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Shallow Packings, Semialgebraic Set Systems, Macbeath Regions, and Polynomial Partitioning

  • Kunal Dutta
  • Arijit Ghosh
  • Bruno Jartoux
  • Nabil H. MustafaEmail author
Article

Abstract

Given a set system \((X, \mathcal {R})\) such that every pair of sets in \(\mathcal {R}\) have large symmetric difference, the Shallow Packing Lemma gives an upper bound on \(|\mathcal {R}|\) as a function of the shallow-cell complexity of \(\mathcal {R}\). In this paper, we first present a matching lower bound. Then we give our main theorem, an application of the Shallow Packing Lemma: given a semialgebraic set system \((X, \mathcal {R})\) with shallow-cell complexity \(\varphi (\cdot , \cdot )\) and a parameter \(\epsilon > 0\), there exists a collection, called an \(\epsilon \)-Mnet, consisting of \(O\bigl ( \frac{1}{\epsilon } \,\varphi \bigl ( O\bigl (\frac{1}{\epsilon } \bigr ), O(1)\bigr ) \bigr )\) subsets of X, each of size \(\Omega ( \epsilon |X| )\), such that any \(R \in \mathcal {R}\) with \(|R| \ge \epsilon |X|\) contains at least one set in this collection. We observe that as an immediate corollary an alternate proof of the optimal \(\epsilon \)-net bound follows.

Keywords

Epsilon-nets Haussler’s Packing Lemma Mnets Shallow-cell complexity Shallow Packing Lemma 

Notes

Acknowledgements

We thank the journal reviewers whose feedback substantially improved the content and presentation of this paper.

References

  1. 1.
    Aronov, B., de Berg, M., Ezra, E., Sharir, M.: Improved bounds for the union of locally fat objects in the plane. SIAM J. Comput. 43(2), 543–572 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Aronov, B., Ezra, E., Sharir, M.: Small-size \(\epsilon \)-nets for axis-parallel rectangles and boxes. SIAM J. Comput. 39(7), 3248–3282 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Arya, S., da Fonseca, G.D., Mount, D.M.: Optimal area-sensitive bounds for polytope approximation. In: Proceedings of the 28th Annual ACM Symposium on Computational Geometry (SoCG’12), pp. 363–372. ACM, New York (2012)Google Scholar
  4. 4.
    Arya, S., da Fonseca, G.D., Mount, D.M.: On the combinatorial complexity of approximating polytopes. In: Proceedings of the 32nd International Symposium on Computational Geometry (SoCG’16). LIPIcs. Leibniz International Proceedings in Informatics, vol. 51, pp. 11:1–11:15. Schloss Dagstuhl. Leibniz-Zentrum für Informatik, Wadern (2016)Google Scholar
  5. 5.
    Bárány, I.: Random polytopes, convex bodies, and approximation. In: Weil, W. (ed.) Stochastic Geometry. Lecture Notes in Mathematics, vol. 1892, pp. 77–118. Springer, Berlin (2007)Google Scholar
  6. 6.
    Bárány, I., Larman, D.G.: Convex bodies, economic cap coverings, random polytopes. Mathematika 35, 274–291 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Basu, S., Pollack, R., Roy, M.-F.: Algorithms in Real Algebraic Geometry. Algorithms and Computation in Mathematics, vol. 10. Springer, Berlin (2003)CrossRefzbMATHGoogle Scholar
  8. 8.
    Buzaglo, S., Pinchasi, R., Rote, G.: Topological hypergraphs. In: Pach, J. (ed.) Thirty Essays on Geometric Graph Theory, pp. 71–81. Springer, New York (2013)CrossRefGoogle Scholar
  9. 9.
    Chan, T.M., Grant, E., Könemann, J., Sharpe, M.: Weighted capacitated, priority, and geometric set cover via improved quasi-uniform sampling. In: Proceedings of the 23rd ACM-SIAM Symposium on Discrete Algorithms (SODA’12), pp. 1576–1585. ACM, New York (2012)Google Scholar
  10. 10.
    Chazelle, B.: A note on Haussler’s Packing Lemma (1992). See Section 5.3 from Geometric Discrepancy: An Illustrated Guide by J. MatoušekGoogle Scholar
  11. 11.
    Chekuri, C., Clarkson, K.L., Har-Peled, S.: On the set multicover problem in geometric settings. ACM Trans. Algorithms 9(1), 9:1–9:17 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Clarkson, K.L., Shor, P.W.: Application of random sampling in computational geometry, II. Discrete Comput. Geom. 4(5), 387–421 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Dutta, K., Ezra, E., Ghosh, A.: Two proofs for shallow packings. Discrete Comput. Geom. 56(4), 910–939 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Ezra, E.: A size-sensitive discrepancy bound for set systems of bounded primal shatter dimension. SIAM J. Comput. 45(1), 84–101 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Ezra, E., Aronov, B., Sharir, S.: Improved bound for the union of fat triangles. In: Proceedings of the 22nd ACM-SIAM Symposium on Discrete Algorithms (SODA’11), pp. 1778–1785. SIAM, Philadelphia (2011)Google Scholar
  16. 16.
    Fox, J., Pach, J., Sheffer, A., Suk, A., Zahl, J.: A semi-algebraic version of Zarankiewicz’s problem. J. Eur. Math. Soc. (JEMS) 19(6), 1785–1810 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Guth, L., Katz, N.H.: On the Erdös distinct distances problem in the plane. Ann. Math. 181(1), 155–190 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Haussler, D.: Sphere packing numbers for subsets of the Boolean \(n\)-cube with bounded Vapnik-Chervonenkis dimension. J. Combin. Theory Ser. A 69(2), 217–232 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Kupavskii, A., Mustafa, N.H., Pach, J.: Near-optimal lower bounds for \(\epsilon \)-nets for half-spaces and low complexity set systems. In: Loebl, M., Nešetřil, J., Thomas, R. (eds.) A Journey Through Discrete Mathematics: A Tribute to Jiří Matoušek, pp. 527–541. Springer, Cham (2017)CrossRefGoogle Scholar
  20. 20.
    Li, Y., Long, P.M., Srinivasan, A.: Improved bounds on the sample complexity of learning. J. Comput. System Sci. 62(3), 516–527 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Matoušek, J.: Geometric Discrepancy: An Illustrated Guide. Algorithms and Combinatorics, vol. 18. Springer, Berlin (1999)CrossRefzbMATHGoogle Scholar
  22. 22.
    Matoušek, J.: Lectures on Discrete Geometry. Graduate Texts in Mathematics, vol. 212. Springer, New York (2002)CrossRefGoogle Scholar
  23. 23.
    Matoušek, J., Patáková, Z.: Multilevel polynomial partitions and simplified range searching. Discrete Comput. Geom. 54(1), 22–41 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Matoušek, J., Pach, J., Sharir, M., Sifrony, S., Welzl, E.: Fat triangles determine linearly many holes. SIAM J. Comput. 23(1), 154–169 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Mustafa, N.H.: A simple proof of the shallow packing lemma. Discrete Comput. Geom. 55(3), 739–743 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Mustafa, N.H., Ray, S.: \(\epsilon \)-Mnets: hitting geometric set systems with subsets. Discrete Comput. Geom. 57(3), 625–640 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Mustafa, N.H., Varadarajan, K.: Epsilon-approximations and epsilon-nets. In: Goodman, J.E., O’Rourke, J., Tóth, C.D. (eds.) Handbook of Discrete and Computational Geometry. CRC Press, Boca Raton (2017)Google Scholar
  28. 28.
    Pach, J., Agarwal, P.K.: Combinatorial Geometry. Wiley-Interscience Series in Discrete Mathematics and Optimization. Wiley, New York (1995)Google Scholar
  29. 29.
    Sauer, N.: On the density of families of sets. J. Combin. Theory Ser. A 13(1), 145–147 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Shelah, S.: A combinatorial problem, stability and order for models and theories in infinitary languages. Pacific J. Math. 41, 247–261 (1972)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  • Kunal Dutta
    • 1
  • Arijit Ghosh
    • 2
  • Bruno Jartoux
    • 3
  • Nabil H. Mustafa
    • 3
    Email author
  1. 1.DataShape, INRIA Sophia Antipolis – MéditerranéeSophia AntipolisFrance
  2. 2.The Institute of Mathematical Sciences, HBNIChennaiIndia
  3. 3.Laboratoire d’Informatique Gaspard-MongeUniversité Paris-EstESIEE ParisFrance

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