On-Line Dimension for Posets Excluding Two Long Incomparable Chains

Abstract

Abstract. For a positive integer k, let k + k denote the poset consisting of two disjoint k-element chains, with all points of one chain incomparable with all points of the other. Bosek, Krawczyk and Szczypka showed that for each k ≥ 1, there exists a constant c k so that First Fit will use at most c k w 2 chains in partitioning a poset P of width at most w, provided the poset excludes k+k as a subposet. This result played a key role in the recent proof by Bosek and Krawczyk that O(w 16 log w ) chains are sufficient to partition on-line a poset of width w into chains. This result was the first improvement in Kierstead's exponential bound: (5 w − 1)/4 in nearly 30 years. Subsequently, Joret and Milans improved the Bosek-Krawczyk-Szczypka bound for the performance of First Fit to 8(k − 1) 2 w, which in turn yields the modest improvement to O(w 14 log w ) for the general on-line chain partitioning result. In this paper, we show that this class of posets admits a notion of on-line dimension. Specifically, we show that when k and w are positive integers, there exists an integer t = t(k, w) and an on-line algorithm that will construct an on-line realizer of size t for any poset P having width at most w, provided that the poset excludes k + k as a subposet.

First-Fit is Linear on Posets Excluding Two Long Incomparable 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. This question also appears in the survey of Bosek et al [1] and in a recent paper of Felsner, Krawczyk, and Trotter [7].…”

Minors and Dimension

“…, z witness Y k Z. To see (6), observe that X k Y and X k Z imply X Y Z or X Z Y , and the conclusion follows from (5).…”

“…For instance, any on-line algorithm trying to build a realizer of a poset of width w can be forced to use arbitrarily many linear extensions even when w = 3 [16], although posets of width w have dimension at most w [11]. On the other hand, for (k + k)-free posets of width w, Felsner, Krawczyk, and Trotter [5] devised an on-line algorithm that builds a realizer of size bounded in terms of k and w. Whether excluding S d or k + k instead of bounding the height in Theorem 1 or its predecessors keeps the dimension bounded remains a challenging open problem.…”

The Graph of Critical Pairs of a Crown

“…These posets are all 3-irreducible, i.e., they have dimension 3, but the removal of any point lowers the dimension to 2. This special case plays an important role in the on-line notion of dimension (see [6] and [12]). Also, the family of crowns includes the standard examples.…”