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• W1550105891 abstract "Let (X, S) be a set system on an n-point set X. The discrepancy of S is defined as the minimum of the largest deviation from an even split, over all subsets of S e S and two-colorings χ on X. We consider the scenario where, for any subset X' ⊆ X of size m ≤ n and for any parameter 1 ≤ k ≤ m, the number of restrictions of the sets of S to X' of size at most k is only O(md1kd-d1), for fixed integers d > 0 and 1 ≤ d1 ≤ d (this generalizes the standard notion of bounded primal shatter dimension when d1 = d). In this case we show that there exists a coloring χ with discrepancy bound O*(|S|1/2-d1/(2d)n(d1-1)/(2d)), for each S e S, where O*(·) hides a polylogarithmic factor in n. This bound is tight up to a polylogarithmic factor [21, 23] and the corresponding coloring χ can be computed in expected polynomial time using the very recent machinery of Lovett and Meka for constructive discrepancy minimization [20]. Our bound improves and generalizes the bounds obtained from the machinery of Har-Peled and Sharir [15] (and the follow-up work in [27]) for points and halfspaces in d-space for d ≥ 3. Last but not least, we show that our bound yields improved bounds for the size of relative (e, δ)-approximations for set systems of the above kind." @default.
• W1550105891 created "2016-06-24" @default.
• W1550105891 creator A5011840036 @default.
• W1550105891 date "2014-01-05" @default.
• W1550105891 modified "2023-09-26" @default.
• W1550105891 title "A size-sensitive discrepancy bound for set systems of bounded primal shatter dimension" @default.
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• W1550105891 doi "https://doi.org/10.5555/2634074.2634175" @default.
• W1550105891 hasPublicationYear "2014" @default.
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