Acta Informatica

, Volume 44, Issue 7–8, pp 509–523 | Cite as

Solving #SAT using vertex covers

  • Naomi Nishimura
  • Prabhakar Ragde
  • Stefan Szeider
Original Article


We propose an exact algorithm for counting the models of propositional formulas in conjunctive normal form. Our algorithm is based on the detection of strong backdoor sets of bounded size; each instantiation of the variables of a strong backdoor set puts the given formula into a class of formulas for which models can be counted in polynomial time. For the backdoor set detection we utilize an efficient vertex cover algorithm applied to a certain “obstruction graph” that we associate with the given formula. This approach gives rise to a new hardness index for formulas, the clustering-width. Our algorithm runs in uniform polynomial time on formulas with bounded clustering-width. It is known that the number of models of formulas with bounded clique-width, bounded treewidth, or bounded branchwidth can be computed in polynomial time; these graph parameters are applied to formulas via certain (hyper)graphs associated with formulas. We show that clustering-width and the other parameters mentioned are incomparable: there are formulas with bounded clustering-width and arbitrarily large clique-width, treewidth, and branchwidth. Conversely, there are formulas with arbitrarily large clustering-width and bounded clique-width, treewidth, and branchwidth.


Base Class Vertex Cover Conjunctive Normal Form Truth Assignment Primal Graph 
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-Verlag 2007

Authors and Affiliations

  • Naomi Nishimura
    • 1
  • Prabhakar Ragde
    • 1
  • Stefan Szeider
    • 2
  1. 1.School of Computer ScienceUniversity of WaterlooWaterlooCanada
  2. 2.Department of Computer ScienceDurham UniversityDurhamUK

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