On the Beck-Fiala Conjecture for Random Set Systems

Authors Esther Ezra, Shachar Lovett

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Esther Ezra
Shachar Lovett

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Esther Ezra and Shachar Lovett. On the Beck-Fiala Conjecture for Random Set Systems. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 60, pp. 29:1-29:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)


Motivated 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| >= |X| the hereditary discrepancy of (X,Sigma) is with high probability O(sqrt{t log t}); and when |X| >> |\Sigma|^t the hereditary discrepancy of (X,Sigma) is with high probability O(1). The first bound combines the Lovasz Local Lemma with a new argument based on partial matchings; the second follows from an analysis of the lattice spanned by sparse vectors.
  • Discrepancy theory
  • Beck-Fiala conjecture
  • Random set systems


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