Abstract
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.
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References
Babai, L., Banerjee, A., Kulkarni, R., Naik, V.: Evasiveness and the Distribution of Prime Numbers. In: STACS 2010, pp. 71–82 (2010)
Benjamini, I., Kalai, G., Schramm, O.: Noise sensitivity of Boolean functions and its application to percolation. Inst. Hautes tudes Sci. Publ. (MATH), 90 (1999)
Buhrman, H., de Wolf, R.: Complexity measures and decision tree complexity: a survey. Theor. Comput. Sci. 288(1), 21–43 (2002)
Chakrabarti, A., Khot, S., Shi, Y.: Evasiveness of Subgraph Containment and Related Properties. SIAM J. Comput. 31(3), 866–875 (2001)
Haselgrove, C.B.: Some theorems on the analytic theory of numbers. J. London Math. Soc. 36, 273–277 (1951)
Hayes, T.P., Kutin, S., van Melkebeek, D.: The Quantum Black-Box Complexity of Majority. Algorithmica 34(4), 480–501 (2002)
Kulkarni, R.: Evasiveness Through A Circuit Lens. To appear in ITCS 2013 (2013)
Kahn, J., Saks, M.E., Sturtevant, D.: A topological approach to evasiveness. Combinatorica 4(4), 297–306 (1984)
Kushilevitz, E., Mansour, Y.: Learning Decision Trees Using the Fourier Spectrum. SIAM J. Comput. 22(6), 1331–1348 (1993)
Kulkarni, R., Santha, M.: Query complexity of matroids. Electronic Colloquium on Computational Complexity (ECCC) 19, 63 (2012)
Linial, N., Mansour, Y., Nisan, N.: Constant Depth Circuits, Fourier Transform, and Learnability. J. ACM 40(3), 607–620 (1993)
Lutz, F.H.: Some Results Related to the Evasiveness Conjecture. Comb. Theory, Ser. B 81(1), 110–124 (2001)
Montanaro, A., Osborne, T.: On the communication complexity of XOR functions. CoRR abs/0909.3392 (2009)
Nisan, N., Szegedy, M.: On the Degree of Boolean Functions as Real Polynomials. Computational Complexity 4, 301–313 (1994)
Nisan, N., Wigderson, A.: On Rank vs. Communication Complexity. Combinatorica 15(4), 557–565 (1995)
Oliver, R.: Fixed-point sets of group actions on finite acyclic complexes. Comment. Math. Helv. 50, 155–177 (1975)
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)
Rivest, R.L., Vuillemin, J.: On Recognizing Graph Properties from Adjacency Matrices. Theor. Comput. Sci. 3(3), 371–384 (1976)
Robinson, D.J.S.: A Course in the Theory of Groups, 2nd edn. Springer (1996)
Saks, M.E., Wigderson, A.: Probabilistic Boolean Decision Trees and the Complexity of Evaluating Game Trees. In: FOCS 1986, pp. 29–38 (1986)
Shi, Y., Zhang, Z.: Communication Complexities of XOR functions. CoRR abs/0808.1762 (2008)
Zhang, Z., Shi, Y.: On the parity complexity measures of Boolean functions. Theor. Comput. Sci. 411(26-28), 2612–2618 (2010)
Vinogradov, I.M.: The Method of Trigonometrical Sums in the Theory of Numbers. Trav. Inst. Math. Stekloff 10 (1937) (Russian)
Yao, A.C.-C.: Monotone Bipartite Graph Properties are Evasive. SIAM J. Comput. 17(3), 517–520 (1988)
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Kulkarni, R., Qiao, Y., Sun, X. (2013). Any Monotone Property of 3-Uniform Hypergraphs Is Weakly Evasive. In: Chan, TH.H., Lau, L.C., Trevisan, L. (eds) Theory and Applications of Models of Computation. TAMC 2013. Lecture Notes in Computer Science, vol 7876. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-38236-9_21
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DOI: https://doi.org/10.1007/978-3-642-38236-9_21
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