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International Workshop on Algorithms and Computation

WALCOM 2017: WALCOM: Algorithms and Computation pp 359–369Cite as

An Upper Bound for Resolution Size: Characterization of Tractable SAT Instances

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10167))

Abstract

We show the first upper bound for resolution size of a SAT instance by pathwidth of its incidence graph. Namely, we prove that if an incidence graph of an unsatisfiable CNF formula has pathwidth \({\mathrm {pw}}\), the formula can be refuted by a resolution proof with at most \(O^*(3^{\mathrm {pw}})\) clauses. It is known that modern practical SAT-solvers run efficiently for instances which have small and narrow resolution refutations. Resolution size is one of the parameters which make SAT tractable, whereas it is shown that even linearly approximating the resolution size is NP-hard. In contrast, computing graph based parameters such as treewidth or pathwidth is fixed-parameter tractable, and also efficient FPT algorithms for SAT of bounded such parameters are widely researched. However, few explicit connection between these parameters and resolutions or SAT-solvers are known. In this paper, we provide an FPT algorithm for SAT on path decomposition of its incidence graph. The algorithm can construct resolution refutations for unsatisfiable formulas, and analyzing the size of constructed proof gives the new bound.

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Acknowledgements

I would like to thank Hiroshi Imai for helpful advice for writing this paper, and Yoichi Iwata for motivating this work by providing a lot of information about SAT, CDCL, and other related researches.

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Authors and Affiliations

  1. Department of Computer Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo, 113-8656, Japan

    Kensuke Imanishi

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  1. Kensuke Imanishi
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Correspondence to Kensuke Imanishi .

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Editors and Affiliations

  1. Universiti Teknologi Brunei, Bandar Seri Begawan, Brunei Darussalam

    Sheung-Hung Poon

  2. Bangladesh University of Engineering and Technology, Dhaka, Bangladesh

    Md. Saidur Rahman

  3. National Taiwan University, Taipei, Taiwan

    Hsu-Chun Yen

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Imanishi, K. (2017). An Upper Bound for Resolution Size: Characterization of Tractable SAT Instances. In: Poon, SH., Rahman, M., Yen, HC. (eds) WALCOM: Algorithms and Computation. WALCOM 2017. Lecture Notes in Computer Science(), vol 10167. Springer, Cham. https://doi.org/10.1007/978-3-319-53925-6_28

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  • DOI: https://doi.org/10.1007/978-3-319-53925-6_28

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-53924-9

  • Online ISBN: 978-3-319-53925-6

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