All-Pairs Min-Cut in Sparse Networks

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 inputn-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 anO(nlogn) 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 timeO(n2). 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, 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 timeO(n+γ2logγ) and all-pairs min-cut can be solved in timeO(n2+γ4logγ) for sparse networks. The corresponding running times for planar networks areO(n+γlogγ) andO(n2+γ3logγ), respectively. The latter bounds depend on a result of independent interest; outerplanar networks have small “mimicking” networks that are also outerplanar.