On-line Adaptive Chain Covering of Upgrowing Posets

We analyze on-line chain partitioning problem and its variants as a two-person game. One person (Spoiler) builds an on-line poset presenting one point at time. The other one (Algorithm) assigns new point to a chain. Kierstead gave a strategy for Algorithm showing that width w posets can be on-line c...

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Bibliographic Details
Published in:	Discrete mathematics and theoretical computer science Vol. DMTCS Proceedings vol. AF,...; no. Proceedings; pp. 37 - 48
Main Authors:	Bosek, Bartłomiej, Micek, Piotr
Format:	Journal Article Conference Proceeding
Language:	English
Published:	DMTCS 01.01.2005 Discrete Mathematics and Theoretical Computer Science Discrete Mathematics & Theoretical Computer Science
Series:	DMTCS Proceedings
Subjects:	[info.info-hc] computer science [cs]/human-computer interaction [cs.hc] [info.info-lo] computer science [cs]/logic in computer science [cs.lo] [info.info-sc] computer science [cs]/symbolic computation [cs.sc] chain partition Computer Science Human-Computer Interaction Logic in Computer Science on-line algorithm order poset Symbolic Computation chain partition on-line algorithm poset order
ISSN:	1365-8050, 1462-7264, 1365-8050
Online Access:	Get full text
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Summary:	We analyze on-line chain partitioning problem and its variants as a two-person game. One person (Spoiler) builds an on-line poset presenting one point at time. The other one (Algorithm) assigns new point to a chain. Kierstead gave a strategy for Algorithm showing that width w posets can be on-line chain partitioned into $\frac{{5}^{w-1}}{4}$ chains. Felsner proved that if Spoiler presents an upgrowing poset, i.e., each new point is maximal at the moment of its arrival then there is a strategy for Algorithm using at most $\binom{w+1}{2}$ chains and it is best possible. An adaptive variant of this problem allows Algorithm to assign to the new point a set of chains and than to remove some of them (but not all) while covering next points. Felsner stated a hypothesis that in on-line adaptive chain covering of upgrowing posets Algorithm may use smaller number of chains than in non-adaptive version. In this paper we provide an argument suggesting that it is true. We present a class of upgrowing posets in which Spoiler has a strategy forcing Algorithm to use at least $\binom{w+1}{2}$ chains (in non-adaptive version) and Algorithm has a strategy using at most $O(w\sqrt{w})$ chains in adaptive version.
ISSN:	1365-8050 1462-7264 1365-8050
DOI:	10.46298/dmtcs.3473