  We study the complexity of approximate counting Constraint Satisfaction
Problems (#CSPs) in a bounded degree setting. Specifically, given a Boolean
constraint language $\Gamma$ and a degree bound $\Delta$, we study the
complexity of #CSP$_\Delta(\Gamma)$, which is the problem of counting
satisfying assignments to CSP instances with constraints from $\Gamma$ and
whose variables can appear at most $\Delta$ times. Our main result shows that:
(i) if every function in $\Gamma$ is affine, then #CSP$_\Delta(\Gamma)$ is in
FP for all $\Delta$, (ii) otherwise, if every function in $\Gamma$ is in a
class called IM$_2$, then for all sufficiently large $\Delta$,
#CSP$_\Delta(\Gamma)$ is equivalent under approximation-preserving (AP)
reductions to the counting problem #BIS (the problem of counting independent
sets in bipartite graphs) (iii) otherwise, for all sufficiently large $\Delta$,
it is NP-hard to approximate the number of satisfying assignments of an
instance of #CSP$_\Delta(\Gamma)$, even within an exponential factor. Our
result extends previous results, which apply only in the so-called
"conservative" case.
