The determinant bound for discrepancy is almost tight

Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F13%3A10172783" target="_blank" >RIV/00216208:11320/13:10172783 - isvavai.cz</a>
Result on the web
<a href="http://arxiv.org/abs/1101.0767" target="_blank" >http://arxiv.org/abs/1101.0767</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1090/S0002-9939-2012-11334-6" target="_blank" >10.1090/S0002-9939-2012-11334-6</a>

Result language
angličtina
Original language name
The determinant bound for discrepancy is almost tight
Original language description
In 1986 Lovasz, Spencer, and Vesztergombi proved a lower bound for the hereditary a discrepancy of a set system F in terms of determinants of square submatrices of the incidence matrix of F. As shown by an example of Hoffman, this bound can differ from herdisc(F) by a multiplicative factor of order almost log n, where n is the size of the ground set of F. We prove that it never differs by more than O((log n)^3/2), assuming |F| bounded by a polynomial in n. We also prove that if such an F is the union oft systems F_1, . . ., F_t, each of hereditary discrepancy at most D, then herdisc(F) leq O(t^(1/2)(log n)^(3/2) D). For t = 2, this almost answers a question of Sos. The proof is based on a recent algorithmic result of Bansal, which computes low-discrepancy colorings using semidefinite programming.
Czech name
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Czech description
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Type
J<sub>x</sub> - Unclassified - Peer-reviewed scientific article (Jimp, Jsc and Jost)
CEP classification
BA - General mathematics
OECD FORD branch
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Project
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Continuities
I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Publication year
2013
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

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