Lazy Queue Layouts of Posets

TL;DR

This paper studies the queue number of posets based on their width, introduces an improved upper bound using lazy linear extensions, and disproves a longstanding conjecture for posets of width greater than two.

Contribution

It extends lazy linear extension analysis to wider posets, providing a tighter upper bound and disproving the conjecture for width w > 2.

Findings

Upper bound of (w-1)^2 + 1 on queue number for width-w posets

Disproof of Heath and Pemmaraju's conjecture for w > 2

Existence of posets requiring at least w+1 queues in any linear extension

Abstract

We investigate the queue number of posets in terms of their width, that is, the maximum number of pairwise incomparable elements. A long-standing conjecture of Heath and Pemmaraju asserts that every poset of width w has queue number at most w. The conjecture has been confirmed for posets of width w=2 via so-called lazy linear extension. We extend and thoroughly analyze lazy linear extensions for posets of width w > 2. Our analysis implies an upper bound of $(w-1)^2 +1$ on the queue number of width-w posets, which is tight for the strategy and yields an improvement over the previously best-known bound. Further, we provide an example of a poset that requires at least w+1 queues in every linear extension, thereby disproving the conjecture for posets of width w > 2.