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On the Boolean Connectivity Problem for Horn Relations

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Book cover Theory and Applications of Satisfiability Testing – SAT 2007 (SAT 2007)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 4501))

Abstract

Gopalan et al. studied in ICALP06 [17] connectivity properties of the solution-space of Boolean formulas, and investigated complexity issues on the connectivity problems in Schaefer’s framework. A set S of logical relations is Schaefer if all relations in S are either bijunctive, Horn, dual Horn, or affine. They conjectured that the connectivity problem for Schaefer is in \(\mathcal{P}\). We disprove their conjecture by showing that there exists a set S of Horn relations such that the connectivity problem for S is co\(\mathcal{NP}\)-complete. We also show that the connectivity problem for bijunctive relations can be solved in O( min {n|ϕ|, T(n)}) time, where n denotes the number of variables, ϕ denotes the corresponding 2-CNF formula, and T(n) denotes the time needed to compute the transitive closure of a directed graph of n vertices. Furthermore, we investigate a tractable aspect of Horn and dual Horn relations with respect to characteristic sets.

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Authors and Affiliations

  1. Graduate School of Information Science and Technology, University of Tokyo, Tokyo, 113-8656, Japan

    Kazuhisa Makino

  2. Graduate School of Informatics, Kyoto University, Kyoto, 606-8501, Japan

    Suguru Tamaki & Masaki Yamamoto

Authors
  1. Kazuhisa Makino
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  2. Suguru Tamaki
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  3. Masaki Yamamoto
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João Marques-Silva Karem A. Sakallah

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Makino, K., Tamaki, S., Yamamoto, M. (2007). On the Boolean Connectivity Problem for Horn Relations. In: Marques-Silva, J., Sakallah, K.A. (eds) Theory and Applications of Satisfiability Testing – SAT 2007. SAT 2007. Lecture Notes in Computer Science, vol 4501. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-72788-0_20

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  • DOI: https://doi.org/10.1007/978-3-540-72788-0_20

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-72787-3

  • Online ISBN: 978-3-540-72788-0

  • eBook Packages: Computer ScienceComputer Science (R0)

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