The Reversal Ratio of a Poset

TL;DR

This paper introduces the reversal ratio of a poset, studies its properties, and provides probabilistic bounds showing it can be small relative to the size of the poset, with implications for order dimension and width.

Contribution

It defines the reversal ratio of a poset and establishes probabilistic bounds demonstrating it can be bounded by a constant over log k for certain posets.

Findings

Reversal ratio can be as small as C/log k for specific posets.

Probabilistic methods effectively bound the reversal ratio.

Examines bounds of reversal ratio in relation to order dimension and width.

Abstract

Felsner and Reuter introduced the linear extension diameter of a partially ordered set $\mathbf{P}$, denoted $\mbox{led}(\mathbf{P})$, as the maximum distance between two linear extensions of $\mathbf{P}$, where distance is defined to be the number of incomparable pairs appearing in opposite orders (reversed) in the linear extensions. In this paper, we introduce the reversal ratio $RR(\mathbf{P})$ of $\mathbf{P}$ as the ratio of the linear extension diameter to the number of (unordered) incomparable pairs. We use probabilistic techniques to provide a family of posets $\mathbf{P}_k$ on at most $k\log k$ elements for which the reversal ratio $RR(\mathbf{P}_k)\leq C/\log k$, where $C$ is a constant. We also examine the questions of bounding the reversal ratio in terms of order dimension and width.