Advertisement

Abstract

In this article we introduce (combinatorial) multi-color discrepancy and generalize some classical results from 2-color discrepancy theory to c colors. We give a recursive method that constructs c-colorings from approximations to the 2-color discrepancy. This method works for a large class of theorems like the six-standard-deviation theorem of Spencer, the Beck-Fiala theorem and the results of Matoušsek, Welzl and Wernisch for bounded VC-dimension. On the other hand there are examples showing that discrepancy in c colors can not be bounded in terms of two-color discrepancy even if c is a power of 2. For the linear discrepancy version of the Beck-Fiala theorem the recursive approach also fails. Here we extend the method of floating colors to multi-colorings and prove multi-color versions of the the Beck-Fiala theorem and the Barany-Grunberg theorem.

Keywords

Linear Discrepancy Weighted Discrepancy Color Discrepancy Discrepancy Theory Color Version 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [ASE]
    Alon, N., Spencer, J., Erdös, P.: The Probabilistic Method. John Wiley & Sons, Inc., Chichester (1992)zbMATHGoogle Scholar
  2. [BHK]
    Babai, L., Hayes, T.P., Kimmel, P.G.: The cost of the Missing Bit: Communication Complexity with Help. in: 30th STOC, pp. 673–682 (1998)Google Scholar
  3. [BF]
    Beck, J., Fiala, T.: Integer making Theorems. Discrete Applied Mathematics 3, 1–8 (1981)zbMATHCrossRefMathSciNetGoogle Scholar
  4. [BG]
    Barany, I., Grunberg, V.S.: On some combinatorial questions in finite dimensional spaces. Linear Algebra Appl. 41, 1–9 (1981)zbMATHCrossRefMathSciNetGoogle Scholar
  5. [BS_o]
    Beck, J., Sós, V.: Discrepancy Theory. In: Graham, R., Grötschel, M., Lovász, L. (eds.) Handbook of Combinatorics, ch. 26 (1995)Google Scholar
  6. [BSp]
    Beck, J., Spencer, J.: Integral approximation sequences. Math. Programming 30, 88–98 (1984)zbMATHCrossRefMathSciNetGoogle Scholar
  7. [D]
    Doerr, B.: Linear and Hereditary Discrepancy. accepted for publication in Combinatorics, Probability and Computing (1999)Google Scholar
  8. [LSV]
    Lovász, L., Spencer, J., Vesztergombi, K.: Discrepancies of set systems and matrices. European J. Combin. 7, 151–160 (1986)zbMATHMathSciNetGoogle Scholar
  9. [Sp1]
    Spencer, J.: Six Standard Deviation Suffice. Trans. Amer. Math. Soc. 289, 679–706 (1985)zbMATHCrossRefMathSciNetGoogle Scholar
  10. [Sp2]
    Spencer, J.: Ten Lectures on the Probabilistic Method. SIAM, Philadelphia (1987)zbMATHGoogle Scholar
  11. [MWW]
    Matoušek, J., Welzl, E., Wernisch, L.: Discrepancy and approximations for bounded VC-Dimension. Combinatorica 13, 455–466 (1984)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Benjamin Doerr
    • 1
  • Anand Srivastav
    • 1
  1. 1.Mathematisches SeminarChristian-Albrechts-Universität zu KielKielGermany

Personalised recommendations