  Query evaluation over probabilistic databases is a notoriously intractable
problem -- not only in combined complexity, but for many natural queries in
data complexity as well. This motivates the study of probabilistic query
evaluation through the lens of approximation algorithms, and particularly of
combined FPRASes, whose runtime is polynomial in both the query and instance
size. In this paper, we focus on tuple-independent probabilistic databases over
binary signatures, which can be equivalently viewed as probabilistic graphs. We
study in which cases we can devise combined FPRASes for probabilistic query
evaluation in this setting.
  We settle the complexity of this problem for a variety of query and instance
classes, by proving both approximability and (conditional) inapproximability
results. This allows us to deduce many corollaries of possible independent
interest. For example, we show how the results of Arenas et al. on counting
fixed-length strings accepted by an NFA imply the existence of an FPRAS for the
two-terminal network reliability problem on directed acyclic graphs: this was
an open problem until now. We also show that one cannot extend a recent result
of van Bremen and Meel that gives a combined FPRAS for self-join-free
conjunctive queries of bounded hypertree width on probabilistic databases:
neither the bounded-hypertree-width condition nor the self-join-freeness
hypothesis can be relaxed. Finally, we complement all our inapproximability
results with unconditional lower bounds, showing that DNNF provenance circuits
must have at least moderately exponential size in combined complexity.
