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

Abstract

Recognition of minimal unsatisfiable CNF formulas (unsatisfiable CNF formulas which become satisfiable by removing any clause) is NP-hard; it was shown recently that minimal unsatisfiable formulas with n variables and n + k clauses can be recognized in time n O(k). We improve this result and present an algorithm with time complexity O(2kn4) —hence the problem turns out to be fixed-parameter tractable (FTP) in the sense of Downey and Fellows (Parameterized Complexity, Springer Verlag, 1999).Our algorithm gives rise to an FPT parameterization of SAT (“maximum deficiency”) which is incomparable with known FPT parameterizations of SAT like tree-width.