An efficient algorithm for the minimum-range ideal problem

Suppose we are given a partially ordered set, a real-valued weight associated with each element and a positive integer k. We consider the problem which asks us to find an ideal of size k of the partially ordered set such that the range of the weights is minimum. We call this problem the minimum-range ideal problem. This paper shows a new and fast O(n log n + m) algorithm for this problem, where n is the number of elements and m is the smallest number of arcs to represent the partially ordered set. It is also proved that this problem has an Ω(n log n + m) lower bound. This means that the algorithm presented in this paper is optimal.