On the Boolean Connectivity Problem for Horn Relations

  • Kazuhisa Makino
  • Suguru Tamaki
  • Masaki Yamamoto
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4501)


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.


Constraint Satisfaction Problem Transitive Closure Boolean Formula Disjunctive Normal Form Satisfying Assignment 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer Berlin Heidelberg 2007

Authors and Affiliations

  • Kazuhisa Makino
    • 1
  • Suguru Tamaki
    • 2
  • Masaki Yamamoto
    • 2
  1. 1.Graduate School of Information Science and Technology, University of Tokyo, Tokyo, 113-8656Japan
  2. 2.Graduate School of Informatics, Kyoto University, Kyoto, 606-8501Japan

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