An exact algorithm for the Boolean connectivity problem for k-CNF

Authors:
Kazuhisa Makino;Suguru Tamaki;Masaki Yamamoto
Affiliations:
Graduate School of Information Science and Technology, University of Tokyo, Japan;Graduate School of Informatics, Kyoto University, Japan;Department of Informatics, Kwansei-Gakuin University, Japan
Venue:
Theoretical Computer Science
Year:
2011

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Abstract

We present an exact algorithm for a PSPACE-complete problem, denoted by CONNkSAT, which asks whether the solution space for a given k-CNF formula is connected on the n-dimensional hypercube. The problem is known to be PSPACE-complete for k=3, and polynomial solvable for k@?2 (Gopalan et al., 2009) [6]. We show that CONNkSAT for k=3 is solvable in time O((2-@e"k)^n) for some constant @e"k0, where @e"k depends only on k, but not on n. This result is considered to be interesting due to the following fact shown by Calabro [5]: QBF-3-SAT, which is a typical PSPACE-complete problem, is not solvable in time O((2-@e)^n) for any constant @e0, provided that the SAT problem (with no restriction to the clause length) is not solvable in time O((2-@e)^n) for any constant @e0.