The Stochastic Boolean Function Evaluation Problem for Symmetric Boolean Functions

TL;DR

This paper presents two approximation algorithms for the stochastic Boolean function evaluation problem for symmetric functions, achieving logarithmic and block-based approximation factors, and explores cost differences in verification versus evaluation.

Contribution

Introduces two novel approximation algorithms for SBFE on symmetric Boolean functions, with bounds based on function structure and goal-value analysis.

Findings

First algorithm has an $O(\log n)$ approximation ratio.

Second algorithm achieves a $(B-1)$ approximation, where $B$ is the number of blocks.

Verification and evaluation costs can differ for symmetric functions.

Abstract

We give two approximation algorithms solving the Stochastic Boolean Function Evaluation (SBFE) problem for symmetric Boolean functions. The first is an $O(\log n)$-approximation algorithm, based on the submodular goal-value approach of Deshpande, Hellerstein and Kletenik. Our second algorithm, which is simple, is based on the algorithm solving the SBFE problem for $k$-of-$n$ functions, due to Salloum, Breuer, and Ben-Dov. It achieves a $(B-1)$ approximation factor, where $B$ is the number of blocks of 0's and 1's in the standard vector representation of the symmetric Boolean function. As part of the design of the first algorithm, we prove that the goal value of any symmetric Boolean function is less than $n(n+1)/2$. Finally, we give an example showing that for symmetric Boolean functions, minimum expected verification cost and minimum expected evaluation cost are not necessarily equal. This contrasts with a previous result, given by Das, Jafarpour, Orlitsky, Pan and Suresh, which showed that equality holds in the unit-cost case.