On the discrepancy of random low degree set systems

10/08/2018

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by   Nikhil Bansal, et al.

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Motivated by the celebrated Beck-Fiala conjecture, we consider the random setting where there are n elements and m sets and each element lies in t randomly chosen sets. In this setting, Ezra and Lovett showed an O((t t)^1/2) discrepancy bound in the regime when n ≤ m and an O(1) bound when n ≫ m^t. In this paper, we give a tight O(√(t)) bound for the entire range of n and m, under a mild assumption that t = Ω ( m)^2. The result is based on two steps. First, applying the partial coloring method to the case when n = m ^O(1) m and using the properties of the random set system we show that the overall discrepancy incurred is at most O(√(t)). Second, we reduce the general case to that of n ≤ m ^O(1)m using LP duality and a careful counting argument.