Abstract
We prove a parameterized analog of Schaefer?s Dichotomy Theorem: we show that for every finite boolean constraint family F, deciding whether a formula containing constraints from F has a satisfying assignment of weight exactly k is either fixed-parameter tractable (FPT) or W[1]-complete. We give a simple characterization of those constraints that make the problem fixedparameter tractable. The special cases when the formula is restricted to be bounded occurrence, bounded tree-width or planar are also considered, it turns out that in these cases the problem is in FPT for every constraint family F.