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On the Hardness of SAT with Community Structure

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Theory and Applications of Satisfiability Testing – SAT 2016 (SAT 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9710))

Abstract

Recent attempts to explain the effectiveness of Boolean satisfiability (\(\mathsf {SAT}\)) solvers based on conflict-driven clause learning (CDCL) on large industrial benchmarks have focused on the concept of community structure. Specifically, industrial benchmarks have been empirically found to have good community structure, and experiments seem to show a correlation between such structure and the efficiency of CDCL. However, in this paper we establish hardness results suggesting that community structure is not sufficient to explain the success of CDCL in practice. First, we formally characterize a property shared by a wide class of metrics capturing community structure, including “modularity”. Next, we show that the \(\mathsf {SAT}\) instances with good community structure according to any metric with this property are still \(\mathsf {NP}\)-hard. Finally, we also prove that with high probability, random unsatisfiable modular instances generated from the “pseudo-industrial” community attachment model of Giráldez-Cru and Levy have exponentially long resolution proofs. Such instances are therefore hard for CDCL on average, indicating that actual industrial instances easily solved by CDCL may have some other relevant structure not captured by this model.

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Notes

  1. 1.

    Available in the full version [27].

  2. 2.

    Note that as required by its statement, we are applying Lemma 8 to formulas drawn from \(F_k(n,m)\), not to formulas drawn from \(\overline{F}_k(n,m,c,p;m')\). Lemmas 6 and 7 work for any formula, so we may use all three lemmas precisely as proved in [24].

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  27. Mull, N., Fremont, D.J., Seshia, S.A.: On the hardness of SAT with community structure. ArXiv e-prints (2016). http://arxiv.org/abs/1602.08620

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Acknowledgments

The authors thank Vijay Ganesh, Holger Hoos, Zack Newsham, Markus Rabe, Stefan Szeider, and several anonymous reviewers for helpful discussions and comments. This work is supported in part by the National Science Foundation Graduate Research Fellowship Program under Grant No. DGE-1106400, by the Hellman Family Faculty Fund, by gifts from Microsoft and Toyota, and by TerraSwarm, one of six centers of STARnet, a Semiconductor Research Corporation program sponsored by MARCO and DARPA.

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Authors and Affiliations

  1. University of California, Berkeley, USA

    Nathan Mull, Daniel J. Fremont & Sanjit A. Seshia

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  1. Nathan Mull
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  2. Daniel J. Fremont
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  3. Sanjit A. Seshia
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Correspondence to Nathan Mull , Daniel J. Fremont or Sanjit A. Seshia .

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Editors and Affiliations

  1. Aix-Marseille Université, Marseille, France

    Nadia Creignou

  2. Université d'Artois, Lens, France

    Daniel Le Berre

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Mull, N., Fremont, D.J., Seshia, S.A. (2016). On the Hardness of SAT with Community Structure. In: Creignou, N., Le Berre, D. (eds) Theory and Applications of Satisfiability Testing – SAT 2016. SAT 2016. Lecture Notes in Computer Science(), vol 9710. Springer, Cham. https://doi.org/10.1007/978-3-319-40970-2_10

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  • DOI: https://doi.org/10.1007/978-3-319-40970-2_10

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  • Online ISBN: 978-3-319-40970-2

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