Monte Carlo simulation techniques that use function approximations have been successfully applied to approximately price multi-dimensional American options. However, for many pricing problems the time required to obtain accurate estimates can still be prohibitive and this motivates the development of variance reduction techniques. In this paper, we describe a zero-variance or "perfect" control variate and a zero-variance or "perfect" importance sampling distribution to price American options. We also observe the natural connection of the perfect control variate to additive duality and the perfect importance sampling distribution to multiplicative duality in American options. We then discuss how function approximations may be used to approximate the perfect control variate and the perfect importance sampling distribution. Empirically, we observe that both techniques give significant variance reduction on examples of singleand multi-dimensional options.