Adaptivity Gaps for the Stochastic Boolean Function Evaluation Problem

TL;DR

This paper investigates the difference in efficiency between adaptive and non-adaptive strategies for evaluating Boolean functions under uncertainty, establishing lower bounds on the adaptivity gap for various classes of functions.

Contribution

It provides new lower bounds on the adaptivity gap for SBFE across different Boolean function classes, highlighting the limitations of non-adaptive strategies.

Findings

Lower bounds on adaptivity gap range from Ω(log n) to Ω(n/ log n)

Contrast with recent results showing O(1) gaps for symmetric and linear threshold functions

Analysis applies to read-once DNF, read-once formulas, and general DNFs

Abstract

We consider the Stochastic Boolean Function Evaluation (SBFE) problem where the task is to efficiently evaluate a known Boolean function $f$ on an unknown bit string $x$ of length $n$. We determine $f(x)$ by sequentially testing the variables of $x$, each of which is associated with a cost of testing and an independent probability of being true. If a strategy for solving the problem is adaptive in the sense that its next test can depend on the outcomes of previous tests, it has lower expected cost but may take up to exponential space to store. In contrast, a non-adaptive strategy may have higher expected cost but can be stored in linear space and benefit from parallel resources. The adaptivity gap, the ratio between the expected cost of the optimal non-adaptive and adaptive strategies, is a measure of the benefit of adaptivity. We present lower bounds on the adaptivity gap for the SBFE problem for popular classes of Boolean functions, including read-once DNF formulas, read-once formulas, and general DNFs. Our bounds range from $\Omega(\log n)$ to $\Omega(n/\log n)$, contrasting with recent $O(1)$ gaps shown for symmetric functions and linear threshold functions.