Meet of integer partitions
The problem is strongly NP-complete.

Reduction from 3-partition, a strongly NP-complete problem. The multiset $S$, $|S| = 3m$, $\sum_{x \in S} x = n$ can be partitioned into tuples of size three of equal sum if and only if $A \leq B$ where $A = S$ and $B$ is $m$ duplicates of $\frac{n}{m}$. If $A \leq B$, we can have $C = A$. If $A \not\leq B$, we must have $|C| &gt; |A|$, hence a 3-partition exists if and only if a set $C$ of size $|A|$ exists.

As for approximations, a trivial 2-approximation exists, since $\max(|A|, |B|) \leq |C| \leq |A| + |B|$. Construction: Build $C$ by always taking smallest integer in either of $A$ or $B$, then remove it from that set, and subtract it from an arbitrary integer in the other set. The value $|C| + |A| + |B|$ can only decrease over the course of the algorithm. This also shows that the problem is in NP.