Improved lower bounds on the on-line chain partitioning of posets of bounded dimension

TL;DR

This paper establishes improved lower bounds on the number of chains needed by any online algorithm to partition posets of bounded dimension, advancing understanding of the problem's complexity.

Contribution

It proves new lower bounds for online chain partitioning of 2-dimensional and d-dimensional posets, nearly doubling previous bounds for certain classes.

Findings

For 2-dimensional posets, at least roughly twice the Szemerédi bound is necessary.

For posets of dimension d, the lower bound approaches (2 - 1/(d-1)) times the Szemerédi bound.

The results significantly tighten the known lower bounds for online chain partitioning.

Abstract

An on-line chain partitioning algorithm receives a poset, one element at a time, and irrevocably assigns the element to one of the chains. Over 30 years ago, Szemer\'edi proved that any on-line algorithm could be forced to use $\binom{w+1}{2}$ chains to partition a poset of width $w$. The maximum number of chains that can be forced on any on-line algorithm remains unknown. In a survey paper by Bosek et al., it is shown that Szemer\'edi's argument could be improved to obtain a lower bound almost twice as good. Variants of the problem were considered where the class is restricted to posets of bounded dimension or where the poset is presented via a realizer of size $d$. In this paper, we prove two results. First, we prove that any on-line algorithm can be forced to use $(2-o(1))\binom{w+1}{2}$ chains to partition a $2$-dimensional poset of width $w$. Second, we prove that any on-line algorithm can be forced to use $(2-\frac{1}{d-1}-o(1))\binom{w+1}{2}$ chains to partition a poset of width $w$ presented via a realizer of size $d$.