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On-line Chain Partitioning of Up-growing Interval Orders

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Abstract

On-line chain partitioning problem of on-line posets has been open for the past 20 years. The best known on-line algorithm uses \(\frac{5^w-1}{4}\) chains to cover poset of width w. Felsner (Theor. Comput. Sci., 175(2):283–292, 1997) introduced a variant of this problem considering only up-growing posets, i.e. on-line posets in which each new point is maximal at the moment of its arrival. He presented an algorithm using \({\left( {\begin{array}{*{20}c} {{w + 1}} \\ {2} \\ \end{array} } \right)}\) chains for width w posets and proved that his solution is optimal. In this paper, we study on-line chain partitioning of up-growing interval orders. We prove lower bound and upper bound to be 2w−1 for width w posets.

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Authors and Affiliations

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

    Patrick Baier

  2. Algorithmics Research Group, Faculty of Mathematics and Computer Science, Jagiellonian University, Gronostajowa 3, 30-387, Kraków, Poland

    Bartłomiej Bosek & Piotr Micek

Authors
  1. Patrick Baier
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  2. Bartłomiej Bosek
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  3. Piotr Micek
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Corresponding author

Correspondence to Bartłomiej Bosek.

Additional information

Piotr Micek and Bartłomiej Bosek are scholars of the project which is co-financed from the European Social Fund and national budget in the frame of The Integrated Regional Operational Programme.

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Baier, P., Bosek, B. & Micek, P. On-line Chain Partitioning of Up-growing Interval Orders. Order 24, 1–13 (2007). https://doi.org/10.1007/s11083-006-9050-0

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  • DOI: https://doi.org/10.1007/s11083-006-9050-0

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