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Strong Backdoors to Nested Satisfiability

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Theory and Applications of Satisfiability Testing – SAT 2012 (SAT 2012)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 7317))

Abstract

Knuth (1990) introduced the class of nested formulas and showed that their satisfiability can be decided in polynomial time. We show that, parameterized by the size of a smallest strong backdoor set to the base class of nested formulas, computing the number of satisfying assignments of any CNF formula is fixed-parameter tractable. Thus, for any k > 0, the satisfiability problem can be solved in polynomial time for any formula F for which there exists a set B of at most k variables such that for every truth assignment τ to B, the reduced formula F[τ] is nested; moreover, the degree of the polynomial is independent of k.

Our algorithm uses the grid-minor theorem of Robertson and Seymour (1986) to either find that the incidence graph of the formula has bounded treewidth—a case that is solved by model checking for monadic second order logic—or to find many vertex-disjoint obstructions in the incidence graph. For the latter case, new combinatorial arguments are used to find a small backdoor set. Combining both cases leads to an approximation algorithm producing a strong backdoor set whose size is upper bounded by a function of the optimum. Going through all assignments to this set of variables and using Knuth’s algorithm, the satisfiability of the input formula can be decided. With a similar approach, one can also count the number of satisfying assignments of the given formula.

The full version of the paper is available on arXiv [16].

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Author information

Authors and Affiliations

  1. School of Computer Science and Engineering, The University of New South Wales, Sydney, Australia

    Serge Gaspers

  2. Institute of Information Systems, Vienna University of Technology, Vienna, Austria

    Serge Gaspers & Stefan Szeider

Authors
  1. Serge Gaspers
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  2. Stefan Szeider
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Editor information

Editors and Affiliations

  1. Center for Information Technology, Fondazione Bruno Kessler, via Sommarive 18, Povo, 38123, Trento, Italy

    Alessandro Cimatti

  2. Department of Information Engineering and Computer Science, University of Trento, via Sommarive 14, Povo, 38123, Trento, Italy

    Roberto Sebastiani

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Gaspers, S., Szeider, S. (2012). Strong Backdoors to Nested Satisfiability. In: Cimatti, A., Sebastiani, R. (eds) Theory and Applications of Satisfiability Testing – SAT 2012. SAT 2012. Lecture Notes in Computer Science, vol 7317. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31612-8_7

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  • DOI: https://doi.org/10.1007/978-3-642-31612-8_7

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-31611-1

  • Online ISBN: 978-3-642-31612-8

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