# 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} }

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…

## 17 Citations

On the discrepancy of random low degree set systems

- MathematicsSODA
- 2019

A tight bound is given for the entire range of n and m, under a mild assumption that t = Ω(log log m)2 and the overall discrepancy incurred is at most .

Balancing Gaussian vectors in high dimension

- Computer Science, MathematicsCOLT
- 2020

A randomized polynomial-time algorithm is presented that achieves discrepancy $e^{-\Omega(\log^2(n)/m)}$ with high probability, provided that $m \leq O(\sqrt{\log{n}})$.

Improved Algorithms for Combinatorial Discrepancy

- Mathematics
- 2020

Discrepancy theory is a subfield of combinatorics which has branched in Computer Science due to several connections it has to geometric problems, randomized algorithms and complexity theory [13, 8].…

Online Geometric Discrepancy for Stochastic Arrivals with Applications to Envy Minimization

- MathematicsArXiv
- 2019

It is shown that the discrepancy of the above problem is sub-polynomial in $n$ and that no algorithm can achieve a constant discrepancy, and a natural generalization of this problem to $2-dimensions where the points arrive uniformly at random in a unit square is obtained.

Smoothed Analysis of the Koml\'os Conjecture

- Mathematics, Computer Science
- 2022

The well-known Koml´os conjecture states that given n vectors in R d with Euclidean norm at most one, there always exists a ± 1 coloring such that the ℓ ∞ norm of the signed-sum vector is a constant…

The Phase Transition of Discrepancy in Random Hypergraphs

- MathematicsArXiv
- 2021

This work applies the partial colouring lemma of Lovett and Meka to show that w.h.p. has discrepancy O( √ dn/m log(m/n), and characterizes how the discrepancy of each random hypergraph model transitions from â‚ d to o(√ d) as m varies from m = Θ(n) to m n.

Discrepancy in random hypergraph models

- MathematicsArXiv
- 2018

This work studies hypergraph discrepancy in two closely related random models of hypergraphs on $n$ vertices and $m$ hyperedges, and shows that the discrepancy is almost surely at most $1".

The discrepancy of random rectangular matrices

- MathematicsRandom Struct. Algorithms
- 2022

A complete answer to the Beck–Fiala conjecture is given for two natural models: matrices with Bernoulli or Poisson entries, and the discrepancy for any rectangular aspect ratio is characterized.

Prefix Discrepancy, Smoothed Analysis, and Combinatorial Vector Balancing

- MathematicsITCS
- 2022

A well-known result of Banaszczyk in discrepancy theory concerns the preﬁx discrepancy problem (also known as the signed series problem): given a sequence of T unit vectors in R d , ﬁnd ± signs for…

On the discrepancy of random matrices with many columns

- Mathematics, Computer ScienceRandom 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.

## References

SHOWING 1-10 OF 16 REFERENCES

On the Beck‐Fiala conjecture for random set systems

- MathematicsElectron. 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

- Mathematics2014 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

- MathematicsComb.
- 1981

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

- Mathematics2012 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 ScienceRandom 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

- Mathematics
- 1985

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

- Mathematics, Computer Science2010 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

- MathematicsRandom 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, Mathematics2016 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

- MathematicsSTOC
- 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.