Any Monotone Property of 3-Uniform Hypergraphs Is Weakly Evasive

  • Raghav Kulkarni
  • Youming Qiao
  • Xiaoming Sun
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7876)


For a Boolean function f, let D(f) denote its deterministic decision tree complexity, i.e., minimum number of (adaptive) queries required in worst case in order to determine f. In a classic paper, Rivest and Vuillemin [18] show that any non-constant monotone property \(\mathcal{P} : \{0, 1\}^{n \choose 2} \to \{0, 1\}\) of n-vertex graphs has \(D(\mathcal{P}) = \Omega(n^2).\)

We extend their result to 3-uniform hypergraphs. In particular, we show that any non-constant monotone property \(\mathcal{P} : \{0, 1\}^{n \choose 3} \to \{0, 1\}\) of n-vertex 3-uniform hypergraphs has \(D(\mathcal{P}) = \Omega(n^3).\)

Our proof combines the combinatorial approach of Rivest and Vuillemin with the topological approach of Kahn, Saks, and Sturtevant. Interestingly, our proof makes use of Vinogradov’s Theorem (weak Goldbach Conjecture), inspired by its recent use by Babai et. al. [1] in the context of the topological approach. Our work leaves the generalization to k-uniform hypergraphs as an intriguing open question.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Babai, L., Banerjee, A., Kulkarni, R., Naik, V.: Evasiveness and the Distribution of Prime Numbers. In: STACS 2010, pp. 71–82 (2010)Google Scholar
  2. 2.
    Benjamini, I., Kalai, G., Schramm, O.: Noise sensitivity of Boolean functions and its application to percolation. Inst. Hautes tudes Sci. Publ. (MATH), 90 (1999)Google Scholar
  3. 3.
    Buhrman, H., de Wolf, R.: Complexity measures and decision tree complexity: a survey. Theor. Comput. Sci. 288(1), 21–43 (2002)CrossRefzbMATHGoogle Scholar
  4. 4.
    Chakrabarti, A., Khot, S., Shi, Y.: Evasiveness of Subgraph Containment and Related Properties. SIAM J. Comput. 31(3), 866–875 (2001)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Haselgrove, C.B.: Some theorems on the analytic theory of numbers. J. London Math. Soc. 36, 273–277 (1951)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Hayes, T.P., Kutin, S., van Melkebeek, D.: The Quantum Black-Box Complexity of Majority. Algorithmica 34(4), 480–501 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Kulkarni, R.: Evasiveness Through A Circuit Lens. To appear in ITCS 2013 (2013)Google Scholar
  8. 8.
    Kahn, J., Saks, M.E., Sturtevant, D.: A topological approach to evasiveness. Combinatorica 4(4), 297–306 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Kushilevitz, E., Mansour, Y.: Learning Decision Trees Using the Fourier Spectrum. SIAM J. Comput. 22(6), 1331–1348 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Kulkarni, R., Santha, M.: Query complexity of matroids. Electronic Colloquium on Computational Complexity (ECCC) 19, 63 (2012)Google Scholar
  11. 11.
    Linial, N., Mansour, Y., Nisan, N.: Constant Depth Circuits, Fourier Transform, and Learnability. J. ACM 40(3), 607–620 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Lutz, F.H.: Some Results Related to the Evasiveness Conjecture. Comb. Theory, Ser. B 81(1), 110–124 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Montanaro, A., Osborne, T.: On the communication complexity of XOR functions. CoRR abs/0909.3392 (2009)Google Scholar
  14. 14.
    Nisan, N., Szegedy, M.: On the Degree of Boolean Functions as Real Polynomials. Computational Complexity 4, 301–313 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Nisan, N., Wigderson, A.: On Rank vs. Communication Complexity. Combinatorica 15(4), 557–565 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Oliver, R.: Fixed-point sets of group actions on finite acyclic complexes. Comment. Math. Helv. 50, 155–177 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    O’Donnell, R., Saks, M.E., Schramm, O., Servedio, R.A.: Every decision tree has an influential variable. In: FOCS 2005, pp. 31–39 (2005)Google Scholar
  18. 18.
    Rivest, R.L., Vuillemin, J.: On Recognizing Graph Properties from Adjacency Matrices. Theor. Comput. Sci. 3(3), 371–384 (1976)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Robinson, D.J.S.: A Course in the Theory of Groups, 2nd edn. Springer (1996)Google Scholar
  20. 20.
    Saks, M.E., Wigderson, A.: Probabilistic Boolean Decision Trees and the Complexity of Evaluating Game Trees. In: FOCS 1986, pp. 29–38 (1986)Google Scholar
  21. 21.
    Shi, Y., Zhang, Z.: Communication Complexities of XOR functions. CoRR abs/0808.1762 (2008)Google Scholar
  22. 22.
    Zhang, Z., Shi, Y.: On the parity complexity measures of Boolean functions. Theor. Comput. Sci. 411(26-28), 2612–2618 (2010)CrossRefzbMATHGoogle Scholar
  23. 23.
    Vinogradov, I.M.: The Method of Trigonometrical Sums in the Theory of Numbers. Trav. Inst. Math. Stekloff 10 (1937) (Russian)Google Scholar
  24. 24.
    Yao, A.C.-C.: Monotone Bipartite Graph Properties are Evasive. SIAM J. Comput. 17(3), 517–520 (1988)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Raghav Kulkarni
    • 1
  • Youming Qiao
    • 1
  • Xiaoming Sun
    • 2
  1. 1.Centre for Quantum Technologiesthe National University of SingaporeSingapore
  2. 2.Institute of Computing TechnologyChinese Academy of SciencesChina

Personalised recommendations