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#### All-Pairs Min-Cut in Sparse Networks

##### MPS-Authors

Arikati,  Srinivasa Rao
Max Planck Society;

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Chaudhuri,  Shiva
Algorithms and Complexity, MPI for Informatics, Max Planck Society;

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Zaroliagis,  Christos
Algorithms and Complexity, MPI for Informatics, Max Planck Society;

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##### Citation

Arikati, S. R., Chaudhuri, S., & Zaroliagis, C. (1998). All-Pairs Min-Cut in Sparse Networks. Journal of Algorithms, 29, 82-110.

Cite as: http://hdl.handle.net/11858/00-001M-0000-000F-374E-1
##### Abstract
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 $n$-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 $O(n\log n)$ 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 such networks the all-pairs min-cut problem can be solved in time $O(n^2)$. 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, $\gamma$, of the input network. The parameter $\gamma$ varies between 1 and $\Theta(n)$; the algorithms perform well when $\gamma = o(n)$. The value of a min-cut can be found in time $O(n + \gamma^2 \log \gamma)$ and all-pairs min-cut can be solved in time $O(n^2 + \gamma^4 \log \gamma)$ for sparse networks. The corresponding running times for planar networks are $O(n+\gamma \log \gamma)$ and $O(n^2 + \gamma^3 \log \gamma)$, respectively. The latter bounds depend on a result of independent interest: outerplanar networks have small mimicking'' networks which are also outerplanar.