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 κ. In particular, if P has width w, then Bosek, Krawczyk, and Szczypka [SIAM J. Discrete Math., 23 (2010), pp. 1992-1999], 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 (2011), pp. 455-464] gave one of ck 2w. In this paper we prove an upper bound of the form ckw. This is most possible up to the value of c.

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Keywords First-Fit, On-line chain partition
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Journal SIAM Journal on Discrete Mathematics
Dujmović, V, Joret, G. (Gwenaël), & Wood, D. (2012). An improved bound for First-Fit on posets without two long incomparable chains. SIAM Journal on Discrete Mathematics, 26(3), 1068–1075. doi:10.1137/110855806