An easy subexponential bound for online chain partitioning

TL;DR

This paper presents a simplified and improved subexponential upper bound for online chain partitioning of posets, combining previous methods and demonstrating the bounds' limitations.

Contribution

The authors improve the upper bound for online chain partitioning from $w^{14 ext{lg} w}$ to $w^{6.5 ext{lg} w + 7}$ with a simpler proof, integrating prior work.

Findings

Improved the upper bound for online chain partitioning.

Provided examples illustrating the approach's limitations.

Simplified the proof methodology.

Abstract

Bosek and Krawczyk exhibited an online algorithm for partitioning an online poset of width $w$ into $w^{14\lg w}$ chains. We improve this to $w^{6.5 \lg w + 7}$ with a simpler and shorter proof by combining the work of Bosek & Krawczyk with work of Kierstead & Smith on First-Fit chain partitioning of ladder-free posets. We also provide examples illustrating the limits of our approach.