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A subexponential upper bound for the on-line chain partitioning problem

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Abstract

The main question in the on-line chain partitioning problem is to decide whether there exists an on-line algorithm that partitions posets of width at most w into polynomial number of chains — see Trotter’s chapter Partially ordered sets in the Handbook of Combinatorics. So far the best known on-line algorithm of Kierstead used at most (5w − 1)/4 chains; on the other hand Szemerédi proved that any on-line algorithm requires at least \(\left( {\begin{array}{*{20}c} {w + 1} \\ 2 \\ \end{array} } \right)\) chains. These results were obtained in the early eighties and since then no progress in the general case has been done.

We provide an on-line algorithm that partitions posets of width w into at most w 13logw chains. This yields the first subexponential upper bound for the on-line chain partitioning problem.

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Authors and Affiliations

  1. Theoretical Computer Science, Jagiellonian University, ul. prof. Stanisława Łojasiewicza 6, 30-348, Kraków, Poland

    Bartłomiej Bosek & Tomasz Krawczyk

Authors
  1. Bartłomiej Bosek
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  2. Tomasz Krawczyk
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Correspondence to Bartłomiej Bosek.

Additional information

This work was supported by Polish Ministry of Science and Higher Education Grant (MNiSW) No. N206492338.

Extended abstract of this article has been published in proceedings of 51th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2010.

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Bosek, B., Krawczyk, T. A subexponential upper bound for the on-line chain partitioning problem. Combinatorica 35, 1–38 (2015). https://doi.org/10.1007/s00493-014-2908-7

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