  Constraint satisfaction problems (or CSPs) have been extensively studied in,
for instance, artificial intelligence, database theory, graph theory, and
statistical physics. From a practical viewpoint, it is beneficial to
approximately solve those CSPs. When one tries to approximate the total number
of truth assignments that satisfy all Boolean-valued constraints for
(unweighted) Boolean CSPs, there is a known trichotomy theorem by which all
such counting problems are neatly classified into exactly three categories
under polynomial-time (randomized) approximation-preserving reductions. In
contrast, we obtain a dichotomy theorem of approximate counting for
complex-weighted Boolean CSPs, provided that all complex-valued unary
constraints are freely available to use. It is the expressive power of free
unary constraints that enables us to prove such a stronger, complete
classification theorem. This discovery makes a step forward in the quest for
the approximation-complexity classification of all counting CSPs. To deal with
complex weights, we employ proof techniques of factorization and arity
reduction along the line of solving Holant problems. Moreover, we introduce a
novel notion of T-constructibility that naturally induces
approximation-preserving reducibility. Our result also gives an approximation
analogue of the dichotomy theorem on the complexity of exact counting for
complex-weighted Boolean CSPs.
