An Asymptotically Tight Bound on the Number of Relevant Variables in a Bounded Degree Boolean function

Abstract

We prove that there is a constant C ≤ 6.614 such that every Boolean function of degree at most d (as a polynomial over ℝ) is a C·2d-junta, i.e., it depends on at most C·2d variables. This improves the d·2d-1 upper bound of Nisan and Szegedy [Computational Complexity 4 (1994)].

The bound of C·2d is tight up to the constant C, since a read-once decision tree of depth d depends on all 2d - 1 variables. We slightly improve this lower bound by constructing, for each positive integer d, a function of degree d with 3·2d-1 - 2 relevant variables. A similar construction was independently observed by Shinkar and Tal.

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Acknowledgements

We thank Avishay Tal for helpful discussions, and for sharing his python code for exhaustive search of Boolean functions on few variables. We also thank Yuval Filmus for pointing out the implications for Boolean functions on the slice, and Jake Lee Wellens for identifying a technical error in Proposition 2.1 from a previous version of this paper. We finally thank the referees for their detailed and insightful comments.

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Correspondence to John Chiarelli.

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Supported in part by the Simons Foundation under award 332622.

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Chiarelli, J., Hatami, P. & Saks, M. An Asymptotically Tight Bound on the Number of Relevant Variables in a Bounded Degree Boolean function. Combinatorica 40, 237–244 (2020). https://doi.org/10.1007/s00493-019-4136-7

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Mathematics Subject Classification (2010)

  • 05D05
  • 06E30
  • 26C05