# On the discrepancy of random low degree set systems

@inproceedings{Bansal2019OnTD, title={On the discrepancy of random low degree set systems}, author={N. Bansal and Raghu Meka}, booktitle={SODA}, year={2019} }

htmlabstractMotivated by the celebrated Beck-Fiala conjecture, we consider the random setting where there are n elements and m sets and each element lies in t randomly chosen sets. [...] Key Method First, applying the partial coloring method to the case when n=mlogO(1)m and using the properties of the random set system we show that the overall discrepancy incurred is at most O(√t). Second, we reduce the general case to that of n≤mlogO(1)m using LP duality and a careful counting argument. Expand

#### Topics from this paper

#### 8 Citations

The Phase Transition of Discrepancy in Random Hypergraphs

- Mathematics, Computer Science
- ArXiv
- 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. Expand

Improved Algorithms for Combinatorial Discrepancy

- 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].… Expand

On the discrepancy of random matrices with many columns

- Mathematics, Computer Science
- Random Struct. Algorithms
- 2020

It is proved that the failure probability is inverse polynomial in $m, n$ and some well-motivated parameters of the random variable, and the analogous bounds for the discrepancy in arbitrary norms are obtained. Expand

The Discrepancy of Random Rectangular Matrices

- Computer Science, Mathematics
- ArXiv
- 2021

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. Expand

Online Geometric Discrepancy for Stochastic Arrivals with Applications to Envy Minimization

- Mathematics, Computer Science
- ArXiv
- 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. Expand

A spectral bound on hypergraph discrepancy

- Mathematics, Computer Science
- ICALP
- 2020

The discrepancy of a random $t$-regular hypergraph with n vertices and m edges is almost surely $O(\sqrt{t} + \lambda)$ as $n$ grows. Expand

Balancing Gaussian vectors in high dimension

- Mathematics, Computer Science
- COLT
- 2020

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

MAX CUT in Weighted Random Intersection Graphs and Discrepancy of Sparse Random Set Systems

- Computer Science, Mathematics
- ArXiv
- 2020

The average case analysis of Weighted Max Cut is studied, assuming the input is a weighted random intersection graph, and a (weak) bipartization algorithm is proposed for the case of m=n, p=\frac{\Theta(1)}{n}$ that can be used to find a 2-coloring with minimum discrepancy in $\Sigma$. Expand

#### References

SHOWING 1-10 OF 19 REFERENCES

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

- Computer Science, Mathematics
- SODA
- 2019

This paper introduces discrepancy bounds based on Fourier analysis on random set systems and proves that in the regime of $n \geq \Theta(m^2\log(m)$ the discrepancy is at most $1$ with high probability. Expand

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… Expand

Constructive Discrepancy Minimization by Walking on the Edges

- Computer Science, Mathematics
- FOCS
- 2012

A new randomized algorithm to find a coloring as in Spencer's result based on a restricted random walk called Edge-Walk is given, giving a new proof of Spencer's theorem and the {\sl partial coloring lemma]. Expand

An Algorithm for Komlós Conjecture Matching Banaszczyk's Bound

- Mathematics, Computer Science
- SIAM J. Comput.
- 2019

An efficient algorithm is given that finds a coloring with discrepancy O((t log n)1/2), matching the best known nonconstructive bound for the problem due to Banaszczyk, and gives an algorithmic O(log 1/2n) bound. Expand

On the Beck-Fiala Conjecture for Random Set Systems

- Computer Science, 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. Expand

Discrepancy of Set-systems and Matrices

- Computer Science, Mathematics
- Eur. J. Comb.
- 1986

Various mathematically more tractable variants of the discrepancy of a set-system are introduced, and the notion is extended to a general matrix, and formed as a problem of covering the unit cube by convex bodies. Expand

Deterministic Discrepancy Minimization via the Multiplicative Weight Update Method

- Mathematics, Computer Science
- IPCO
- 2017

A well-known theorem of Spencer shows that any set system with n sets over n elements admits a coloring of discrepancy \(O(\sqrt{n})\). While the original proof was non-constructive, recent progress… Expand

Geometric Discrepancy: An Illustrated Guide

- Mathematics
- 2009

1. Introduction 1.1 Discrepancy for Rectangles and Uniform Distribution 1.2 Geometric Discrepancy in a More General Setting 1.3 Combinatorial Discrepancy 1.4 On Applications and Connections 2.… Expand

"Integer-making" theorems

- Computer Science, Mathematics
- Discret. Appl. Math.
- 1981

It is proved that given any family F of subsets of X having maximum degree n [cardinality n], one can find integers αi, i=1,2,… so that f(n) − 1 and g (n)≤c(n log n) 1 2 are proved. Expand

A Panorama of Discrepancy Theory

- Mathematics
- 2014

Preface.- Classical and Geometric Discrepancy.- Upper Bounds in Classical Discrepancy Theory.- Roth's Orthogonal Function Method in Discrepancy Theory and Some New Connections.- Irregularities of… Expand