I am wondering if there are any results on the following version of the assignment problem. We are given a set of jobs $J$ and a set of workers $W$, and for each job $j$ and worker $w$ we are given the expertise of the worker for the job $\omega(w,j)$. The goal is to select a subset of jobs $S\subseteq J$ and assign exactly two workers to each job $j \in S$ while maximizing the total expertise.
That is, if $x_{w, j} = 1$ indicates that the worker $w$ has been assigned to job $j$, solve the following program:
$$ maximize \sum_{j \in S} x_{w, j} \cdot \omega(w, j) $$ s.t. $$ \sum_{j\in S} x_{w, j} \leq 1, \forall w\in W $$ $$ \sum_{w\in W} x_{w, j} = 2, \forall j \in S $$ $$ x_{w, j} \in \{0, 1\} $$
Edit:
It seems like my explanation is not very clear so I am adding a simple example. Consider the following graph where dotted edges have weight 0 and solid edges have weight 1. Worker $i$ has expertise $1$ in job $i$ and expertise $0$ in the other job.
The two possible solutions both have value 1: either assign both workers to job 1, or assign both workers to job 2. Observe that assigning worker 1 to job 1 and worker 2 to job 2 would result in a solution of value 2, but it is not a valid solution since a job must be assigned exactly 2 workers.
