CDAM: Computational, Discrete and Applicable
Mathematics@LSE


CDAM Research Report, LSECDAM200719August 2007 
Partitioning Posets
Viresh Patel
Given a poset P=(X, <), a partition X_{1},..., X_{k} of X is called an ordered partition of P if, whenever x ∈ X_{i} and y ∈ X_{j} 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 (m1)/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.A PDF file (193 kB) with the full contents of this report can be downloaded by clicking here.
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