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 provoked by our work.