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Trade-offs Between Time and Memory in a Tighter Model of CDCL SAT Solvers

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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

A long line of research has studied the power of conflict-driven clause learning (CDCL) and how it compares to the resolution proof system in which it searches for proofs. It has been shown that CDCL can polynomially simulate resolution even with an adversarially chosen learning scheme as long as it is asserting. However, the simulation only works under the assumption that no learned clauses are ever forgotten, and the polynomial blow-up is significant. Moreover, the simulation requires very frequent restarts, whereas the power of CDCL with less frequent or entirely without restarts remains poorly understood. With a view towards obtaining results with tighter relations between CDCL and resolution, we introduce a more fine-grained model of CDCL that captures not only time but also memory usage and number of restarts. We show how previously established strong size-space trade-offs for resolution can be transformed into equally strong trade-offs between time and memory usage for CDCL, where the upper bounds hold for CDCL without any restarts using the standard 1UIP clause learning scheme, and the (in some cases tightly matching) lower bounds hold for arbitrarily frequent restarts and arbitrary clause learning schemes.

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Notes

  1. 1.

    We mention for completeness that there is also a total space measure counting the number of literals in memory, which has been studied in, e.g., [1, 13, 15, 16], but for our purposes clause space seems like a more relevant measure to focus on.

  2. 2.

    In fact, our results hold for any UIP scheme, but for simplicity we focus on 1UIP, which is anyway dominant in practice.

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Acknowledgements

We are grateful to the anonymous SAT conference reviewers for detailed comments that helped improve the exposition in this paper.

The third author performed this work while at KTH Royal Institute of Technology, and most of the work of the second and fourth author was done while visiting KTH. The first, third, fifth, and sixth author were funded by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007–2013) / ERC grant agreement no. 279611 as well as by Swedish Research Council grant 621-2012-5645. The third author was also supported by the European Research Council under the European Union’s Horizon 2020 Research and Innovation Programme/ERC grant agreement no. 648276.

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Authors and Affiliations

  1. KTH Royal Institute of Technology, Stockholm, Sweden

    Jan Elffers, Jakob Nordström & Marc Vinyals

  2. Ludwig-Maximilians-Universität München, Munich, Germany

    Jan Johannsen

  3. Universitat Politècnica de Catalunya, Barcelona, Spain

    Massimo Lauria

  4. École Normale Supérieure, Paris, France

    Thomas Magnard

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  1. Jan Elffers
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Correspondence to Jakob Nordström .

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  1. Aix-Marseille Université, Marseille, France

    Nadia Creignou

  2. Université d'Artois, Lens, France

    Daniel Le Berre

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Elffers, J., Johannsen, J., Lauria, M., Magnard, T., Nordström, J., Vinyals, M. (2016). Trade-offs Between Time and Memory in a Tighter Model of CDCL SAT Solvers. 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_11

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  • DOI: https://doi.org/10.1007/978-3-319-40970-2_11

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