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Alternation, Sparsity and Sensitivity: Combinatorial Bounds and Exponential Gaps

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10743)

Abstract

The well-known Sensitivity Conjecture regarding combinatorial complexity measures on Boolean functions states that for any Boolean function \(f:\{0,1\}^n\rightarrow \{0,1\}\), block sensitivity of f is polynomially related to sensitivity of f (denoted by \(\mathsf {s}(f)\)). From the complexity theory side, the Xor Log-Rank Conjecture states that for any Boolean function, \(f:\{0,1\}^n\rightarrow \{0,1\}\) the communication complexity of a related function \(f^{\oplus }:\{0,1\}^n\times \{0,1\}^n\rightarrow \{0,1\}\), (defined as \(f^{\oplus }(x,y) = f(x \oplus y)\)) is bounded by polynomial in logarithm of the sparsity of f (the number of non-zero Fourier coefficients for f, denoted by \(\mathsf {sparsity}(f)\)). Both the conjectures play a central role in the domains in which they are studied.

A recent result of Lin and Zhang (2017) implies that to confirm the above two conjectures it suffices to upper bound alternation of f (denoted \(\mathsf {alt}(f)\)) for all Boolean functions f by polynomial in \(\mathsf {s}(f)\) and logarithm of \(\mathsf {sparsity}(f)\), respectively. In this context, we show the following results:

  • We show that there exists a family of Boolean functions for which \(\mathsf {alt}(f)\) is at least exponential in \(\mathsf {s}(f)\) and \(\mathsf {alt}(f)\) is at least exponential in \(\log \mathsf {sparsity}(f)\). Enroute to the proof, we also show an exponential gap between \(\mathsf {alt}(f)\) and the decision tree complexity of f, which might be of independent interest.

  • As our main result, we show that, despite the above exponential gap between \(\mathsf {alt}(f)\) and \(\log \mathsf {sparsity}(f)\), the Xor Log-Rank Conjecture is true for functions with the alternation upper bounded by \({\mathsf {poly}}(\log n)\). It is easy to observe that the Sensitivity Conjecture is also true for this class of functions.

  • The starting point for the above result is the observation (derived from Lin and Zhang (2017)) that for any Boolean function f, \(\mathsf {deg}(f) \le \mathsf {alt}(f)\mathsf {deg_{{\mathbb {F}}_2}}(f)^2\) where \(\mathsf {deg}(f)\) and \(\mathsf {deg_{{\mathbb {F}}_2}}(f)\) are the degrees of f over \({\mathbb {R}}\) and \({\mathbb {F}}_2\). We give two further applications of this bound: (1) We show that Boolean functions with bounded alternation have high sparsity (\(\varOmega (\sqrt{\mathsf {deg}(f)})\)), thus partially answering a question of Kulkarni and Santha (2013). (2) We observe that the above relation improves the upper bound for influence to \(\mathsf {deg_{{\mathbb {F}}_2}}(f)^2 \cdot \mathsf {alt}(f)\) improving Guo and Komargodski (2017).

Notes

Acknowledgments

The authors would like to thank the anonymous reviewers for constructive comments which improved the presentation of the paper.

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Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.Department of Computer Science and EngineeringIndian Institute of Technology MadrasChennaiIndia

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