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 languageEnglish (US)
    Pages (from-to)1-13
    Number of pages13
    JournalDiscrete Applied Mathematics
    Volume238
    DOIs
    StatePublished - Mar 31 2018

    Keywords

    • Boolean functions
    • Read-once formulas
    • Submodularity

    ASJC Scopus subject areas

    • Discrete Mathematics and Combinatorics
    • Applied Mathematics

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  • Cite this

    Bach, E., Dusart, J., Hellerstein, L., & Kletenik, D. (2018). Submodular goal value of Boolean functions. Discrete Applied Mathematics, 238, 1-13. https://doi.org/10.1016/j.dam.2017.10.022