On-line partitioning of width w posets into w^O(log log w) chains

TL;DR

This paper introduces an improved online algorithm for partitioning width w posets into significantly fewer chains, reducing the number from w^{O(log w)} to w^{O(log log w)}.

Contribution

The paper presents a novel online algorithm that partitions width w posets into w^{O(log log w)} chains, improving the previous bounds.

Findings

Reduces the number of chains from w^{O(log w)} to w^{O(log log w)}.

Algorithm runs in w^{O(√w)}n time, efficient for large posets.

Advances the theoretical understanding of online poset partitioning.

Abstract

An on-line chain partitioning algorithm receives the elements of a poset one at a time, and when an element is received, irrevocably assigns it to one of the chains. In this paper, we present an on-line algorithm that partitions posets of width $w$ into $w^{O(\log{\log{w}})}$ chains. This improves over previously best known algorithms using $w^{O(\log{w})}$ chains by Bosek and Krawczyk and by Bosek, Kierstead, Krawczyk, Matecki, and Smith. Our algorithm runs in $w^{O(\sqrt{w})}n$ time, where $w$ is the width and $n$ is the size of a presented poset.