Inimizing DNF formulas and ACd 0 circuits given a truth table

Eric Allender, Lisa Hellerstein, Paul McCabe, Toniann Pitassi, Michael Saks

    Research output: Chapter in Book/Report/Conference proceedingConference contribution

    Abstract

    For circuit classes R, the fundamental computational problem Min-R asks for the minimum R-size of a Boolean function presented as a truth table. Prominent examples of this problem include Min-DNF, which asks whether a given Boolean function presented as a truth table has a k-term DNF, and Min-Circuit (also called MCSP), which asks whether a Boolean function presented as a truth table has a size k Boolean circuit. We present a new reduction proving that Min-DNF is NP-complete. It is significantly simpler than the known reduction of Masek [31], which is from Circuit-SAT. We then give a more complex reduction, yielding the result that Min-DNF cannot be approximated to within a factor smaller than (log N)γ, for some constant γ > 0, assuming that NP is not contained in quasipolynomial time. The standard greedy algorithm for Set Cover is often used in practice to approximate Min-DNF. The question of whether Min-DNF can be approximated to within a factor of o(log N) remains open, but we construct an instance of Min-DNF on which the solution produced by the greedy algorithm is Ω(log N) larger than optimal. Finally, we extend known hardness results for Min-TCd 0 to obtain new hardness results for Min-ACd 0, under cryptographic assumptions.

    Original languageEnglish (US)
    Title of host publicationProceedings - Twenty-First Annual IEEE Conference on Computational Complexity, CCC 2006
    Pages237-251
    Number of pages15
    Volume2006
    DOIs
    StatePublished - 2006
    Event21st Annual IEEE Conference on Computational Complexity, CCC 2006 - Prague, Czech Republic
    Duration: Jul 16 2006Jul 20 2006

    Other

    Other21st Annual IEEE Conference on Computational Complexity, CCC 2006
    CountryCzech Republic
    CityPrague
    Period7/16/067/20/06

    Fingerprint

    Truth table
    Boolean Functions
    Boolean functions
    Greedy Algorithm
    Hardness
    Networks (circuits)
    Boolean Circuits
    Set Cover
    NP-complete problem
    Term

    ASJC Scopus subject areas

    • Computational Mathematics

    Cite this

    Allender, E., Hellerstein, L., McCabe, P., Pitassi, T., & Saks, M. (2006). Inimizing DNF formulas and ACd 0 circuits given a truth table. In Proceedings - Twenty-First Annual IEEE Conference on Computational Complexity, CCC 2006 (Vol. 2006, pp. 237-251). [1663741] https://doi.org/10.1109/CCC.2006.27

    Inimizing DNF formulas and ACd 0 circuits given a truth table. / Allender, Eric; Hellerstein, Lisa; McCabe, Paul; Pitassi, Toniann; Saks, Michael.

    Proceedings - Twenty-First Annual IEEE Conference on Computational Complexity, CCC 2006. Vol. 2006 2006. p. 237-251 1663741.

    Research output: Chapter in Book/Report/Conference proceedingConference contribution

    Allender, E, Hellerstein, L, McCabe, P, Pitassi, T & Saks, M 2006, Inimizing DNF formulas and ACd 0 circuits given a truth table. in Proceedings - Twenty-First Annual IEEE Conference on Computational Complexity, CCC 2006. vol. 2006, 1663741, pp. 237-251, 21st Annual IEEE Conference on Computational Complexity, CCC 2006, Prague, Czech Republic, 7/16/06. https://doi.org/10.1109/CCC.2006.27
    Allender E, Hellerstein L, McCabe P, Pitassi T, Saks M. Inimizing DNF formulas and ACd 0 circuits given a truth table. In Proceedings - Twenty-First Annual IEEE Conference on Computational Complexity, CCC 2006. Vol. 2006. 2006. p. 237-251. 1663741 https://doi.org/10.1109/CCC.2006.27
    Allender, Eric ; Hellerstein, Lisa ; McCabe, Paul ; Pitassi, Toniann ; Saks, Michael. / Inimizing DNF formulas and ACd 0 circuits given a truth table. Proceedings - Twenty-First Annual IEEE Conference on Computational Complexity, CCC 2006. Vol. 2006 2006. pp. 237-251
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