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Algorithms for Discrepancy, Matchings, and Approximations: Fast, Simple, and Practical
We propose a randomized algorithm which, for any $d>0$ and $(X,\mathcal S)$ with dual shatter function $\pi^*(k)=O(k^d)$, returns a coloring with expected discrepancy $O\left({\sqrt{|X|^{1-1/d}\log|\mathcal S|}}\right)$ (this bound is tight) in time $\tilde O\left({|\mathcal S|\cdot|X|^{1/d}+|X|^{2+1/d}}\right)$, improving upon the previous-best time of $O\left(|\mathcal S|\cdot|X|^3\right)$ by at least a factor of $|X|^{2-1/d}$ when $|\mathcal S|\geq|X|$.
Optimal Approximations Made Easy
The fundamental result of Li, Long, and Srinivasan on approximations of set systems has become a key tool across several communities such as learning theory, algorithms, computational geometry, combinatorics and data analysis.
Optimal Bounds on the VC-dimension
The VC-dimension of a set system is a way to capture its complexity and has been a key parameter studied extensively in machine learning and geometry communities.
