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Evaluation of Monotone DNF Formulas

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Abstract

Stochastic boolean function evaluation (SBFE) is the problem of determining the value of a given boolean function f on an unknown input x, when each bit \(x_i\) of x can only be determined by paying a given associated cost \(c_i\). Further, x is drawn from a given product distribution: for each \(x_i\), \(\mathbf{Pr}[x_i=1] = p_i\) and the bits are independent. The goal is to minimize the expected cost of evaluation. In this paper, we study the complexity of the SBFE problem for classes of DNF formulas. We consider both exact and approximate versions of the problem for subclasses of DNF, for arbitrary costs and product distributions, and for unit costs and/or the uniform distribution.

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Notes

  1. We note that Kaplan et al. actually define the problem slightly differently. They require the evaluation strategy to output a proof of the function value upon termination. In the case of a tautological DNF formula, they would require testing of the variables in one term of the DNF in order to output that term as a certificate.

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Acknowledgments

Sarah R. Allen was partially supported by an NSF Graduate Research Fellowship under Grant 0946825 and by NSF Grant CCF-1116594. Lisa Hellerstein was partially supported by NSF Grants 1217968 and 0917153. Devorah Kletenik was partially supported by NSF Grant 0917153. Tonguç Ünlüyurt was partially supported by TUBITAK 2219 programme. Part of this research was performed while Tonguç Ünlüyurt was visiting faculty at the NYU School of Engineering and Sarah Allen and Devorah Kletenik were students there. We thank anonymous referees for their helpful suggestions.

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Authors and Affiliations

  1. Carnegie Mellon University, 5000 Forbes Ave, Pittsburgh, PA, 15213, USA

    Sarah R. Allen

  2. NYU School of Engineering, 2 Metrotech Center, Brooklyn, NY, 11201, USA

    Lisa Hellerstein

  3. Brooklyn College, 2900 Bedford Avenue, Brooklyn, NY, 11210, USA

    Devorah Kletenik

  4. Sabanci University, Orhanli, Tuzla, 34956, Istanbul, Turkey

    Tonguç Ünlüyurt

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  1. Sarah R. Allen
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  2. Lisa Hellerstein
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  3. Devorah Kletenik
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  4. Tonguç Ünlüyurt
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Correspondence to Devorah Kletenik.

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Allen, S.R., Hellerstein, L., Kletenik, D. et al. Evaluation of Monotone DNF Formulas. Algorithmica 77, 661–685 (2017). https://doi.org/10.1007/s00453-015-0092-9

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