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arXiv · 1810.03807

A Dichotomy Theorem for First-Fit Chain Partitions

Kevin G. Milans·Michael C. Wigal
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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 $\textrm{FF}(w,Q) \ge 2^w - 1$.

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Kevin G. Milans, Michael C. Wigal. 2018-10-09. A Dichotomy Theorem for First-Fit Chain Partitions. https://arxiv.org/abs/1810.03807

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