Optimal partitions

Consider a set of numbersZ={z ₁≥z ₂≥...≥z _n} and a functionf defined on subsets ofZ. LetP be a partition ofZ into disjoint subsetsS _i, say,g of them. The cost ofP is defined as $$C(P) = \sum\limits_{i = 1}^g {f(S_i )} .$$ By definition, in anordered partition, every pair of subsets has the property that the numbers in one subset are all greater than or equal to every number in the other subset. The problem of minimizingC(P) over all ordered partitions is called the optimal ordered partition problem. While no efficient method is known for solving the general optimal partition problem, the optimal ordered partition problem can be solved in quadratic time by dynamic programming. In this paper, we study the conditions onf under which an optimal ordered partition is indeed an optimal partition. In particular, we present an additive model and a multiplicative model for the functionf and give conditions such that the optimal partition problem can be reduced to the optimal ordered partition problem. We illustrate our results by applying them on problems which have been investigated previously in the literature.