The ε-t-Net Problem

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

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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


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

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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)


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.

Subject Classification

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


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  1. Pankaj K. Agarwal, János Pach, and Micha Sharir. State of the union (of geometric objects). In Proc. Joint Summer Research Conference on Discrete and Computational Geometry: 20 Years Later, Contemporary Mathematics 452, AMS, pages 9-48, 2008. Google Scholar
  2. Noga Alon, Graham Brightwell, H.A. Kierstead, A.V. Kostochka, and Peter Winkler. Dominating sets in k-majority tournaments. Journal of Combinatorial Theory, Series B, 96(3):374-387, 2006. URL:
  3. Noga Alon, Bruno Jartoux, Chaya Keller, Shakhar Smorodinsky, and Yelena Yuditsky. The Epsilon-t-Net Problem, 2020. URL:
  4. Boris Aronov, Anirudh Donakonda, Esther Ezra, and Rom Pinchasi. On pseudo-disk hypergraphs, 2018. URL:
  5. Boris Aronov, Esther Ezra, and Micha Sharir. Small-size ε-nets for axis-parallel rectangles and boxes. SIAM Journal on Computing, 39(7):3248-3282, 2010. URL:
  6. Sunil Arya, Guilherme Dias da Fonseca, and David M. Mount. Polytope approximation and the Mahler volume. In Yuval Rabani, editor, Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012, Kyoto, Japan, January 17-19, 2012, pages 29-42. SIAM, 2012. URL:
  7. Patrick Assouad. Densité et dimension. Annales de l'Institut Fourier, 33(3):233-282, 1983. URL:
  8. Amos Beimel. Secret-sharing schemes: A survey. In Yeow Meng Chee, Zhenbo Guo, San Ling, Fengjing Shao, Yuansheng Tang, Huaxiong Wang, and Chaoping Xing, editors, Coding and Cryptology - Third International Workshop, IWCC 2011, Qingdao, China, May 30-June 3, 2011. Proceedings, volume 6639 of Lecture Notes in Computer Science, pages 11-46. Springer, 2011. URL:
  9. Anselm Blumer, Andrzej Ehrenfeucht, David Haussler, and Manfred K. Warmuth. Learnability and the Vapnik-Chervonenkis dimension. J. ACM, 36(4):929-965, 1989. URL:
  10. Hervé Brönnimann, Bernard Chazelle, and Jiří Matoušek. Product range spaces, sensitive sampling, and derandomization. SIAM Journal on Computing, 28(5):1552-1575, 1999. URL:
  11. Chris Calabro. The Exponential Complexity of Satisfiability Problems. Phd thesis, University of California, San Diego, 2009. URL:
  12. Timothy M. Chan. Improved deterministic algorithms for linear programming in low dimensions. ACM Trans. Algorithms, 14(3):30:1-30:10, June 2018. URL:
  13. Bernard Chazelle and Jiří Matoušek. On linear-time deterministic algorithms for optimization problems in fixed dimension. Journal of Algorithms, 21(3):579-597, 1996. URL:
  14. Kenneth L. Clarkson and Peter W. Shor. Application of random sampling in computational geometry, II. Discrete & Computational Geometry, 4:387-421, 1989. URL:
  15. Richard M. Dudley. Notes on empirical processes. Lecture notes, second printing, 2000. Google Scholar
  16. Kunal Dutta, Arijit Ghosh, Bruno Jartoux, and Nabil H. Mustafa. Shallow packings, semialgebraic set systems, Macbeath regions, and polynomial partitioning. Discrete & Computational Geometry, 61(4):756-777, 2019. URL:
  17. Nicolas Grelier, Saeed Gh. Ilchi, Tillmann Miltzow, and Shakhar Smorodinsky. On the VC-dimension of convex sets and half-spaces, 2019. URL:
  18. David Haussler and Emo Welzl. Epsilon-nets and simplex range queries. Discrete & Computational Geometry, 2:127-151, 1987. URL:
  19. Klara Kedem, Ron Livne, János Pach, and Micha Sharir. On the union of Jordan regions and collision-free translational motion amidst polygonal obstacles. Discrete & Computational Geometry, 1:59-71, 1986. URL:
  20. Chaya Keller and Shakhar Smorodinsky. Conflict-free coloring of intersection graphs of geometric objects. Discrete & Computational Geometry, June 2019. URL:
  21. Balázs Keszegh. Coloring intersection hypergraphs of pseudo-disks. Discrete & Computational Geometry, October 2019. URL:
  22. János Komlós, János Pach, and Gerhard Woeginger. Almost tight bounds for ε-nets. Discrete & Computational Geometry, 7(2):163-173, February 1992. URL:
  23. Chung L. Liu. Introduction to Combinatorial Mathematics. McGraw-Hill, New York, 1968. Google Scholar
  24. Jiří Matoušek. Approximations and optimal geometric divide-and-conquer. Journal of Computer and System Sciences, 50(2):203-208, 1995. URL:
  25. Jiří Matoušek. Geometric Discrepancy: An Illustrated Guide. Number 18 in Algorithms and Combinatorics. Springer, Berlin, New York, 1999. URL:
  26. Jiří Matoušek. Lectures on Discrete Geometry. Springer-Verlag New York, Inc., Secaucus, NJ, USA, 2002. URL:
  27. Nabil H. Mustafa and Saurabh Ray. ε-Mnets: Hitting geometric set systems with subsets. Discrete & Computational Geometry, 57(3):625-640, 2017. URL:
  28. Nabil H. Mustafa and Kasturi Varadarajan. Epsilon-approximations and epsilon-nets. In Jacob E. Goodman, Joseph O'Rourke, and Csaba D. Tóth, editors, Handbook of Discrete and Computational Geometry, 3rd Edition, pages 1241-1267. CRC Press, 2018. Google Scholar
  29. János Pach and Gábor Tardos. Tight lower bounds for the size of epsilon-nets. J. Amer. Math. Soc., 26(3):645-658, 2013. URL:
  30. Evangelia Pyrga and Saurabh Ray. New existence proofs for ε-nets. In Proceedings of the Twenty-fourth Annual Symposium on Computational Geometry, SCG '08, pages 199-207, New York, NY, USA, 2008. ACM. URL:
  31. Yuval Rabani and Amir Shpilka. Explicit construction of a small ε-net for linear threshold functions. SIAM Journal on Computing, 39(8):3501-3520, 2010. URL:
  32. Rajiv Raman and Saurabh Ray. Planar support for non-piercing regions and applications. In Yossi Azar, Hannah Bast, and Grzegorz Herman, editors, 26th Annual European Symposium on Algorithms, ESA 2018, August 20-22, 2018, Helsinki, Finland, volume 112 of LIPIcs, pages 69:1-69:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018. URL:
  33. Norbert Sauer. On the density of families of sets. Journal of Combinatorial Theory, Series A, 13(1):145-147, 1972. URL:
  34. Adi Shamir. How to share a secret. Communications of the ACM, 22(11):612-613, 1979. URL:
  35. Saharon Shelah. A combinatorial problem; stability and order for models and theories in infinitary languages. Pacific J. Math., 41(1):247-261, 1972. URL:
  36. Vladimir N. Vapnik and Alexei Ya. Chervonenkis. On the uniform convergence of relative frequencies of events to their probabilities. Theory of Probability and its Applications, 16(2):264-280, 1971. Google Scholar
  37. Emo Welzl. Partition trees for triangle counting and other range searching problems. In Proceedings of the Fourth Annual Symposium on Computational Geometry (Urbana, IL, 1988), pages 23-33. ACM, New York, 1988. URL:
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