Online Deposited in DRO : 14

Recognition of minimal unsatisfiable CNF formulas (unsatisfiable CNF formulas which become satisfiable if any clause is removed) is a classical DP -complete problem. It was shown recently that minimal unsatisfiable formulas with n variables and n + k clauses can be recognized in time nO(k). We improve this result and present an algorithm with time complexity O(2kn4); hence the problem turns out to be fixed-parameter tractable (FPT) in the sense of Downey and Fellows (Parameterized Complexity, 1999). Our algorithm gives rise to a fixed-parameter tractable parameterization of the satisfiability problem: If for a given CNF formula F , the number of clauses in each of its subsets exceeds the number of variables occuring in the subset at most by k, then we can decide in time O(2kn3) whether F is satisfiable; k is called the maximum deficiency of F and can be efficiently computed by means of graph matching algorithms. Known parameters for fixed-parameter tractable satisfiability decision are tree-width or related to tree-width. Tree-width and maximum deficiency are incomparable in the sense that we can find formulas with constant maximum deficiency and arbitrarily high tree-width, and formulas where the converse prevails.