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The ε-t-Net Problem

Authors Noga Alon, Bruno Jartoux , Chaya Keller, Shakhar Smorodinsky , Yelena Yuditsky

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ACM Subject Classification
  • Mathematics of computing → Hypergraphs
  • Theory of computation → Computational geometry
Keywords
  • epsilon-nets
  • geometric hypergraphs
  • VC-dimension
  • linear union complexity

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Abstract

We study a natural generalization of the classical ε-net problem (Haussler - Welzl 1987), which we call the ε-t-net problem: Given a hypergraph on n vertices and parameters t and ε ≥ t/n, find a minimum-sized family S of t-element subsets of vertices such that each hyperedge of size at least ε n contains a set in S. When t=1, this corresponds to the ε-net problem.
We prove that any sufficiently large hypergraph with VC-dimension d admits an ε-t-net of size O((1+log t)d/ε log 1/ε). For some families of geometrically-defined hypergraphs (such as the dual hypergraph of regions with linear union complexity), we prove the existence of O(1/ε)-sized ε-t-nets. 
We also present an explicit construction of ε-t-nets (including ε-nets) for hypergraphs with bounded VC-dimension. In comparison to previous constructions for the special case of ε-nets (i.e., for t=1), it does not rely on advanced derandomization techniques. To this end we introduce a variant of the notion of VC-dimension which is of independent interest.

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Noga Alon, Bruno Jartoux, Chaya Keller, Shakhar Smorodinsky, and Yelena Yuditsky. The ε-t-Net Problem. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 5:1-5:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.SoCG.2020.5

Author Details

Noga Alon
  • Princeton University, NJ, USA
  • Tel Aviv University, Israel
Bruno Jartoux
  • Ben-Gurion University of the Negev, Be'er-Sheva, Israel
Chaya Keller
  • Ariel University, Ariel, Israel
Shakhar Smorodinsky
  • Ben-Gurion University of the Negev, Be'er-Sheva, Israel
Yelena Yuditsky
  • Ben-Gurion University of the Negev, Be'er-Sheva, Israel

Funding

  • Alon, Noga: Research supported in part by NSF grant DMS-1855464, ISF grant 281/17, GIF grant G-1347-304.6/2016, and the Simons Foundation.
  • Jartoux, Bruno: Research supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant agreement No. 678765) and by Grant 635/16 from the Israel Science Foundation.
  • Keller, Chaya: Part of the research was done when the author was at the Technion, Israel, and was supported by Grant 409/16 from the Israel Science Foundation.
  • Smorodinsky, Shakhar: Research partially supported by Grant 635/16 from the Israel Science Foundation.
  • Yuditsky, Yelena: Research supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant agreement No. 678765) and by Grant 635/16 from the Israel Science Foundation.

Acknowledgements

The authors are grateful to Adi Shamir for fruitful suggestions regarding the application of ε-t-nets to secret sharing.

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