Approximating the Influence of a monotone Boolean function in O(\sqrt{n}) query complexity

TL;DR

This paper introduces an efficient randomized algorithm to approximate the total influence of monotone Boolean functions within a multiplicative factor, achieving near-optimal query complexity of O(√n) for certain influence ranges.

Contribution

The paper presents a new O(√n log n / I[f]) query algorithm for approximating influence and establishes lower bounds, demonstrating near-optimality in query complexity for monotone Boolean functions.

Findings

Algorithm approximates influence within (1±ε) factor.

Query complexity is nearly optimal for influence I[f] = Ω(1).

Lower bounds match the algorithm's complexity for certain influence ranges.

Abstract

The {\em Total Influence} ({\em Average Sensitivity) of a discrete function is one of its fundamental measures. We study the problem of approximating the total influence of a monotone Boolean function \ifnum\plusminus=1 $f: \{\pm1\}^n \longrightarrow \{\pm1\}$, \else $f: \bitset^n \to \bitset$, \fi which we denote by $I[f]$. We present a randomized algorithm that approximates the influence of such functions to within a multiplicative factor of $(1\pm \eps)$ by performing $O(\frac{\sqrt{n}\log n}{I[f]} \poly(1/\eps)) $ queries. % \mnote{D: say something about technique?} We also prove a lower bound of % $\Omega(\frac{\sqrt{n/\log n}}{I[f]})$ $\Omega(\frac{\sqrt{n}}{\log n \cdot I[f]})$ on the query complexity of any constant-factor approximation algorithm for this problem (which holds for $I[f] = \Omega(1)$), % and $I[f] = O(\sqrt{n}/\log n)$), hence showing that our algorithm is almost optimal in terms of its dependence on $n$. For general functions we give a lower bound of $\Omega(\frac{n}{I[f]})$, which matches the complexity of a simple sampling algorithm.