ECCC-Report TR15-190https://eccc.weizmann.ac.il/report/2015/190Comments and Revisions published for TR15-190en-usFri, 27 Nov 2015 15:38:15 +0200
Paper TR15-190
| On the Beck-Fiala Conjecture for Random Set Systems |
Shachar Lovett,
Esther Ezra
https://eccc.weizmann.ac.il/report/2015/190Motivated by the Beck-Fiala conjecture, we study discrepancy bounds for random sparse set systems. Concretely, these are set systems $(X,\Sigma)$, where each element $x \in X$ lies in $t$ randomly selected sets of $\Sigma$, where $t$ is an integer parameter. We provide new bounds in two regimes of parameters. We show that when $|\Sigma| \ge |X|$ the hereditary discrepancy of $(X,\Sigma)$ is with high probability $O(\sqrt{t \log t})$; and when $|X| \gg |\Sigma|^t$ the hereditary discrepancy of $(X,\Sigma)$ is with high probability $O(1)$. The first bound combines the Lov{\'a}sz Local Lemma with a new argument based on partial matchings; the second follows from an analysis of the lattice spanned by sparse vectors.
Fri, 27 Nov 2015 15:38:15 +0200https://eccc.weizmann.ac.il/report/2015/190