We're given a set of items $A=\{1,\dots,m\}$ and a set of people $B=\{1,\dots,n\}$. Each person has a preference ordering for the items in $A$. Each item in $A$ has a specific positive cost for each specific person in $B$: item $i$'s cost to person $j$ is $c_{ij}$.
For any non-empty subset $T \subseteq A$ of items, we can imagine each person in $B$ choosing exactly one item in $T$: their top preference out of all the items in $T$ according to their preference ordering. (Different people can choose the same item.) Let $c(T)$ be the total cost of the items selected in this way.
Given the costs $c_{ij}$, I'd like to find a non-empty subset $T \subseteq A$ that makes $c(T)$ as small as possible. What is the complexity of this task?
It looks like a NP-Complete problem to me. It seems quite similar to a subset-sum or travel salesman problem, but I'm not sure. Do you have any suggestion for a reduction or a polynomial-time algorithm?
