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All-pairs min-cut in sparse networks

Arikati, Srinivasa R. and Chaudhuri, Shiva and Zaroliagis, Christos D.

March 1996, 27 pages.

Status: available - back from printing

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 times4 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.

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  AUTHOR = {Arikati, Srinivasa R. and Chaudhuri, Shiva and Zaroliagis, Christos D.},
  TITLE = {All-pairs min-cut in sparse networks},
  TYPE = {Research Report},
  INSTITUTION = {Max-Planck-Institut f{\"u}r Informatik},
  ADDRESS = {Im Stadtwald, D-66123 Saarbr{\"u}cken, Germany},
  NUMBER = {MPI-I-96-1-007},
  MONTH = {March},
  YEAR = {1996},
  ISSN = {0946-011X},