# Submodular goal value of Boolean functions

Eric Bach, Jérémie Dusart, Lisa Hellerstein, Devorah Kletenik

Research output: Contribution to journalArticle

### Abstract

Recently, Deshpande et al. introduced a new measure of the complexity of a Boolean function. We call this measure the "goal value" of the function. The goal value of f is defined in terms of a monotone, submodular utility function associated with f. As shown by Deshpande et al., proving that a Boolean function f has small goal value can lead to a good approximation algorithm for the Stochastic Boolean Function Evaluation problem for f. Also, if f has small goal value, it indicates a close relationship between two other measures of the complexity of f, its average-case decision tree complexity and its average-case certificate complexity. In this paper, we explore the goal value measure in detail. We present bounds on the goal values of arbitrary and specific Boolean functions, and present results on properties of the measure. We compare the goal value measure to other, previously studied, measures of the complexity of Boolean functions. Finally, we discuss a number of open questions suggested by our work.

Original language English (US) Discrete Applied Mathematics https://doi.org/10.1016/j.dam.2017.10.022 Accepted/In press - Jan 1 2017

### Fingerprint

Boolean functions
Boolean Functions
Function evaluation
Approximation algorithms
Decision trees
Submodular Function
Certificate
Utility Function
Decision tree
Approximation Algorithms
Monotone
Evaluation
Arbitrary

### Keywords

• Boolean functions
• Submodularity

### ASJC Scopus subject areas

• Discrete Mathematics and Combinatorics
• Applied Mathematics

### Cite this

Bach, E., Dusart, J., Hellerstein, L., & Kletenik, D. (Accepted/In press). Submodular goal value of Boolean functions. Discrete Applied Mathematics. https://doi.org/10.1016/j.dam.2017.10.022

Submodular goal value of Boolean functions. / Bach, Eric; Dusart, Jérémie; Hellerstein, Lisa; Kletenik, Devorah.

In: Discrete Applied Mathematics, 01.01.2017.

Research output: Contribution to journalArticle

Bach, Eric ; Dusart, Jérémie ; Hellerstein, Lisa ; Kletenik, Devorah. / Submodular goal value of Boolean functions. In: Discrete Applied Mathematics. 2017.
title = "Submodular goal value of Boolean functions",
abstract = "Recently, Deshpande et al. introduced a new measure of the complexity of a Boolean function. We call this measure the {"}goal value{"} of the function. The goal value of f is defined in terms of a monotone, submodular utility function associated with f. As shown by Deshpande et al., proving that a Boolean function f has small goal value can lead to a good approximation algorithm for the Stochastic Boolean Function Evaluation problem for f. Also, if f has small goal value, it indicates a close relationship between two other measures of the complexity of f, its average-case decision tree complexity and its average-case certificate complexity. In this paper, we explore the goal value measure in detail. We present bounds on the goal values of arbitrary and specific Boolean functions, and present results on properties of the measure. We compare the goal value measure to other, previously studied, measures of the complexity of Boolean functions. Finally, we discuss a number of open questions suggested by our work.",
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AU - Bach, Eric

AU - Dusart, Jérémie

AU - Hellerstein, Lisa

AU - Kletenik, Devorah

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N2 - Recently, Deshpande et al. introduced a new measure of the complexity of a Boolean function. We call this measure the "goal value" of the function. The goal value of f is defined in terms of a monotone, submodular utility function associated with f. As shown by Deshpande et al., proving that a Boolean function f has small goal value can lead to a good approximation algorithm for the Stochastic Boolean Function Evaluation problem for f. Also, if f has small goal value, it indicates a close relationship between two other measures of the complexity of f, its average-case decision tree complexity and its average-case certificate complexity. In this paper, we explore the goal value measure in detail. We present bounds on the goal values of arbitrary and specific Boolean functions, and present results on properties of the measure. We compare the goal value measure to other, previously studied, measures of the complexity of Boolean functions. Finally, we discuss a number of open questions suggested by our work.

AB - Recently, Deshpande et al. introduced a new measure of the complexity of a Boolean function. We call this measure the "goal value" of the function. The goal value of f is defined in terms of a monotone, submodular utility function associated with f. As shown by Deshpande et al., proving that a Boolean function f has small goal value can lead to a good approximation algorithm for the Stochastic Boolean Function Evaluation problem for f. Also, if f has small goal value, it indicates a close relationship between two other measures of the complexity of f, its average-case decision tree complexity and its average-case certificate complexity. In this paper, we explore the goal value measure in detail. We present bounds on the goal values of arbitrary and specific Boolean functions, and present results on properties of the measure. We compare the goal value measure to other, previously studied, measures of the complexity of Boolean functions. Finally, we discuss a number of open questions suggested by our work.

KW - Boolean functions

KW - Submodularity

UR - http://www.scopus.com/inward/record.url?scp=85036652784&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85036652784&partnerID=8YFLogxK

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