Parameterized complexity of constraint satisfaction problems

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 (TPT,) or W[l]-complete. We give a simple characterization of those constraints that make the problem fixed-parameter tractable. The special cases when the formula is restricted to be bounded occurrence, bounded treewidth or planar are also considered, it turns out that in these cases the problem is in FPT for every constraint family, F.