# A Fourier-Analytic Approach for the Discrepancy of Random Set Systems

@inproceedings{Hoberg2019AFA,
title={A Fourier-Analytic Approach for the Discrepancy of Random Set Systems},
author={Rebecca Hoberg and Thomas Rothvoss},
booktitle={SODA},
year={2019}
}
• Published in SODA 1 June 2018
• Mathematics, Computer Science
One of the prominent open problems in combinatorics is the discrepancy of set systems where each element lies in at most $t$ sets. The Beck-Fiala conjecture suggests that the right bound is $O(\sqrt{t})$, but for three decades the only known bound not depending on the size of the set system has been $O(t)$. Arguably we currently lack techniques for breaking that barrier. In this paper we introduce discrepancy bounds based on Fourier analysis. We demonstrate our method on random set systems…
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## References

SHOWING 1-10 OF 16 REFERENCES
On the Beck‐Fiala conjecture for random set systems
• Mathematics
Electron. Colloquium Comput. Complex.
• 2015
The first bound combines the Lov{\'a}sz Local Lemma with a new argument based on partial matchings; the second follows from an analysis of the lattice spanned by sparse vectors.
Constructive Discrepancy Minimization for Convex Sets
• T. Rothvoss
• Mathematics
2014 IEEE 55th Annual Symposium on Foundations of Computer Science
• 2014
It is shown that for any symmetric convex set K with measure at least e-n/500, the following algorithm finds a point y ∈ K ∩ [-1, 1]n with Ω(n) coordinates in ±1: (1) take a random Gaussian vector x, (2) compute the point y in K∩ [- 1, 1)n that is closest to x.
Roth’s estimate of the discrepancy of integer sequences is nearly sharp
It is proved that R(N)=N1/4+o(1) thus showing that Roth’s original lower bound was essentially best possible, and the notion ofdiscrepancy of hypergraphs is introduced and derive an upper bound from which the above result follows.
Constructive Discrepancy Minimization by Walking on the Edges
• Mathematics
2012 IEEE 53rd Annual Symposium on Foundations of Computer Science
• 2012
A new randomized algorithm to find a coloring as in Spencer's result based on a restricted random walk which is “truly” constructive in that it does not appeal to the existential arguments, giving a new proof of Spencer's theorem and the partial coloring lemma.
On the discrepancy of random matrices with many columns
• Mathematics, Computer Science
Random Struct. Algorithms
• 2020
It is proved that for n at least polynomial in m, with high probability the ℓ∞‐discrepancy is at most twice theℓ ∞‐covering radius of the integer span of the support of the random variable.
Six standard deviations suffice
Given n sets on n elements it is shown that there exists a two-coloring such that all sets have discrepancy at most Knl/2, K an absolute constant. This improves the basic probabilistic method with
Constructive Algorithms for Discrepancy Minimization
• N. Bansal
• Mathematics, Computer Science
2010 IEEE 51st Annual Symposium on Foundations of Computer Science
• 2010
The main idea in the algorithms is to produce a coloring over time by letting the color of the elements perform a random walk (with tiny increments) starting from 0 until they reach $\pm 1$.
Phase transition and finite-size scaling for the integer partitioning problem
• Mathematics
Random Struct. Algorithms
• 2001
This paper considers the problem of partitioning n randomly chosen integers between 1 and 2 m into two subsets such that the discrepancy, the absolute value of the difference of their sums, is minimized and proves that with high probability the optimum partition is unique, and that the optimum discrepancy is bounded.
An Algorithm for Komlós Conjecture Matching Banaszczyk's Bound
• Computer Science, Mathematics
2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)
• 2016
An efficient algorithm is given that finds a coloring with discrepancy O((t log n)1/2), matching the best known non-constructive bound for the problem due to Banaszczyk, and gives an algorithmic O(log 1/2 n) bound.
The Gram-Schmidt walk: a cure for the Banaszczyk blues
• Mathematics
STOC
• 2018
This paper gives an efficient randomized algorithm to find a ± 1 combination of the vectors which lies in cK for c>0 an absolute constant, which leads to new efficient algorithms for several problems in discrepancy theory.