A Tighter Relation Between Hereditary Discrepancy and Determinant Lower Bound

Abstract

In seminal work, Lovász, Spencer, and Vesztergombi [European J. Combin., 1986] proved a lower bound for the hereditary discrepancy of a matrix A ∈ Rm×n in terms of the maximum |det(B)|1/k over all k × k submatrices B of A. We show algorithmically that this determinant lower bound can be off by at most a factor of O( √ log(m) · log(n)), improving over the previous bound of O(log(mn)· √ log(n)) given by Matoušek [Proc. of the AMS, 2013]. Our result immediately implies herdisc(F1 ∪ F2) ≤ O( √ log(m) · log(n)) · max(herdisc(F1), herdisc(F2)), for any two set systems F1,F2 over [n] satisfying |F1 ∪ F2| = m. Our bounds are tight up to constants when m = O(poly(n)) due to a construction of Pálvölgyi [Discrete Comput. Geom., 2010] or the counterexample to Beck’s three permutation conjecture by Newman, Neiman and Nikolov [FOCS, 2012]. University of Washington, Seattle, USA. jhtdavid@cs.washington.edu. University of Washington, Seattle, USA. voreis@cs.washington.edu.