Let $A$ and $B$ be two sets with $|A| \ge |B|$, and let $f: C \to \mathbb{R}$ be a nonnegative function with $C \subseteq A×B$. Furthermore, let $D \subseteq C$ be a matching in $C$ such that $D$ provides all the elements of $B$ (through their ordered pairs). How to get
$$D^{*}=\underset{D \subseteq C}{\mathrm{argmin}} \sum_{d \in D}f(d)?$$
Similar question has been raised in A variant of assignment problem (different sizes of sets), but the unequal size problem has not been asnwered.