An Improved Bound for First-Fit on Posets Without Two Long Incomparable Chains

TL;DR

This paper improves the theoretical upper bound on the number of chains First-Fit uses to partition certain posets, showing it is proportional to the product of width and chain length, which is optimal up to a constant.

Contribution

It establishes a tighter upper bound of ckw for First-Fit on posets without two long incomparable chains, improving previous bounds of ck^{2}w and ck w.

Findings

New bound ckw for First-Fit on specific posets

Bound is proven to be optimal up to a constant

Advances understanding of chain partitioning efficiency

Abstract

It is known that the First-Fit algorithm for partitioning a poset P into chains uses relatively few chains when P does not have two incomparable chains each of size k. In particular, if P has width w then Bosek, Krawczyk, and Szczypka (SIAM J. Discrete Math., 23(4):1992--1999, 2010) proved an upper bound of ckw^{2} on the number of chains used by First-Fit for some constant c, while Joret and Milans (Order, 28(3):455--464, 2011) gave one of ck^{2}w. In this paper we prove an upper bound of the form ckw. This is best possible up to the value of c.