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On-Line Chain Partitions of Orders: A Survey

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Abstract

On-line chain partition is a two-player game between Spoiler and Algorithm. Spoiler presents, point by point, a partially ordered set. Algorithm assigns incoming points (immediately and irrevocably) to the chains which constitute a chain partition of the order. The value of the game for orders of width w is a minimum number val(w) such that Algorithm has a strategy using at most val(w) chains on orders of width at most w. There are many recent results about variants of the general on-line chain partition problem. With this survey we attempt to give an overview over the state of the art in this field. As particularly interesting aspects of the article we see:

  • The sketch of the proof for the new sub-exponential upper bound of Bosek and Krawczyk: \({\rm{val}}(w)\leqslant w^{16\lg(w)}\).

  • The new lower bound: \({\rm{val}}(w)\geqslant(2-o(1))\binom{w+1}{2}\).

  • The inclusion of some simplified proofs of previously published results.

  • The comprehensive account on variants of the problem for interval orders.

  • The new lower bound for 2-dimensional up-growing orders.

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Authors and Affiliations

  1. Algorithmics Research Group, Faculty of Mathematics and Computer Science, Jagiellonian University, Lojasiewicza 6, 30-348, Kraków, Poland

    Bartłomiej Bosek, Tomasz Krawczyk, Grzegorz Matecki & Piotr Micek

  2. Diskrete Mathematik, Institut für Mathematik, Technische Universität Berlin, Strasse des 17. Juni 136, 10623, Berlin, Germany

    Stefan Felsner

  3. Embedded Systems Lab, University of Passau, Innstrasse 43, 94032, Passau, Germany

    Kamil Kloch

Authors
  1. Bartłomiej Bosek
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  2. Stefan Felsner
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  3. Kamil Kloch
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  4. Tomasz Krawczyk
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  5. Grzegorz Matecki
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  6. Piotr Micek
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Correspondence to Piotr Micek.

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Bosek, B., Felsner, S., Kloch, K. et al. On-Line Chain Partitions of Orders: A Survey. Order 29, 49–73 (2012). https://doi.org/10.1007/s11083-011-9197-1

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  • DOI: https://doi.org/10.1007/s11083-011-9197-1

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