The determinant bound for discrepancy is almost tight

Author:
Jiří Matoušek

Journal:
Proc. Amer. Math. Soc. **141** (2013), 451-460

MSC (2010):
Primary 05D99

Published electronically:
June 18, 2012

MathSciNet review:
2996949

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Abstract | References | Similar Articles | Additional Information

Abstract: In 1986 Lovász, Spencer, and Vesztergombi proved a lower bound for the hereditary discrepancy of a set system in terms of determinants of square submatrices of the incidence matrix of . As shown by an example of Hoffman, this bound can differ from by a multiplicative factor of order almost , where is the size of the ground set of . We prove that it never differs by more than , assuming bounded by a polynomial in . We also prove that if such an is the union of systems , each of hereditary discrepancy at most , then . For , this almost answers a question of Sós. The proof is based on a recent algorithmic result of Bansal, which computes low-discrepancy colorings using semidefinite programming.

**[Ban10]**N. Bansal.

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Additional Information

**Jiří Matoušek**

Affiliation:
Department of Applied Mathematics and Institute of Theoretical Computer Science (ITI), Charles University, Malostranské nám. 25, 118 00 Praha 1, Czech Republic

DOI:
http://dx.doi.org/10.1090/S0002-9939-2012-11334-6

Received by editor(s):
January 4, 2011

Received by editor(s) in revised form:
July 2, 2011

Published electronically:
June 18, 2012

Additional Notes:
The author was partially supported by the ERC Advanced Grant No. 267165.

Communicated by:
Jim Haglund

Article copyright:
© Copyright 2012
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.