Strong backdoors to nested satisfiability

  • Authors:

  • Serge Gaspers;Stefan Szeider

  • Affiliations:

  • School of Computer Science and Engineering, The University of New South Wales, Sydney, Australia, Institute of Information Systems, Vienna University of Technology, Vienna, Austria;Institute of Information Systems, Vienna University of Technology, Vienna, Austria

  • Venue:
  • SAT'12 Proceedings of the 15th international conference on Theory and Applications of Satisfiability Testing
  • Year:
  • 2012

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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 k0, 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.