On the Gap Between Hereditary Discrepancy and the Determinant Lower Bound

Abstract

SIAM Journal on Discrete Mathematics, Volume 38, Issue 2, Page 1222-1238, June 2024.
Abstract. The determinant lower bound of Lovász, Spencer, and Vesztergombi [European J. Combin., 7 (1986), pp. 151–160] is a general way to prove lower bounds on the hereditary discrepancy of a set system. In their paper, Lovász, Spencer, and Vesztergombi asked if hereditary discrepancy can also be bounded from above by a function of the determinant lower bound. This was answered in the negative by Hoffman, and the largest known multiplicative gap between the two quantities for a set system of [math] subsets of a universe of size [math] is on the order of [math]. On the other hand, building upon work of Matoušek [Proc. Amer. Math. Soc., 141 (2013), pp. 451–460], Jiang and Reis [in Proceedings of the Symposium on Simplicity in Algorithms (SOSA), SIAM, Philadelphia, 2022, pp. 308–313] showed that this gap is always bounded up to constants by [math]. This is tight when [math] is polynomial in [math] but leaves open the case of large [math]. We show that the bound of Jiang and Reis is tight for nearly the entire range of [math]. Our proof amplifies the discrepancy lower bounds of a set system derived from the discrete Haar basis via Kronecker products.