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

An Efficient $\varepsilon$-BIC to BIC Transformation and Its Application to Black-Box Reduction in Revenue Maximization