Strong backdoors to nested satisfiability

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. © 2012 Springer-Verlag.