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We consider the black-box reduction from multi-dimensional revenue maximization to virtual welfare maximization. Cai et al. [12, 13, 14, 15] show a polynomial-time approximation-preserving re-duction, however, the mechanism produced by their reduction is only approximately Bayesian incentive compatible (ε-BIC). We provide two new polynomial time transformations that convert anyε-BICmechanism to an exactly BIC mechanism with only a negligible revenue loss. (i) Our first transformation applies to any mechanism design setting with downward-closed outcome space and only requires sample accessto the agents’ type distributions. (ii) 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 originalε-BIC mechanism. Otherε-BIC to BICtransformations for revenue exist in the literature but all require exponential time to run in either 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.