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

Chiarelli, John; Hatami, Pooya; Saks, Michael

doi:10.1007/s00493-019-4136-7

Cited by 15 publications

(24 citation statements)

References 8 publications

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“…The Nisan-Szegedy Theorem says that a degree-k Boolean-valued function on the Hamming cube is a k2 k -junta. (We remark that the smallest quantity γ 2 (k) that can replace k2 k here is now known [7] [25], an analogous result for functions on Hamming slices was shown; they conjectured a similar result for functions on multislices. We resolve this conjecture, following the structure of their proof.…”

Section: Nisan-szegedy Theorem On the Multislicementioning

confidence: 55%

Log-Sobolev inequality for the multislice, with applications

Filmus

O’Donnell

2022

Electron. J. Probab.

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Let κ ∈ N + satisfy κ1 + • • • + κ = n, and let Uκ denote the multislice of all strings u ∈ [ ] n having exactly κi coordinates equal to i, for all i ∈ [ ]. Consider the Markov chain on Uκ where a step is a random transposition of two coordinates of u. We show that the log-Sobolev constant κ for the chain satisfieswhich is sharp up to constants whenever is constant. From this, we derive some consequences for small-set expansion and isoperimetry in the multislice, including a KKL Theorem, a Kruskal-Katona Theorem for the multislice, a Friedgut Junta Theorem, and a Nisan-Szegedy Theorem.

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Section: Nisan-szegedy Theorem On the Multislicementioning

confidence: 55%

Log-Sobolev inequality for the multislice, with applications

Filmus

O’Donnell

2022

Electron. J. Probab.

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“…Our main combinatorial result is that set systems corresponding to the monomials of boolean functions have small hitting sets. (Note that an upper bound on the hitting set in terms of degree was known before [8].) Theorem 1.1.…”

Section: Resultsmentioning

confidence: 99%

Log-rank and lifting for AND-functions

Knop

Lovett

McGuire

et al. 2021

Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing

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Let 𝑓 : {0, 1} 𝑛 → {0, 1} be a boolean function, and let 𝑓 ∧ (𝑥, 𝑦) = 𝑓 (𝑥 ∧𝑦) denote the AND-function of 𝑓 , where 𝑥 ∧𝑦 denotes bit-wise AND. We study the deterministic communication complexity of 𝑓 ∧ and show that, up to a log 𝑛 factor, it is bounded by a polynomial in the logarithm of the real rank of the communication matrix of 𝑓 ∧ . This comes within a log 𝑛 factor of establishing the log-rank conjecture for AND-functions with no assumptions on 𝑓 . Our result stands in contrast with previous results on special cases of the logrank conjecture, which needed significant restrictions on 𝑓 such as monotonicity or low F 2 -degree. Our techniques can also be used to prove (within a log 𝑛 factor) a lifting theorem for AND-functions, stating that the deterministic communication complexity of 𝑓 ∧ is polynomially related to the AND-decision tree complexity of 𝑓 .The results rely on a new structural result regarding boolean functions 𝑓 : {0, 1} 𝑛 → {0, 1} with a sparse polynomial representation, which may be of independent interest. We show that if the polynomial computing 𝑓 has few monomials then the set system of the monomials has a small hitting set, of size poly-logarithmic in its sparsity. We also establish extensions of this result to multi-linear polynomials 𝑓 : {0, 1} 𝑛 → R with a larger range. CCS CONCEPTS• Theory of computation → Communication complexity; Oracles and decision trees.

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“…In light of the previous subsections, it seems natural to expect that one should be able to prove a bound S(f ) = O(1) for any f using a similar inductive argument, thereby improving Simon's theorem (in the same sense that [4] improved Nisan-Szegedy's bound.) However, choosing a good H to restrict for S is tricky business -neither choice from the previous two subsections will work in general here.…”

Section: Sensitivitymentioning

confidence: 95%

“…By Fact 3.3, M satisfies the inequality (3.2) for any set H ⊆ [n] of restricted coordinates. Next we introduce three explicit families of RRCMs, the first of which (deg i ) was introduced in [4]:…”

Section: Restriction-reducing Coordinate Measuresmentioning

confidence: 99%

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Relationships between the number of inputs and other complexity measures of Boolean functions

Wellens¹

2020

Preprint

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We generalize and extend the ideas in a recent paper of Chiarelli, Hatami and Saks to prove new bounds on the number of relevant variables for boolean functions in terms of a variety of complexity measures. Our approach unifies and refines all previously known bounds of this type. We also improve Nisan and Szegedy's well-known inequality bs(f ) ≤ deg(f ) 2 by a constant factor, thereby improving Huang's recent proof of the sensitivity conjecture by the same constant.

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

Cited by 15 publications

References 8 publications

Log-Sobolev inequality for the multislice, with applications

Log-Sobolev inequality for the multislice, with applications

Log-rank and lifting for AND-functions

Relationships between the number of inputs and other complexity measures of Boolean functions

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