problem with
a special case of the Steiner tree packing problem. The third formulation is based on
the notion of laminar separations. We utilize upper and lower bounds obtained by
heuristic algorithms, reduction techniques and cutting planes to increase the efficiency
of our models. We use all three models for the branchwidth problem on hypergraphs as
well. We compare the performance of three models both on graphs and hypergraphs.
Furthermore we use the third model for rank-width problem and also offer a
heuristic for finding good rank decompositions. We provide computational results for
this problem, which can be a basis of comparison for future formulations
Roteamento multicast multisessão: modelos e algoritmos
Multicast Technology has been studied over the last two decades and It has shown to be a
good approach to save network resources. Many approaches have been considered to solve
the multicast routing problem considering only one session and one source to attending
session‘s demand, as well, multiple sessions with more than one source per session. In
this thesis, the multicast routing problem is explored taking in consideration the models
and the algorithms designed to solve it when where multiple sessions and sources. Two
new models are proposed with different focuses. First, a mono-objective model optimizing
residual capacity, Z, of the network subject to a budget is designed and the objective is to
maximize Z. Second, a multi-objective model is designed with three objective functions:
cost, Z and hops counting. Both models consider multisession scenario with one source
per session. Besides, a third model is examined. This model was designed to optimize
Z in a scenario with multiple sessions with support to more than one source per session.
An experimental analysis was realized over the models considered. For each model, a set
of algorithms were designed. First, an ACO, a Genetic algorithm, a GRASP and an ILS
algorithm were designed to solve the mono-objective model – optimizing Z subject to a
budget. Second, a set of algorithm were designed to solve the multi-objective model. The
classical approaches were used: NSGA2, ssNSGA2, SMS-EMOA, GDE3 and MOEA/D.
In addition, a transgenetic algorithm was designed to solve the problem and it was compared
against the classical approaches. This algorithm considers the use of subpopulations
during the evolution. Each subpopulation is based on a solution construction operator
guided by one of the objective functions. Some solutions are considered as elite solutions
and they are considered to be improved by a transposon operator. Eight versions of the
transgenetic algorithm were evaluated. Third, an algorithm was designed to solve the
problem with multiple sessions and multiple sources per sessions. This algorithm is based
on Voronoi Diagrams and it is called MMVD. The algorithm designed were evaluated on
large experimental analysis. The sample generated by each algorithm on the instances
were evaluated based on non-parametric statistical tests. The analysis performed indicates
that ILS and Genetic algorithm have outperformed the ACO and GRASP. The comparison between ILS and Genetic has shown that ILS has better processing time performance.
In the multi-objective scenario, the version of Transgenetic called cross0 has
shown to be statistically better than the other algorithms in most of the instances based
on the hypervolume and addictive/multiplicative epsilon quality indicators. Finally, the
MMVD algorithm has shown to be better than the algorithm from literature based on the
experimental analysis performed for the model with multiple session and multiple sources
per session.A tecnologia multicast tem sido amplamente estudada ao longo dos anos e apresenta-se
como uma solução para melhor utilização dos recursos da rede. Várias abordagens já
foram avaliadas para o problema de roteamento desde o uso de uma sessão com apenas
uma fonte a um cenário com múltiplas sessões e múltiplas fontes por sessão. Neste trabalho,
é feito um estudo dos modelos matemáticos para o problema com múltiplas sessões
e múltiplas fontes. Dois modelos matemáticos foram propostos: uma versão multissessão
mono-objetivo que visa a otimização da capacidade residual sujeito a um limite de
custo e uma versão multiobjetivo com três funções-objetivo. Ambos os modelos levam em
conta o cenário multissessão com uma fonte por sessão. Além disso, um estudo algorÃtmico
foi realizado sobre um modelo da literatura que utiliza múltiplas fontes por sessão.
Três conjuntos de algoritmos foram propostos. O primeiro conjunto trata do problema
mono-objetivo proposto e considera as abordagens ACO, Genético, GRASP e ILS. O segundo
conjunto consiste dos algoritmos propostos para o modelo multiobjetivo. Foram
projetados os seguintes algoritmos: NSGA2, ssNSGA2, GDE3, MOEA/D e SMS-EMOA.
Além disso, foi projetado um algoritmo transgenético com subpopulações baseadas em
operadores de criação de solução direcionados por objetivos do problema. Também foi
utilizado o conceito de soluções de elite. No total, 8 versões do algoritmo transgenético foram
avaliadas. O terceiro conjunto de algoritmos consiste da heurÃstica MMVD proposta
para o modelo da literatura com múltiplas fontes por sessão. Esta heurÃstica é baseada no
uso de diagramas de Voronoi. O processo experimental foi realizado com amplo número
de instâncias configuradas de modo a avaliar diferentes situações. Os resultados foram
comparados utilizando métodos estatÃsticos não-paramétricos. A análise final indicou que
o ILS e o Genético obtiveram resultados muito similares, entretanto o ILS possui melhor
tempo de processamento. A versão cross0 do algoritmo transgenético obteve o melhor
resultado em praticamente todos os cenários avaliados. A heurÃstica MMVD obteve excelentes
resultados sobre algoritmos da literatura