Assume you have $n$ independent binary variables $\{x_1,\dots,x_n\}$ and for each variable $x_i$ you know that its value is equal to $1$ with a probability $p_i$. I would like to enumerate the joint assignments of these variables in such a way that in $k$ steps I capture as much volume of the joint probability mass function as possible. Does there exists some approximation algorithm which runs faster than the algorithm which enumerates the assignments according to their joint probability $\Pi p_i$. Or something related?