Diametral Pairs of Linear Extensions

TL;DR

This paper investigates the maximum distance between linear extensions of a poset, establishes NP-completeness of computing this diameter, and explores properties of diametral pairs, including counterexamples to previous conjectures and classes where certain reversing properties hold.

Contribution

It proves NP-completeness of the linear extension diameter problem, provides polynomial algorithms for width-3 posets, and refutes a conjecture about critical pairs in diametral pairs.

Findings

Computing the linear extension diameter is NP-complete in general.

Polynomial-time solution exists for posets of width 3.

Counterexample disproves the conjecture that diametral pairs always reverse a critical pair.

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 NP-complete 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.