Ancestor tree for arbitrary multi-terminal cut functions

Abstract

In many applications, a function is defined on the cuts of a network. In the max-flow min-cut theorem, the function on a cut is simply the sum of all capacities of edges across the cut, and we want the minimum value of a cut separating a given pair of nodes. To find the minimum cuts separating$(\begin{array}{*{20}c} n \\ 2 \\ \end{array} )$ pairs of nodes, we only needn − 1 computations to construct the cut-tree. In general, we can define arbitrary values associated with all cuts in a network, and assume that there is a routine which gives the minimum cut separating a pair of nodes. To find the minimum cuts separating$(\begin{array}{*{20}c} n \\ 2 \\ \end{array} )$ pairs of nodes, we also only needn − 1 routine calls to construct a binary tree which gives all$(\begin{array}{*{20}c} n \\ 2 \\ \end{array} )$minimum partitions. The binary tree is analogous to the cut-tree of Gomory and Hu.