  We consider the black-box reduction from multi-dimensional revenue
maximization to virtual welfare maximization. Cai et al. show a polynomial-time
approximation-preserving reduction, however, the mechanism produced by their
reduction is only approximately Bayesian incentive compatible
($\varepsilon$-BIC). We provide two new polynomial time transformations that
convert any $\varepsilon$-BIC mechanism to an exactly BIC mechanism with only a
negligible revenue loss.
  Our first transformation applies to any mechanism design setting with
downward-closed outcome space and only requires sample access to the agents'
type distributions. Our second transformation applies to the fully general
outcome space, removing the downward-closed assumption, but requires full
access to the agents' type distributions. Both transformations only require
query access to the original $\varepsilon$-BIC mechanism. Other
$\varepsilon$-BIC to BIC transformations for revenue exist in the literature
but all require exponential time to run in both of the settings we consider. As
an application of our transformations, we improve the reduction by Cai et al.
to generate an exactly BIC mechanism.
