Diametral Pairs of Linear Extensions

Abstract

Given a finite poset P, we consider pairs of linear extensions of P with maximal distance, where the distance between two linear extensions L 1 , L 2 is the number of pairs of elements of P appearing in different orders in L 1 and L 2 . A diametral pair maximizes the distance among all pairs of linear extensions of P. Felsner and Reuter defined the linear extension diameter of P as the distance between a diametral pair of linear extensions.We show that computing the linear extension diameter is NPcomplete in general, but can be solved in polynomial time for posets of width 3.Felsner and Reuter conjectured that, in every diametral pair, at least one of the linear extensions reverses a critical pair. We construct a counterexample to this conjecture. On the other hand, we show that a slightly stronger property holds for many classes of posets: We call a poset diametrally reversing if, in every diametral pair, both linear extensions reverse a critical pair. Among other results we show that interval orders and 3-layer posets are diametrally reversing. From the latter it follows that almost all posets are diametrally reversing.

Complete Valuations on Finite Distributive Lattices

“…Then (3) follows from last two equalities and ( 1). ( 4) follows by duality from (2). ✷ Proof (of Theorem 3.1).…”

“…The construction of a complete valuation on a lattice from a realizer of the underlying poset, that we'll give in Section 4, has some points in common with the construction of a pair of diametral linear extensions, as defined, for instance, in [2] , [4] , [6]. The order induced by a complete valuation, which is lexicographic with respect to one of the linear extensions of the poset, is a linear extension of the lattice; the diametrally opposite linear extension would be given by another complete valuation, whose weight function counts chains in the opposite direction.…”

Drawing Order Diagrams Through Two-Dimension Extension

“…It is known to be N P-complete to decide whether an ordered set can be altered to be two-dimensional by inserting k pairs [1]. Hence, we propose an algorithm that tackles the problem for approximating the corresponding optimization problem.…”

“…The transitive incomparability graph of this ordered set has 206 vertices. By removing pairs (5,9), (3,13), (6,15), (17,15), (5,15), (5,13), (11,10), (9,6), (5,6), (4,15), (1,8), (3,15), (11,12), (2,8), (11,7), (0,15), (1,16), the transitive incomparability graph of this ordered set becomes bipartite. However, if we once again compute the transitive incomparability graph of the new ordered set, we see that it is not bipartite, i.e., the new ordered set is once again not two-dimensional.…”

The Reversal Ratio of a Poset

“…More generally, in [5], Felsner and Massow give a polynomial-time algorithm for computing the linear extension diameter of downset lattices of 2-dimensional posets; an analysis of their proof shows that RR(P) ≥ 1/2 for all such posets. By contrast, Brightwell and Massow [4] showed that it is NPcomplete to compute the linear extension diameter of an arbitrary poset.…”