32,356 research outputs found
All-Pairs Min-Cut in Sparse Networks
Algorithms are presented for the all-pairs min-cut problem in bounded treewidth, planar, and sparse networks. The approach used is to preprocess the input n-vertex network so that afterward, the value of a min-cut between any two vertices can be efficiently computed. A tradeoff is shown between the preprocessing time and the time taken to compute min-cuts subsequently. In particular, after an Onlog Ž n. preprocessing of a bounded tree-width network, it is possible to find the value of a min-cut between any two vertices in constant time. This implies that for Ž 2 such networks the all-pairs min-cut problem can be solved in time On.. This algorithm is used in conjunction with a graph decomposition technique of Frederickson to obtain algorithms for sparse and planar networks. The running times depend upon a topological property, �, of the input network. The parameter � varies between 1 and �Ž. n; the algorithms perform well when � � on. Ž. The value Ž 2 of a min-cut can be found in time On� � log �. and all-pairs min-cut can be Ž 2 4 solved in time On � � log �. for sparse networks. The corresponding runnin
All-pairs min-cut in sparse networks
Algorithms are presented for the all-pairs min-cut problem in bounded tree-width, planar and sparse networks. The approach used is to preprocess the input -vertex network so that, afterwards, the value of a min-cut between any two vertices can be efficiently computed. A tradeoff is shown between the preprocessing time and the time taken to compute min-cuts subsequently. In particular, after an preprocessing of a bounded tree-width network, it is possible to find the value of a min-cut between any two vertices in constant time. This implies that for such networks the all-pairs min-cut problem can be solved in time . This algorithm is used in conjunction with a graph decomposition technique of Frederickson to obtain algorithms for sparse and planar networks. The running times depend upon a topological property, , of the input network. The parameter varies between 1 and ; the algorithms perform well when . The value of a min-cut can be found in time and all-pairs min-cut can be solved in time for sparse networks. The corresponding running times4 for planar networks are and , respectively. The latter bounds depend on a result of independent interest: outerplanar networks have small ``mimicking'' networks which are also outerplanar
Community detection and graph partitioning
Many methods have been proposed for community detection in networks. Some of
the most promising are methods based on statistical inference, which rest on
solid mathematical foundations and return excellent results in practice. In
this paper we show that two of the most widely used inference methods can be
mapped directly onto versions of the standard minimum-cut graph partitioning
problem, which allows us to apply any of the many well-understood partitioning
algorithms to the solution of community detection problems. We illustrate the
approach by adapting the Laplacian spectral partitioning method to perform
community inference, testing the resulting algorithm on a range of examples,
including computer-generated and real-world networks. Both the quality of the
results and the running time rival the best previous methods.Comment: 5 pages, 2 figure
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