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Algoritmy pro řezy v grafech
Pecsők, Ján ; Kolman, Petr (advisor) ; Tiwary, Hans Raj (referee)
Graph-partitioning problems can be generically defined as a family of problems in which we are asked to partition a graph into two or more components. We present overview of methods and concepts used to find best graph partitions according to several criteria. We prove duality of multi-commodity flow and sparsest cut problem due to work of Leighton and Rao by describing algorithm using a Linear programming relaxation and a geometric embedding. Then we present the work of Arora, Rao and Vazirani (ARV) and their algorithm based on Semidefinite programming relaxation and a geometric embedding. We also explain the concept of expander flows first introduced in the work of ARV. One section of our work is devoted to the spectral graph theory, introducing the concepts of the spectral gap, random walks, conductance and relations between them. We connect the ideas of expander flows and spectral theory in chapter about so called Cut-Matching game framework. Finally we present the performance results of our implementation of the Leighton-Rao and the Cut-Matching game algorithms. Powered by TCPDF (www.tcpdf.org)
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Multicriteria graph partitioning
Houška, Ondřej ; Tůma, Miroslav (advisor) ; Hnětynková, Iveta (referee)
The thesis is about graph partitioning and applications of graph partitioning in paral- lel algorithms for solving big sparse linear equations. The problem of graph partitioning is thorougly described and standard graph partitioning algorithms are explained. The appli- cation part is focusing on the Conjugate Gradient method preconditioned by a variant of incomplete Cholesky factorization based on drop tolerance. The role of graph partitioning in the problem decomposition is described and a load balancing problem is studied. 1
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Algoritmy pro řezy v grafech
Pecsők, Ján ; Kolman, Petr (advisor) ; Tiwary, Hans Raj (referee)
Graph-partitioning problems can be generically defined as a family of problems in which we are asked to partition a graph into two or more components. We present overview of methods and concepts used to find best graph partitions according to several criteria. We prove duality of multi-commodity flow and sparsest cut problem due to work of Leighton and Rao by describing algorithm using a Linear programming relaxation and a geometric embedding. Then we present the work of Arora, Rao and Vazirani (ARV) and their algorithm based on Semidefinite programming relaxation and a geometric embedding. We also explain the concept of expander flows first introduced in the work of ARV. One section of our work is devoted to the spectral graph theory, introducing the concepts of the spectral gap, random walks, conductance and relations between them. We connect the ideas of expander flows and spectral theory in chapter about so called Cut-Matching game framework. Finally we present the performance results of our implementation of the Leighton-Rao and the Cut-Matching game algorithms. Powered by TCPDF (www.tcpdf.org)
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