Alternation, Sparsity and Sensitivity: Combinatorial Bounds and Exponential Gaps

TLDR: Two conjectures regarding combinatorial complexity measures on Boolean functions and the number of non-zero Fourier coefficients for f are studied play a central role in the domains in which they are studied.

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.