On the Gap between Hereditary Discrepancy and the Determinant Lower Bound
SIAM Journal on Discrete Mathematics(2023)
摘要
The determinant lower bound of Lovasz, Spencer, and Vesztergombi [European
Journal of Combinatorics, 1986] is a powerful general way to prove lower bounds
on the hereditary discrepancy of a set system. In their paper, Lovasz, Spencer,
and Vesztergombi asked if hereditary discrepancy can also be bounded from above
by a function of the hereditary discrepancy. This was answered in the negative
by Hoffman, and the largest known multiplicative gap between the two quantities
for a set system of m substes of a universe of size n is on the order of
max{log n, √(log m)}. On the other hand, building on work of
Matoušek [Proceedings of the AMS, 2013], recently Jiang and Reis [SOSA,
2022] showed that this gap is always bounded up to constants by
√(log(m)log(n)). This is tight when m is polynomial in n, but
leaves open what happens for large m. We show that the bound of Jiang and
Reis is tight for nearly the entire range of m. Our proof relies on a
technique of amplifying discrepancy via taking Kronecker products, and on
discrepancy lower bounds for a set system derived from the discrete Haar basis.
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