Minimal Unsatisfiable Formulas with Bounded Clause-Variable Difference are Fixed-Parameter Tractable

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. We improve this result and present an algorithm with time complexity O(2n). 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 rise 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(2n) 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.