Assignment Problem -- finding the $k$ agents with the best assignment I have a question that I have been thinking about. Suppose we have $n$ agents, $m$ tasks, a cost matrix with $M_{ij}$ being the cost of agent $i$ performing task $j$, and are given a value $k \leq n$. How can we find the $k$ unique agents, who when each optimally allocated a unique task, result in a minimum total cost? Can this be related to the assignment problem? Thank you very much for any guidance or assistance.
The assignment problem can be extended to solve this problem. The regular problem without the $k$ restriction can be solved by building a Minimum Cost Maximum Flow network is as follow:


We have a source $S$ a sink $T$ and the corresponding bipartite graph in the middle. Note that each edge has two values $f/c$ which denote maximum flow allowed through this edge and cost of each flow unit. Now, the problem is that we want to allow at most $k$ units of flow from $S$ to $T$. To do this, just duplicate node $S$ into $S_1$ and $S_2$ and put an edge among them such that cost of each unit is 0 (no penalty) and maximum flow allowed is $k$. The new graph will look like this:



The minimum cost maximum flow from $S_1$ to $T$ is the solution to the original problem for a fixed $k$.