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Solving #SAT using vertex covers

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Abstract

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.

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Authors and Affiliations

  1. School of Computer Science, University of Waterloo, Waterloo, ON, N2L 3G1, Canada

    Naomi Nishimura & Prabhakar Ragde

  2. Department of Computer Science, Durham University, Durham, DH1 3LE, UK

    Stefan Szeider

Authors
  1. Naomi Nishimura
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  2. Prabhakar Ragde
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  3. Stefan Szeider
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Correspondence to Naomi Nishimura.

Additional information

Research supported by the Natural Science and Engineering Research Council of Canada, the Nuffield Foundation (NAL/01012/G), and the Engineering and Physical Sciences Research Council of the UK (EP/E001394/1). A preliminary and shortened version of this paper appeared in the proceedings of SAT 2006, LNCS 4121, pp. 396–409 (2006)

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Nishimura, N., Ragde, P. & Szeider, S. Solving #SAT using vertex covers. Acta Informatica 44, 509–523 (2007). https://doi.org/10.1007/s00236-007-0056-x

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