Multiterminal xcut problems

Abstract

Ani-j xcut of a setV={1, ...,n} is defined to be a partition ofV into two disjoint nonempty subsets such that bothi andj are contained in the same subset. When partitions are associated with costs, we define thei-j xcut problem to be the problem of computing ani-j xcut of minimum cost. This paper contains a proof that the\((\begin{array}{*{20}c} n \\ 2 \\ \end{array} )\) minimum xcut problems have at mostn distinct optimal solution values. These solutions can be compactly represented by a set ofn partitions in such a way that the optimal solution to any of the problems can be found inO(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.