Let $A=\left\{ a_{i}\right\} $ be a sequence of $n$ positive numbers such that $\sum a_{i}=1$. We define $C\left(A\right)=\left\{ \left\{ b_{i}\right\} \subset\left\{ 1,2..,n\right\} :\sum a_{b_{i}}=\frac{1}{2}\right\} $ . We say that a set $B\subset2^{\left\{ 1,2,..,n\right\} }$ is feasible, if and only if there is some $A$, such that $B=C\left(A\right)$.
Which are sufficient and necessary conditions for characterizing feasible sets?
Thanks for your attention.
I think that a nice characterization is unlikely to exist, since the decision version of this problem, i.e., the problem of deciding whether a feasible set exists seems to be equivalent to the partition problem.