On the beck-fiala conjecture for random set systems

Esther Ezra; Shachar Lovett

doi:10.4230/LIPIcs.APPROX-RANDOM.2016.59

On the beck-fiala conjecture for random set systems

Esther Ezra, Shachar Lovett

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

7 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 O(p t log t); 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
Title of host publication	Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 19th International Workshop, APPROX 2016 and 20th International Workshop, RANDOM 2016
Editors	Klaus Jansen, Claire Mathieu, Jose D. P. Rolim, Chris Umans
Publisher	Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)	9783959770187
DOIs	https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2016.59
State	Published - 1 Sep 2016
Externally published	Yes
Event	19th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2016 and the 20th International Workshop on Randomization and Computation, RANDOM 2016 - Paris, France Duration: 7 Sep 2016 → 9 Sep 2016

Publication series

Name	Leibniz International Proceedings in Informatics, LIPIcs
Volume	60
ISSN (Print)	1868-8969

Conference

Conference	19th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2016 and the 20th International Workshop on Randomization and Computation, RANDOM 2016
Country/Territory	France
City	Paris
Period	7/09/16 → 9/09/16

Bibliographical note

Funding Information:
E.E is supported by an NSF CAREER award 1553354. S.L. is supported by an NSF CAREER award 1350481 and a Sloan fellowship.

Keywords

Beck-Fiala conjecture
Discrepancy theory
Random set systems

Access to Document

10.4230/LIPIcs.APPROX-RANDOM.2016.59

Cite this

Ezra, E., & Lovett, S. (2016). On the beck-fiala conjecture for random set systems. In K. Jansen, C. Mathieu, J. D. P. Rolim, & C. Umans (Eds.), Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 19th International Workshop, APPROX 2016 and 20th International Workshop, RANDOM 2016 (Leibniz International Proceedings in Informatics, LIPIcs; Vol. 60). Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2016.59

Ezra, Esther ; Lovett, Shachar. / On the beck-fiala conjecture for random set systems. Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 19th International Workshop, APPROX 2016 and 20th International Workshop, RANDOM 2016. editor / Klaus Jansen ; Claire Mathieu ; Jose D. P. Rolim ; Chris Umans. Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 2016. (Leibniz International Proceedings in Informatics, LIPIcs).

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Ezra, E & Lovett, S 2016, On the beck-fiala conjecture for random set systems. in K Jansen, C Mathieu, JDP Rolim & C Umans (eds), Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 19th International Workshop, APPROX 2016 and 20th International Workshop, RANDOM 2016. Leibniz International Proceedings in Informatics, LIPIcs, vol. 60, Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 19th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2016 and the 20th International Workshop on Randomization and Computation, RANDOM 2016, Paris, France, 7/09/16. https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2016.59

On the beck-fiala conjecture for random set systems. / Ezra, Esther; Lovett, Shachar.
Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 19th International Workshop, APPROX 2016 and 20th International Workshop, RANDOM 2016. ed. / Klaus Jansen; Claire Mathieu; Jose D. P. Rolim; Chris Umans. Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 2016. (Leibniz International Proceedings in Informatics, LIPIcs; Vol. 60).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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Ezra E, Lovett S. On the beck-fiala conjecture for random set systems. In Jansen K, Mathieu C, Rolim JDP, Umans C, editors, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 19th International Workshop, APPROX 2016 and 20th International Workshop, RANDOM 2016. Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. 2016. (Leibniz International Proceedings in Informatics, LIPIcs). doi: 10.4230/LIPIcs.APPROX-RANDOM.2016.59

On the beck-fiala conjecture for random set systems

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