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Nonuniform Boolean constraint satisfaction problems with cardinality constraint

We study the computational complexity of Boolean constraint satisfaction problems with cardinality constraint. A Galois connection between clones and coclones has received a lot of attention in the context of complexity considerations for constraint satisfaction problems. This connection does not seem to help when considering constraint satisfaction problems that support in addition a cardinality constraint. We prove that a similar Galois connection, involving a weaker closure operator and partial polymorphisms, can be applied to such problems. Thus, we establish dichotomies for the decision as well as for the counting problems in Schaefer's framework.