The determinant bound for discrepancy is almost tight

Abstract

In 1986 Lovász, Spencer, and Vesztergombi proved a lower bound for the hereditary discrepancy of a set system F \mathcal {F} in terms of determinants of square submatrices of the incidence matrix of F \mathcal {F} . As shown by an example of Hoffman, this bound can differ from h e r d i s c ( F ) \mathrm {herdisc}(\mathcal {F}) by a multiplicative factor of order almost log ⁡ n \log n , where n n is the size of the ground set of F \mathcal {F} . We prove that it never differs by more than O ( ( log ⁡ n ) 3 / 2 ) O((\log n)^{3/2}) , assuming | F | |\mathcal {F}| bounded by a polynomial in  n n . We also prove that if such an F \mathcal {F} is the union of t t systems F 1 , … , F t \mathcal {F}_1,\ldots ,\mathcal {F}_t , each of hereditary discrepancy at most D D , then h e r d i s c ( F ) ≤ O ( t ( log ⁡ n ) 3 / 2 D ) \mathrm {herdisc}(\mathcal {F})\le O(\sqrt t (\log n)^{3/2}D) . For t = 2 t=2 , 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.