  We study the complexity of approximating the number of answers to a small
query $\varphi$ in a large database $\mathcal{D}$. We establish an exhaustive
classification into tractable and intractable cases if $\varphi$ is a
conjunctive query with disequalities and negations:
  $\bullet$ If there is a constant bound on the arity of $\varphi$, and if the
randomised Exponential Time Hypothesis (rETH) holds, then the problem has a
fixed-parameter tractable approximation scheme (FPTRAS) if and only if the
treewidth of $\varphi$ is bounded.
  $\bullet$ If the arity is unbounded and we allow disequalities only, then the
problem has an FPTRAS if and only if the adaptive width of $\varphi$ (a width
measure strictly more general than treewidth) is bounded; the lower bound
relies on the rETH as well.
  Additionally we show that our results cannot be strengthened to achieve a
fully polynomial randomised approximation scheme (FPRAS): We observe that,
unless $\mathrm{NP} =\mathrm{RP}$, there is no FPRAS even if the treewidth (and
the adaptive width) is $1$. However, if there are neither disequalities nor
negations, we prove the existence of an FPRAS for queries of bounded fractional
hypertreewidth, strictly generalising the recently established FPRAS for
conjunctive queries with bounded hypertreewidth due to Arenas, Croquevielle,
Jayaram and Riveros (STOC 2021).
