  Model counting, or counting the satisfying assignments of a Boolean formula,
is a fundamental problem with diverse applications. Given #P-hardness of the
problem, developing algorithms for approximate counting is an important
research area. Building on the practical success of SAT-solvers, the focus has
recently shifted from theory to practical implementations of approximate
counting algorithms. This has brought to focus new challenges, such as the
design of auditable approximate counters that not only provide an approximation
of the model count, but also a certificate that a verifier with limited
computational power can use to check if the count is indeed within the promised
bounds of approximation.
  Towards generating certificates, we start by examining the best-known
deterministic approximate counting algorithm that uses polynomially many calls
to a $\Sigma_2^P$ oracle. We show that this can be audited via a $\Sigma_2^P$
oracle with the query constructed over $n^2 \log^2 n$ variables, where the
original formula has $n$ variables. Since $n$ is often large, we ask if the
count of variables in the certificate can be reduced -- a crucial question for
potential implementation. We show that this is indeed possible with a tradeoff
in the counting algorithm's complexity. Specifically, we develop new
deterministic approximate counting algorithms that invoke a $\Sigma_3^P$
oracle, but can be certified using a $\Sigma_2^P$ oracle using certificates on
far fewer variables: our final algorithm uses only $n \log n$ variables. Our
study demonstrates that one can simplify auditing significantly if we allow the
counting algorithm to access a slightly more powerful oracle. This shows for
the first time how audit complexity can be traded for complexity of approximate
counting.
