All-pairs min-cut in sparse networks

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 preproeessing 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 rain-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 7 of the input network. The parameter 7 varies between 1 and Θ(n); the algorithms perform well when γ = o(n). The value of a rain-cut can be found in time O(n + γ2log 7) and all-pairs min-cut can be solved in time O(n2+ γ4log γ).