Partitioning Posets

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.

• Type: Journal
• Year: 2008
• Where: ORDER
• Authors: Viresh Patel
• Keywords: ORDER 2008, Ordered Partition, Poset, Total Number