On finding short resolution refutations and small unsatisfiable subsets

We consider the parameterized problems of whether a given set of clauses can be refuted within k resolution steps, and whether a given set of clauses contains an unsatisfiable subset of size at most k. We show that both problems are complete for the class W[1], the first level of the W-hierarchy of fixed-parameter intractable problems. Our results remain true if restricted to 3-SAT instances and/or to various restricted versions of resolution including tree-like resolution, input resolution, and read-once resolution. Applying a metatheorem of Frick and Grohe, we show that, restricted to classes of sets of clauses of locally bounded treewidth, the considered problems are fixed-parameter tractable. For example, the problems are fixed-parameter tractable for planar CNF formulas.