Assigning Balls to Bins with Constraints on Which Ball to Go to Which Bin?
You can reduce the Exact Cover problem to your problem. The elements in the Exact Cover problem corresponds to the balls in your problem. For each subset $T$ in the Exact Cover problem, we construct a bin $i$ with $S_i=\{\emptyset, T\}$. Then there exists an exact cover if and only if all balls can be put into the bins. Hence, your problem is NP-hard.