Assignment problem with multiple workers for each job 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.
This problem is a special case of the b-matching problem, and hence can be solved in polynomial time. Extensive information on b-matchings can be found for instance in the book:

László Lovász and Michael D. Plummer:
Matching Theory
ISBN-10: 0-8218-4759-7
ISBN-13: 978-0-8218-4759-6