First-Fit Algorithm for the On-Line Chain Partitioning Problem

On-Line Dimension for Posets Excluding Two Long Incomparable Chains

“…In [10], it is shown that on a width 2 poset having O(n 2 ) points, First Fit can be forced to use n chains. Nevertheless, Bosek, Krawczyk and Szczypka [4] showed that First Fit works surprisingly well in partitioning posets into chains provided they exclude two long incomparable chains. Theorem 2.6.…”

“…However, it was noted in [4] that the inequality in Theorem 2.6 might not be tight, and quite recently, this issue has been settled by Joret and Milans [9] with the following strengthening. We note that the elegant argument given by Joret and Milans is an extension of the column labeling method introduced by Pemmaraju, Raman and Varadarajan [17].…”

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

“…Using the positive integers for colors, First-Fit colors x with the least j such that x and all elements previously assigned color j form a chain. It is known that, for general posets, the number of chains used by First-Fit is not bounded by a function of w. In fact, Kierstead [9] showed that First-Fit uses arbitrarily many chains on posets of width 2 (see also [4]). …”

“…When r and s are at least two, the family of (r + s)-free posets contains the family of interval orders. Bosek, Krawczyk, and Szczypka [4] showed that when r ≥ s, First-Fit partitions every (r + s)-free poset into at most (3r − 2)(w − 1)w + w chains. They asked whether First-Fit uses only a linear number of chains, in terms of w, on (r + s)-free posets, as it does on interval orders.…”

Reverse Mathematics and Grundy colorings of graphs

“…, G n }). We say that K is a natural class 1) if K = Forb(F), where F consists only of connected finite graphs. Thus, it makes sense to refer to natural classes within RCA 0 .…”