Alternation, Sparsity and Sensitivity: Combinatorial Bounds and Exponential Gaps

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).