Alternation, Sparsity and Sensitivity: Combinatorial Bounds and Exponential Gaps

• Krishnamoorthy Dinesh
• Jayalal Sarma
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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Authors and Affiliations

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