We consider the black-box reduction from multidimensional revenue maximization to virtual welfare maximization. Cai et al. [12,13,14,15] show a polynomialtime approximation-preserving reduction, however, the mechanism produced by their reduction is only approximately Bayesian incentive compatible (ε-BIC). We provide two new polynomial time transformations that convert any ε-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 ε-BIC mechanism. Other ε-BIC to BIC transformations for revenue exist in the literature [23,36,18] 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. [12,13,14,15] to generate an exactly BIC mechanism.

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