  The degree of a CSP instance is the maximum number of times that a variable
may appear in the scope of constraints. We consider the approximate counting
problem for Boolean CSPs with bounded-degree instances, for constraint
languages containing the two unary constant relations {0} and {1}. When the
maximum degree is at least 25 we obtain a complete classification of the
complexity of this problem. It is exactly solvable in polynomial-time if every
relation in the constraint language is affine. It is equivalent to the problem
of approximately counting independent sets in bipartite graphs if every
relation can be expressed as conjunctions of {0}, {1} and binary implication.
Otherwise, there is no FPRAS unless NP=RP. For lower degree bounds, additional
cases arise in which the complexity is related to the complexity of
approximately counting independent sets in hypergraphs.
