First-Fit is Linear on Posets Excluding Two Long Incomparable Chains

TL;DR

This paper proves that the First-Fit algorithm efficiently partitions (r + s)-free posets into a bounded number of chains proportional to their width, solving a longstanding open problem.

Contribution

It establishes a linear bound on the number of chains needed by First-Fit for (r + s)-free posets, where r,s ≥ 2, advancing understanding of poset chain partitioning.

Findings

First-Fit partitions (r + s)-free posets into at most 8(r-1)(s-1)w chains.

The bound is linear in the width w of the poset.

This resolves an open problem from 2010.

Abstract

A poset is (r + s)-free if it does not contain two incomparable chains of size r and s, respectively. We prove that when r and s are at least 2, the First-Fit algorithm partitions every (r + s)-free poset P into at most 8(r-1)(s-1)w chains, where w is the width of P. This solves an open problem of Bosek, Krawczyk, and Szczypka (SIAM J. Discrete Math., 23(4):1992--1999, 2010).