Is it possible to efficiently find the elements that make up a sum $S$ given a set of number sets which determine the sum elements? It seems like an NP-Complete problem, though I might miss something.
Given $N$ equal sized (different) number sets $E_n$ ($0<n<N$) we sum $N$ elements $e_i$ such that $S = \sum^{N}_{i=0} e_i$ with $e_i \in E_i$.
Now, can we find $\{e_0, ..., e_N\}$ that make up $S$ given $S$ and $\{E_0,...,E_N\}$?
It seems closely related to the subset sum problem, though, there is slightly more information available. Can we exploit that to efficiently break down $S$ into the summed elements?