Sensitivity Conjecture and Log-rank Conjecture for functions with small alternating numbers

TL;DR

This paper extends the validity of the Sensitivity and Log-rank Conjectures to functions with small alternating numbers, enhancing understanding of these conjectures for a broader class of functions.

Contribution

It generalizes known results for monotone functions to functions with limited alternating values on monotone paths, advancing the study of these conjectures.

Findings

Sensitivity Conjecture holds for functions with small alternating numbers.

Log-rank Conjecture applies to functions with small alternating numbers.

Deepens understanding of complexity conjectures for non-monotone functions.

Abstract

The Sensitivity Conjecture and the Log-rank Conjecture are among the most important and challenging problems in concrete complexity. Incidentally, the Sensitivity Conjecture is known to hold for monotone functions, and so is the Log-rank Conjecture for $f(x \wedge y)$ and $f(x\oplus y)$ with monotone functions $f$, where $\wedge$ and $\oplus$ are bit-wise AND and XOR, respectively. In this paper, we extend these results to functions $f$ which alternate values for a relatively small number of times on any monotone path from $0^n$ to $1^n$. These deepen our understandings of the two conjectures, and contribute to the recent line of research on functions with small alternating numbers.