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 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(n²+γ⁴ log γ).

This work was partially supported by the EU ESPRIT Basic Research Action No. 7141 (ALCOM II).