Better Algorithms for Satisfiability Problems for Formulas of Bounded Rank-width

  • Authors:

  • Robert Ganian;Petr Hliněný;Jan Obdržálek

  • Affiliations:

  • Faculty of Informatics, Masaryk University, Botanická 68a, Brno, Czech Republic. {ganian, hlineny, obdrzalek}@fi.muni.cz;Faculty of Informatics, Masaryk University, Botanická 68a, Brno, Czech Republic. {ganian, hlineny, obdrzalek}@fi.muni.cz;Faculty of Informatics, Masaryk University, Botanická 68a, Brno, Czech Republic. {ganian, hlineny, obdrzalek}@fi.muni.cz

  • Venue:
  • Fundamenta Informaticae - MFCS & CSL 2010 Satellite Workshops: Selected Papers
  • Year:
  • 2013

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Abstract

We provide a parameterized algorithm for the propositional model counting problem #SAT, the runtime of which has a single-exponential dependency on the rank-width of the signed graph of a formula. That is, our algorithm runs in time $\cal{O}t^3 \cdot 2^{3tt+1/2} \cdot \vert\phi\vert$ for a width-t rank-decomposition of the input φ, and can be of practical interest for small values of rank-width. Previously, analogical algorithms have been known --e.g. [Fischer, Makowsky, and Ravve] --with a single-exponential dependency on the clique-width k of the signed graph of a formula with a given k-expression. Our algorithm presents an exponential runtime improvement over the worst-case scenario of the previous one, since clique-width reaches up to exponentially higher values than rankwidth. We also provide an algorithm for the MAX-SAT problem along the same lines.