On the Beck-Fiala conjecture for random set systems

Esther Ezra; Shachar Lovett

doi:10.1002/rsa.20810

On the Beck-Fiala conjecture for random set systems

Esther Ezra
, Shachar Lovett

Department of Computer Science and Artificial Intelligence

Georgia Institute of Technology
University of California at San Diego

Research output: Contribution to journal › Article › peer-review

13 Scopus citations

Abstract

Motivated by the Beck-Fiala conjecture, we study discrepancy bounds for random sparse set systems. Concretely, these are set systems (X,Σ), where each element x ∈ X lies in t randomly selected sets of Σ, where t is an integer parameter. We provide new bounds in two regimes of parameters. We show that when |Σ| ≥ |X| the hereditary discrepancy of (X,Σ) is with high probability (Formula presented.); and when |X| ≫ |Σ|^t the hereditary discrepancy of (X,Σ) is with high probability O(1). The first bound combines the Lovász Local Lemma with a new argument based on partial matchings; the second follows from an analysis of the lattice spanned by sparse vectors.

Original language	English
Pages (from-to)	665-675
Number of pages	11
Journal	Random Structures and Algorithms
Volume	54
Issue number	4
DOIs	https://doi.org/10.1002/rsa.20810
State	Published - Jul 2019

Bibliographical note

Publisher Copyright:
© 2018 Wiley Periodicals, Inc.

Funding

The authors wish to thank Aravind Srinivasan for presenting this problem during a discussion at the IMA Workshop on the Power of Randomness in Computation. Esther Ezra is supported by an NSF CAREER award CCF-15-53354 and by Grant 824/17 from the Israel Science Foundation. Shachar Lovett is supported by an NSF CAREER award 1350481, CCF award 1614023 and a Sloan fellowship. A preliminary version of this paper appeared in the 20th International Workshop on Randomization and Computation (RANDOM’2016). information: This research was supported by the NSF CAREER; CCF-15-53354, Israel Science Foundation; 824/17 The authors wish to thank Aravind Srinivasan for presenting this problem during a discussion at the IMA Workshop on the Power of Randomness in Computation. Esther Ezra is supported by an NSF CAREER award CCF-15-53354 and by Grant 824/17 from the Israel Science Foundation. Shachar Lovett is supported by an NSF CAREER award 1350481, CCF award 1614023 and a Sloan fellowship. A preliminary version of this paper appeared in the 20th International Workshop on Randomization and Computation (RANDOM'2016).

Funders	Funder number
NSF CAREER
National Science Foundation	824/17, CCF-15-53354
Israel Science Foundation	1614023, 1350481

Keywords

beck-fiala conjecture
discrepancy theory
random set systems

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10.1002/rsa.20810

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On the Beck-Fiala conjecture for random set systems

Abstract

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Keywords

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