Instead of "sum-free" sets, consider sets where $S\subset S+S$. This is trivially satisfied when $0 \in S$. Is there a subset of $S$ with sum $0$?

Question

As an example, here is a set $S$ with $S\subset S+S$ and $|S|=8$ and having subsets of four elements whose sum is zero, but no smaller subsets have this property: $S=\{-14,-13,-11,-7,1,2,4,8\}$. This can be easily generalized to sets having $2n$ elements, containing the powers of two, up to $2^n$ and suitable negative elements, allowing subsets of $n$ elements with sum zero, but no smaller subsets have this property. In general, it is easy to find a sequence (or multi-set) of size at most $|S|/2$ having sum zero. So the question is if we could actually find a subset of $S$ with zero sum.