In seminal work, Lov\'asz, Spencer, and Vesztergombi [European J. Combin., 1986] proved a lower bound for the hereditary discrepancy of a matrix $A \in \mathbb{R}^{m \times n}$ in terms of the maximum $|\det(B)|^{1/k}$ over all $k \times k$ submatrices $B$ of $A$. We show algorithmically that this determinant lower bound can be off by at most a factor of $O(\sqrt{\log (m) \cdot \log (n)})$, improving over the previous bound of $O(\log(mn) \cdot \sqrt{\log (n)})$ given by Matou\v{s}ek [Proc. of the AMS, 2013]. Our result immediately implies $\mathrm{herdisc}(\mathcal{F}_1 \cup \mathcal{F}_2) \leq O(\sqrt{\log (m) \cdot \log (n)}) \cdot \max(\mathrm{herdisc}(\mathcal{F}_1), \mathrm{herdisc}(\mathcal{F}_2))$, for any two set systems $\mathcal{F}_1, \mathcal{F}_2$ over $[n]$ satisfying $|\mathcal{F}_1 \cup \mathcal{F}_2| = m$. Our bounds are tight up to constants when $m = O(\mathrm{poly}(n))$ due to a construction of P\'alv\"olgyi [Discrete Comput. Geom., 2010] or the counterexample to Beck's three permutation conjecture by Newman, Neiman and Nikolov [FOCS, 2012].
翻译:在初始工作中, Lov\'asz, Spencer, 和 Vesztergombi [European J. 组合, 1986] 证明了一个基质($A\ in\mathbb{R ⁇ m\time n}$) 的遗传差异的下限, 最大值為$det( B)\\\\ k}$ ktime, $B$。 我们从逻辑上显示, 这个基点下限最多可以由 $( sqrt) (m) (xxxxxxxxxxal) (m) (xxxxxxxxxxxxxal) (xxxxxxxxxx) (xxxxxxxxxxxxxxxal) (xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx