Minimal unsatisfiable formulas with bounded clause-variable difference are fixed-parameter tractableJournal of Computer and System Sciences, vol. 69, no. 4, pp. 656-674, 2004. Download: [paper ps] [paper pdf] Abstract:The deficiency of a propositional formula F in CNF with n variables and m clauses is defined as m-n. It is known that minimal unsatisfiable formulas (unsatisfiable formulas which become satisfiable by removing any clause) have positive deficiency. Recognition of minimal unsatisfiable formulas is NP-hard, and it was shown recently that minimal unsatisfiable formulas with deficiency k can be recognized in time n^{O(k)}. We improve this result and present an algorithm with time complexity O(2^k n^4). Whence the problem is fixed-parameter tractable in the sense of R.G. Downey and M.R. Fellows, Parameterized Complexity, Springer, New York, 1999.
Our algorithm gives raise to a fixed-parameter tractable
parameterization of the satisfiability problem: If the maximum
deficiency over all subsets of a formula F is at most k, then we can
decide in time O(2^k n^3) whether F is satisfiable (and we certify the
decision by providing either a satisfying truth assignment or a
regular resolution refutation). Known parameters for fixed-parameter
tractable satisfiability decision are tree-width or related to
tree-width. In contrast to tree-width (which is NP-hard to compute)
the maximum deficiency can be calculated efficiently by graph matching
algorithms. We exhibit an infinite class of formulas where maximum
deficiency outperforms tree-width (and related parameters), as well as
an infinite class where the converse prevails.
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