  Constraint satisfaction problems have been studied in numerous fields with
practical and theoretical interests. In recent years, major breakthroughs have
been made in a study of counting constraint satisfaction problems (or #CSPs).
In particular, a computational complexity classification of bounded-degree
#CSPs has been discovered for all degrees except for two, where the "degree" of
an input instance is the maximal number of times that each input variable
appears in a given set of constraints. Despite the efforts of recent studies,
however, a complexity classification of degree-2 #CSPs has eluded from our
understandings. This paper challenges this open problem and gives its partial
solution by applying two novel proof techniques--T_{2}-constructibility and
parametrized symmetrization--which are specifically designed to handle
"arbitrary" constraints under randomized approximation-preserving reductions.
We partition entire constraints into four sets and we classify the
approximation complexity of all degree-2 #CSPs whose constraints are drawn from
two of the four sets into two categories: problems computable in
polynomial-time or problems that are at least as hard as #SAT. Our proof
exploits a close relationship between complex-weighted degree-2 #CSPs and
Holant problems, which are a natural generalization of complex-weighted #CSPs.
