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.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Alon, N., Spencer, J., Erdös, P.: The Probabilistic Method. John Wiley & Sons, Inc., Chichester (1992)
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)
Beck, J., Fiala, T.: Integer making Theorems. Discrete Applied Mathematics 3, 1–8 (1981)
Barany, I., Grunberg, V.S.: On some combinatorial questions in finite dimensional spaces. Linear Algebra Appl. 41, 1–9 (1981)
Beck, J., Sós, V.: Discrepancy Theory. In: Graham, R., Grötschel, M., Lovász, L. (eds.) Handbook of Combinatorics, ch. 26 (1995)
Beck, J., Spencer, J.: Integral approximation sequences. Math. Programming 30, 88–98 (1984)
Doerr, B.: Linear and Hereditary Discrepancy. accepted for publication in Combinatorics, Probability and Computing (1999)
Lovász, L., Spencer, J., Vesztergombi, K.: Discrepancies of set systems and matrices. European J. Combin. 7, 151–160 (1986)
Spencer, J.: Six Standard Deviation Suffice. Trans. Amer. Math. Soc. 289, 679–706 (1985)
Spencer, J.: Ten Lectures on the Probabilistic Method. SIAM, Philadelphia (1987)
Matoušek, J., Welzl, E., Wernisch, L.: Discrepancy and approximations for bounded VC-Dimension. Combinatorica 13, 455–466 (1984)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1999 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Doerr, B., Srivastav, A. (1999). Approximation of Multi-Color Discrepancy. In: Hochbaum, D.S., Jansen, K., Rolim, J.D.P., Sinclair, A. (eds) Randomization, Approximation, and Combinatorial Optimization. Algorithms and Techniques. RANDOM APPROX 1999 1999. Lecture Notes in Computer Science, vol 1671. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-48413-4_5
Download citation
DOI: https://doi.org/10.1007/978-3-540-48413-4_5
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-66329-4
Online ISBN: 978-3-540-48413-4
eBook Packages: Springer Book Archive