The problem: A company has $n$ roles to be filled. There are $m$ candidates. Each candidate $c$ the company hires may fulfill one or more roles. Each assignment of a candidate $c$ to a role $r$ produces a profit $p_{c,r}$. Each candidate $c$ has a fixed salary requirement, $S_c$. The salary requirement $S_c$ does not depend on the number of roles or type of role that candidate $c$ is assigned. There is a total salary budget, $B$. The sum of salaries of candidates who are hired cannot exceed $B$. Find an assignment of candidates to roles that maximizes profit without violating the budget constraint.
My question: is there a name for this problem, and if so, where can I find resources about its solution?
It is very similar to the generalized assignment problem, but in the GAP, each candidate has a budget, e.g. a time budget, for completing tasks / fulfilling roles. In this problem, the candidate's time budget is replaced with a salary budget for hiring candidates.
Let binary decision variable $x_{cr}$ indicate whether candidate $c$ fulfills role $r$, and let binary decision variable $y_c$ indicate whether candidate $c$ is hired. The problem is to maximize $$\sum_{c,r} p_{cr} x_{cr}$$ subject to \begin{align} \sum_c x_{cr} &= 1 &&\text{for all $r$} \tag1 \\ x_{cr} &\le y_c &&\text{for all $c,r$} \tag2 \\ \sum_c S_c y_c &\le B \tag3 \end{align} Constraint $(1)$ forces each role to be fulfilled by exactly one candidate. Constraint $(2)$ forces candidate $c$ to be hired if that candidate fulfills any role. Constraint $(3)$ enforces the salary budget.
This is like a budget-constrained uncapacitated facility location problem where the candidates are facilities and the roles are customers, each with demand $1$. (Negate the objective function to switch between minimization and maximization.) In the special case $S_c \equiv 1$, this is the uncapacitated $p$-median problem.