### 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) |
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Journal | Discrete Applied Mathematics |

DOIs | |

State | Accepted/In press - Jan 1 2017 |

### Fingerprint

### Keywords

- Boolean functions
- Read-once formulas
- Submodularity

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Applied Mathematics

### Cite this

*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.

Research output: Contribution to journal › Article

*Discrete Applied Mathematics*. https://doi.org/10.1016/j.dam.2017.10.022

}

TY - JOUR

T1 - Submodular goal value of Boolean functions

AU - Bach, Eric

AU - Dusart, Jérémie

AU - Hellerstein, Lisa

AU - Kletenik, Devorah

PY - 2017/1/1

Y1 - 2017/1/1

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 - Read-once formulas

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

U2 - 10.1016/j.dam.2017.10.022

DO - 10.1016/j.dam.2017.10.022

M3 - Article

JO - Discrete Applied Mathematics

JF - Discrete Applied Mathematics

SN - 0166-218X

ER -