Document Open Access Logo

Shattered Sets and the Hilbert Function

Authors Shay Moran, Cyrus Rashtchian

PDF
Thumbnail PDF

File

LIPIcs.MFCS.2016.70.pdf
  • Filesize: 0.84 MB
  • 14 pages

Document Identifiers

Author Details

Shay Moran
Cyrus Rashtchian

Cite AsGet BibTex

Shay Moran and Cyrus Rashtchian. Shattered Sets and the Hilbert Function. In 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 58, pp. 70:1-70:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)
https://doi.org/10.4230/LIPIcs.MFCS.2016.70

Abstract

We study complexity measures on subsets of the boolean hypercube and exhibit connections between algebra (the Hilbert function) and combinatorics (VC theory). These connections yield results in both directions. Our main complexity-theoretic result demonstrates that a large and natural family of linear program feasibility problems cannot be computed by polynomial-sized constant-depth circuits. Moreover, our result applies to a stronger regime in which the hyperplanes are fixed and only the directions of the inequalities are given as input to the circuit. We derive this result by proving that a rich class of extremal functions in VC theory cannot be approximated by low-degree polynomials. We also present applications of algebra to combinatorics. We provide a new algebraic proof of the Sandwich Theorem, which is a generalization of the well-known Sauer-Perles-Shelah Lemma. Finally, we prove a structural result about downward-closed sets, related to the Chvatal conjecture in extremal combinatorics.
Keywords
  • VC dimension
  • shattered sets
  • sandwich theorem
  • Hilbert function
  • polynomial method
  • linear programming
  • Chvatal's conjecture
  • downward-closed sets

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

References

  1. Noga Alon. Combinatorial Nullstellensatz. Combinatorics Probability and Computing, 8(1):7-30, 1999. Google Scholar
  2. Noga Alon, Peter Frankl, and Vojtech Rödl. Geometrical Realization of Set Systems and Probabilistic communication Complexity. In FOCS, pages 277-280. IEEE, 1985. Google Scholar
  3. Noga Alon, Shay Moran, and Amir Yehudayoff. Sign rank, VC dimension and spectral gaps. Electronic Colloquium on Computational Complexity (ECCC), 21:135, 2014. URL: http://eccc.hpi-web.de/report/2014/135.
  4. Ian Anderson. Combinatorics of Finite sets. Bull. Amer. Math. Soc., 1988. Google Scholar
  5. Richard P. Anstee, Lajos Rónyai, and Attila Sali. Shattering news. Graphs and Combinatorics, 18(1):59-73, 2002. URL: http://dx.doi.org/10.1007/s003730200003.
  6. J. Aspnes, R. Beigel, M. Furst, and S. Rudich. The expressive power of voting polynomials. Combinatorica, 14(2):135-148, 1994. URL: http://dx.doi.org/10.1007/BF01215346.
  7. L Babai and P Frankl. Linear Algebra Methods in Combinatorics. University of Chicago, 1992. Google Scholar
  8. Hans-Jürgen Bandelt, Victor Chepoi, Andreas W. M. Dress, and Jack H. Koolen. Combinatorics of lopsided sets. Eur. J. Comb., 27(5):669-689, 2006. Google Scholar
  9. Richard Beigel. The Polynomial Method in Circuit Complexity. In In Proceedings of the 8th IEEE Structure in Complexity Theory Conference. Citeseer, 1993. Google Scholar
  10. C Berge. A Theorem Related to the Chvátal conjecture. In Proceedings, 5th British Combinatorial Conference, 1976. Google Scholar
  11. Anna Bernasconi and Lavinia Egidi. Hilbert Function and Complexity Lower Bounds for Symmetric Boolean Functions. Information and Computation, 153(1):1-25, 1999. Google Scholar
  12. Abhishek Bhowmick and Shachar Lovett. Nonclassical Polynomials as a Barrier to Polynomial Lower Bounds. In 30th Conference on Computational Complexity, page 72, 2015. Google Scholar
  13. Béla Bollobás and Imre Leader. Sums in the grid. Discrete Mathematics, 162(1-3):31-48, 1996. URL: http://dx.doi.org/10.1016/S0012-365X(96)00303-2.
  14. Béla Bollobás and A. J. Radcliffe. Defect sauer results. J. Comb. Theory, Ser. A, 72(2):189-208, 1995. Google Scholar
  15. Béla Bollobás, A. J. Radcliffe, and Leader I. Reverse kleitman inequalities. Proc. London Math. Soc., Ser. A, (3) 58:153-168, 1989. Google Scholar
  16. Vašek Chvátal and Donald Knuth. Selected Combinatorial Research Problems. Stanford University, 1972. Google Scholar
  17. David P Dobkin and Steven P Reiss. The Complexity of Linear Programming. Theoretical Computer Science, 11:1-18, 1980. Google Scholar
  18. A.W.M. Dress. Towards a theory of holistic clustering. DIMACS Ser. Discrete Math. Theoret. Comput. Sci., 37 Amer. Math. Soc.:271-289, 1997. Google Scholar
  19. Zeev Dvir. On the size of Kakeya sets in finite fields. Journal of the American Mathematical Society, 22(4):1093-1097, 2009. Google Scholar
  20. Per Enflo. On the nonexistence of uniform homeomorphisms betweenl p-spaces. Arkiv för Matematik, 8(2):103-105, 1970. URL: http://dx.doi.org/10.1007/BF02589549.
  21. Bernd Gärtner and Emo Welzl. Vapnik-Chervonenkis dimension and (pseudo-) hyperplane arrangements. Discrete &Computational Geometry, 12(1):399-432, 1994. Google Scholar
  22. Oded Goldreich, Shafi Goldwasser, Eric Lehman, Dana Ron, and Alex Samorodnitsky. Testing monotonicity. Combinatorica, 20(3):301-337, 2000. URL: http://dx.doi.org/10.1007/s004930070011.
  23. Ben Joseph Green and Terence Tao. Freiman’s theorem in finite fields via extremal set theory. Combinatorics, Probability & Computing, 18(3):335-355, 2009. URL: http://dx.doi.org/10.1017/S0963548309009821.
  24. Leonid Gurvits. Linear algebraic proofs of VC-dimension based inequalities. In Computational Learning Theory, pages 238-250. Springer, 1997. Google Scholar
  25. Larry Guth and Nets Hawk Katz. On the Erdos Distinct Distances Problem in the Plane. Annals of Mathematics, 181(1):155-190, 2015. Google Scholar
  26. Stasys Jukna. Extremal Combinatorics: with Applications in Computer Science. Springer Science &Business Media, 2011. Google Scholar
  27. Daniel J. Kleitman. Families of non-disjoint subsets. Journal of Combinatorial Theory, 1(1):153-155, 1966. URL: http://dx.doi.org/10.1016/S0021-9800(66)80012-1.
  28. László Kozma and Shay Moran. Shattering, graph orientations, and connectivity. Electr. J. Comb., 20(3):P44, 2013. URL: http://www.combinatorics.org/ojs/index.php/eljc/article/view/v20i3p44.
  29. J. Lawrence. Lopsided sets and orthant-intersection by convex sets. Pac. J. Math., 104(1):155-173, 1983. Google Scholar
  30. Jiří Matoušek. Lectures on discrete geometry, volume 212. Springer New York, 2002. Google Scholar
  31. Tamás Mészáros. S-extremal set systems and Gröbner Bases. Master’s thesis, Budapest University of Technology and Economics, 2009. Google Scholar
  32. Tamás Mészáros and Lajos Rónyai. Shattering-extremal set systems of VC dimension at most 2. Electr. J. Comb., 21(4):P4.30, 2014. URL: http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i4p30.
  33. S. Moran. Shattering Extremal Systems. Master’s thesis, Saarland University, Saarbrücken, Germany, 2012. Google Scholar
  34. Shay Moran, Amir Shpilka, Avi Wigderson, and Amir Yehudayoff. Teaching and compressing for low vc-dimension. In FOCS, 2015. URL: http://arxiv.org/abs/1502.06187.
  35. Shay Moran and Manfred K. Warmuth. Labeled compression schemes for extremal classes. CoRR, abs/1506.00165, 2015. URL: http://arxiv.org/abs/1506.00165.
  36. A. Pajor. Sous-espaces lⁿ₁ des espaces de banach. Travaux en Cours. Hermann, Paris, 1985. Google Scholar
  37. Alexander A Razborov. Lower bounds on the size of bounded depth circuits over a complete basis with logical addition. Mathematical Notes, 41(4):333-338, 1987. Google Scholar
  38. Zachary Remscrim. The Hilbert Function, Algebraic Extractors, and Recursive Fourier Sampling. ECCC Technical Report TR 16-020, 2016. Google Scholar
  39. Micha Sharir, Adam Sheffer, and Joshua Zahl. Improved Bounds for Incidences Between Points and Circles. Combinatorics, Probability and Computing, 24(03):490-520, 2015. Google Scholar
  40. R. Smolensky. On Representations by Low-degree Polynomials. In FOCS, 1993. Google Scholar
  41. Roman Smolensky. Algebraic methods in the theory of lower bounds for Boolean circuit complexity. In Proceedings of the nineteenth annual ACM symposium on Theory of computing, pages 77-82. ACM, 1987. Google Scholar
  42. Roman Smolensky. Well-Known Bound for the VC-Dimension Made Easy. Computational Complexity, 6(4):299-300, 1997. URL: http://dx.doi.org/10.1007/BF01270383.
  43. Richard P Stanley. Introduction to Hyperplane Arrangements. Lecture notes, IAS/Park City Mathematics Institute, 2004. Google Scholar
  44. Terence Tao. Algebraic Combinatorial Geometry: the Polynomial Method in Arithmetic Combinatorics, Incidence Combinatorics, and Number Theory. EMS Surveys in Mathematical Sciences, 1(1):1-46, 2014. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact