A size-sensitive discrepancy bound for set systems of bounded primal shatter dimension

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Abstract

Let (X, δ) be a set system on an n-point set X. The discrepancy of δ is defined as the minimum of the largest deviation from an even split, over all subsets of δ ∈ δ 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 δ to X' of size at most k is only 0(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(d 1-1)(2d), for each S ∈ δ, 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 (ε, δ)- Approximations for set systems of the above kind.

Original languageEnglish
Title of host publicationProceedings of the 25th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014
PublisherAssociation for Computing Machinery
Pages1378-1388
Number of pages11
ISBN (Print)9781611973389
DOIs
StatePublished - 2014
Externally publishedYes
Event25th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014 - Portland, OR, United States
Duration: 5 Jan 20147 Jan 2014

Publication series

NameProceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms

Conference

Conference25th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014
Country/TerritoryUnited States
CityPortland, OR
Period5/01/147/01/14

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