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

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.

This is a preview of subscription content, access via your institution.

References

  1. [1]

    P. Baier, B. Bosek and P. Micek: On-line chain partitioning of up-growing interval orders, Order 24 (2007), 1–13.

    Article  MATH  MathSciNet  Google Scholar 

  2. [2]

    B. Bosek: On-line chain partitioning approach to scheduling, Ph.D. thesis, Jagiellonian University, 2008.

    Google Scholar 

  3. [3]

    B. Bosek, S. Felsner, K. Kloch, T. Krawczyk, G. Matecki and P. Micek: On-line chain partitions of orders: survey, Order 29 (2012), 49–73.

    Article  MATH  MathSciNet  Google Scholar 

  4. [4]

    B. Bosek, H. A. Kierstead, T. Krawczyk, G. Matecki and M. E. Smith: Regular posets and on-line chain partitioning, manuscript, 2013.

    Google Scholar 

  5. [5]

    B. Bosek and T. Krawczyk: On-line chain partitioning of 2-dimensional orders, unpublished result, 2010.

    Google Scholar 

  6. [6]

    B. Bosek, T. Krawczyk and G. Matecki: First—fit coloring of incomparability graphs, SIAM J. Discrete Math. 27 (2013), 126–140.

    Article  MATH  MathSciNet  Google Scholar 

  7. [7]

    B. Bosek, T. Krawczyk and E. Szczypka: First—fit algorithm for the on-line chain partitioning problem, SIAM J. Discrete Math. 23 (2010), 1992–1999.

    Article  MathSciNet  Google Scholar 

  8. [8]

    P. Broniek: On-line chain partitioning as a model for real-time scheduling, in: Proceedings of the Second Workshop on Computational Logic and Applications (CLA 2004) (Amsterdam), Electron. Notes Theor. Comput. Sci. 140, Elsevier, 2005, 15–29 (electronic).

    Article  MathSciNet  Google Scholar 

  9. [9]

    M. Chrobak and M. ślusarek: On some packing problem related to dynamic storage allocation, RAIRO Inform. Théor. Appl. 22 (1988), 487–499.

    MATH  Google Scholar 

  10. [10]

    V. DujmoviĆ, G. Joret and D. R. Wood: An Improved Bound for First-Fit on Posets Without Two Long Incomparable Chains, SIAM J. Discrete Math. 26 (2012), 1068–1075.

    Article  MATH  MathSciNet  Google Scholar 

  11. [11]

    S. Felsner: On-line chain partitions of orders, Theoret. Comput. Sci. 175 (1997), 283–292.

    Article  MATH  MathSciNet  Google Scholar 

  12. [12]

    S. Felsner, K. Kloch, G. Matecki and P. Micek: On-line Chain Partitions of Up-growing Semi-orders, Order 30 (2013), 85–101.

    Article  MATH  MathSciNet  Google Scholar 

  13. [13]

    G. Joret and K. Milans: First—fit is linear on posets excluding two long incomparable chains, Order 28 (2011), 455–464.

    Article  MATH  MathSciNet  Google Scholar 

  14. [14]

    H. A. Kierstead: An e ective version of Dilworth’s theorem, Trans. Amer. Math. Soc. 268 (1981), 63–77.

    MATH  MathSciNet  Google Scholar 

  15. [15]

    H. A. Kierstead: Recursive ordered sets, Combinatorics and ordered sets (Arcata, Calif., 1985), Contemp. Math. 57, Amer. Math. Soc., Providence, RI, 1986, 75–102.

    Google Scholar 

  16. [16]

    H. A. Kierstead, G. F. McNulty and W. T. Trotter: A theory of recursive dimension for ordered sets, Order 1 (1984), 67–82.

    Article  MATH  MathSciNet  Google Scholar 

  17. [17]

    H. A. Kierstead and M. E. Smith: On First—fit coloring of ladder-free posets, European J. Combin. 34 (2013), 474–489.

    Article  MATH  MathSciNet  Google Scholar 

  18. [18]

    H. A. Kierstead and W. T. Trotter: An extremal problem in recursive combi-natorics, in: Proceedings of the Twelfth Southeastern Conference on Combinatorics, Graph Theory and Computing, Vol. II (Baton Rouge, La., 1981), 33, 1981, 143–153.

    MathSciNet  Google Scholar 

  19. [19]

    P. Micek: On-line chain partitioning of semi-orders, Ph.D. thesis, Jagiellonian University, 2008.

    Google Scholar 

  20. [20]

    W. T. Trotter: Partially ordered sets, Handbook of combinatorics, 1, Elsevier, Amsterdam, 1995, 433–480.

    Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

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.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

Mathematics Subject Classication (2000)

  • 68W27
  • 05C57
  • 06A07