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Consider a central problem in randomized approximation schemes that use aMonte Carlo approach. Given a sequence of independent, identically distributedrandom variables $X_1,X_2,\ldots$ with mean $\mu$ and standard deviation atmost $c \mu$, where $c$ is a known constant, and $\epsilon,\delta > 0$, createan estimate $\hat \mu$ for $\mu$ such that $\text{P}(|\hat \mu - \mu| >\epsilon \mu) \leq \delta$. This technique has been used for buildingrandomized approximation schemes for the volume of a convex body, the permanentof a nonnegative matrix, the number of linear extensions of a poset, thepartition function of the Ising model and many other problems. Existing methodsuse (to the leading order) $19.35 (c/\epsilon)^2 \ln(\delta^{-1})$ samples.This is the best possible number up to the constant factor, and it is an openquestion as to what is the best constant possible. This work gives an easy toapply estimate that only uses $6.96 (c/\epsilon)^2 \ln(\delta^{-1})$ samples inthe leading order.

Improving Monte Carlo randomized approximation schemes