An optimal (ϵ,δ)‐randomized approximation scheme for the mean of random variables with bounded relative variance

Abstract

Randomized approximation algorithms for many #P‐complete problems (such as the partition function of a Gibbs distribution, the volume of a convex body, the permanent of a {0,1}‐matrix, and many others) reduce to creating random variables X1,X2,… with finite mean μ and standard deviation σ such that μ is the solution for the problem input, and the relative standard deviation |σ/μ| ≤ c for known c. Under these circumstances, it is known that the number of samples from the {Xi} needed to form an (ϵ,δ)‐approximation that satisfies is at least . We present here an easy to implement (ϵ,δ)‐approximation that uses samples. This achieves the same optimal running time as other estimators, but without the need for extra conditions such as bounds on third or fourth moments.