combinatorics – Partitioning a set with minimum weight

Let $$X$$ be a finite collection of natural numbers.($$X$$ is a multiset.)
Consider $$R$$ as the set of all $$(A,B)$$ such that $$X$$ is disjoint union of $$A$$ and $$B$$.
We now assume that $$S$$ be the set of elements $$(A,B)$$ of $$R$$ in such a way that $$Sigma_{ain A} a$$ is minimum among all elements of $$R$$.

What is the lower bound for $$(A,B)in S$$ in terms of the minimum of $$X$$?