On the Boolean connectivity problem for Horn relations

  • Authors:

  • Kazuhisa Makino;Suguru Tamaki;Masaki Yamamoto

  • Affiliations:

  • Graduate School of Information Science and Technology, University of Tokyo, Tokyo, 113-8656, Japan;Graduate School of Informatics, Kyoto University, Kyoto, 606-8501, Japan;Department of Mathematical Sciences, School of Science, Tokai University, Kanagawa, 259-1292, Japan

  • Venue:
  • Discrete Applied Mathematics
  • Year:
  • 2010

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Abstract

Gopalan et al. studied in [P. Gopalan, P.G. Kolaitis, E.N. Maneva, C.H. Papadimitriou, The connectivity of Boolean satisfiability: computational and structural dichotomies, in: Proceedings of the 33rd International Colloquium on Automata, Languages and Programming, ICALP 2006, 2006, pp. 346-357] and [P. Gopalan, P.G. Kolaitis, E.N. Maneva, C.H. Papadimitriou, The connectivity of Boolean satisfiability: computational and structural dichotomies, SIAM J. Comput. 38 (6) (2009) 2330-2355] 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 first conjectured that the connectivity problem for Schaefer is in P. We disprove their conjecture by showing that there exists a set S of Horn relations such that the connectivity problem for S is coNP-complete. We also investigate a tractable aspect of Horn and dual Horn relations with respect to characteristic sets.