A Tighter Relation Between Hereditary Discrepancy and Determinant Lower Bound
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In seminal work, Lovász, Spencer, and Vesztergombi [European J. Combin., 1986] proved a lower bound for the hereditary discrepancy of a matrix A ∈ℝ^m × n in terms of the maximum |(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(ℱ_1 ∪ℱ_2) ≤ O(√(log (m) ·log (n))) ·max(herdisc(ℱ_1), herdisc(ℱ_2)), for any two set systems ℱ_1, ℱ_2 over [n] satisfying |ℱ_1 ∪ℱ_2| = 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].
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