All-pairs Min-cut in Sparse Networks ? (fsttcs'95)

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 eeciently computed. A tradeoo 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 nd 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 networks. The running times depend upon a topological property of the input network. The parameter varies between 1 and (n); the algorithms perform well when = o(n). The value of a min-cut can be found in time O(n + 2 log) and all-pairs min-cut can be solved in time O(n 2 + 4 log).