Partitioning Posets.dvi

Given a poset P = (X,≺), a partition X1, . . . ,Xk of X is called an ordered partition of P if, whenever x ∈ Xi and y ∈ Xj with x ≺ y, then i ≤ j. In this paper, we show that for every poset P = (X,≺) and every integer k ≥ 2, there exists an ordered partition of P into k parts such that the total number of comparable pairs within the parts is at most (m− 1)/k, where m ≥ 1 is the total number of edges in the comparability graph of P . We show that this bound is best possible for k = 2, but we give an improved bound, m/k − c(k)√m, for k ≥ 3, where c(k) is a constant depending only on k. We also show that, given a poset P = (X,≺), we can find an ordered partition of P that minimises the total number of comparable pairs within parts in polynomial time. We prove more general, weighted versions of these results.