Multiterminal xcut problems

An i-j xcut of a set V={1, ..., n} is defined to be a partition of V into two disjoint nonempty subsets such that both i and j are contained in the same subset. When partitions are associated with costs, we define the i-j xcut problem to be the problem of computing an i-j xcut of minimum cost. This paper contains a proof that the {Mathematical expression} minimum xcut problems have at most n distinct optimal solution values. These solutions can be compactly represented by a set of n partitions in such a way that the optimal solution to any of the problems can be found in O(n) time. For a special additive cost function that naturally arises in connection to graphs, some interesting properties of the set of optimal solutions that lead to a very simple algorithm are presented.