# All-pairs min-cut in sparse networks

• Srinivasa R. Arikati
• Shiva Chaudhuri
• Christos D. Zaroliagis
Algorithms
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1026)

## Abstract

Algorithms for the all-pairs min-cut problem in bounded tree-width and sparse networks are presented. The approach used is to preprocess the input network so that, afterwards, the value of a min-cut between any two vertices can be efficiently computed. A tradeoff between the preprocessing time and the time taken to compute min-cuts subsequently is shown. In particular, after 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(n2). This algorithm is used in conjunction with a graph decomposition technique of Frederickson to obtain algorithms for sparse networks. The running times depend upon a topological property γ of the input network. The parameter γ varies between 1 and Θ(n); the algorithms perform well when γ=o(n). The value of a min-cut can be found in time O(n+γ2 log γ) and all-pairs min-cut can be solved in time O(n24 log γ).

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