  A unary constraint (on the Boolean domain) is a function from {0,1} to the
set of real numbers. A free use of auxiliary unary constraints given besides
input instances has proven to be useful in establishing a complete
classification of the computational complexity of approximately solving
weighted counting Boolean constraint satisfaction problems (or #CSPs). In
particular, two special constant unary constraints are a key to an arity
reduction of arbitrary constraints, sufficient for the desired classification.
In an exact counting model, both constant unary constraints are always assumed
to be available since they can be eliminated efficiently using an arbitrary
nonempty set of constraints. In contrast, we demonstrate in an approximate
counting model, that at least one of them is efficiently approximated and thus
eliminated approximately by a nonempty constraint set. This fact directly leads
to an efficient construction of polynomial-time randomized
approximation-preserving Turing reductions (or AP-reductions) from #CSPs with
designated constraints to any given #CSPs composed of symmetric real-valued
constraints of arbitrary arities even in the presence of arbitrary extra unary
constraints.
