A Dichotomy Theorem for First-Fit Chain Partitions

TL;DR

This paper establishes a dichotomy for the number of chains used by the First-Fit algorithm on Q-free posets, showing it is either exponentially bounded or nearly exponential depending on the structure of Q.

Contribution

It characterizes exactly which posets Q lead to polynomially bounded chain partitions under First-Fit, introducing a family nd proving a sharp dichotomy.

Findings

If Q is in amily, FF(w,Q) ounded by 2^{c(\,log w)^2}

If Q is not in amily, FF(w,Q) t least 2^w - 1

Dichotomy theorem for First-Fit chain partitions based on Q structure

Abstract

First-Fit is a greedy algorithm for partitioning the elements of a poset into chains. Let $\textrm{FF}(w,Q)$ be the maximum number of chains that First-Fit uses on a $Q$-free poset of width $w$. A result due to Bosek, Krawczyk, and Matecki states that $\textrm{FF}(w,Q)$ is finite when $Q$ has width at most $2$. We describe a family of posets $\mathcal{Q}$ and show that the following dichotomy holds: if $Q\in\mathcal{Q}$, then $\textrm{FF}(w,Q) \le 2^{c(\log w)^2}$ for some constant $c$ depending only on $Q$, and if $Q\not\in\mathcal{Q}$, then $\textrm{FF}(w,Q) \ge 2^w - 1$.