On the Gap between Hereditary Discrepancy and the Determinant Lower Bound
Lily Li, Aleksandar Nikolov
TL;DR
This paper sharpens the understanding of the gap between hereditary discrepancy and the determinant lower bound for set systems. It proves that, for m in the range [n, 2^{n^{1−ε}}], there exist incidence matrices A with the ratio of hereditery discrepancy to the determinant lower bound reaching at least a constant multiple of √(log m · log n), up to constants, by amplifying discrepancy through Kronecker products with a carefully chosen Haar-basis matrix A_k that has detlb(A_k)=O(1) and disc1(A_k) ≳ √k. Conversely, it shows a universal upper bound herdisc(A)/detlb(A) ≲ √n for all A via a deep connection between volume bounds and discrepancy, leveraging volLB, volLB^*, and recent partial-coloring results, plus a simpler VC-dimension-based argument for incidence matrices. Together, these results establish nearly tight separations across a broad spectrum of m, and introduce amplification techniques and Haar-basis constructions that may influence constructive discrepancy bounds and related combinatorial encoding problems.
Abstract
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 $\max\{\log n, \sqrt{\log m}\}$. 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 $\sqrt{\log(m)\log(n)}$. 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.