Suppose we have $k$ jobs, $J=\left\{ j_{1},\ldots j_{k}\right\}$ and $n$ agents, $A=\left\{ a_{1},\ldots a_{n}\right\} $. Each assignment has associated with it a subset of agents which can perform it, this information given by some $f:J\to2^{A}$ and each agent has a cost associated with using him given by $c:A\to\mathbb{N}$. We are looking for an assignment $g:J\to A$ ($g$ must be legal in the sense that $g\left(j\right)\in f\left(j\right)$ for all jobs) The cost function for an assignment $g$ is $\sum_{a\in Im\left(g\right)}c\left(a\right)$. So using any agent holds constant cost no matter how many assignments he ends performing.
Formulated as a decision problem, we have input $\left\langle J,A,f,c,k\right\rangle$ and the problem is deciding whether there exists a valid assignment with cost at most $k$.
The NP-hard problems I know about (i.e. Have proven myself) are
• SAT/3-SAT
• MAX-2-SAT
• Existence of a clique of size $k$ in an undirected graph
• Existence of Hamiltonian paths (All versions - with or without given start/end vertices, directed/undirected, path/cycle)
• Vertex cover of size $k$
• Dominating sets of size $k$
I have managed to prove reductions from the given problem to some of these, out of the hope that might give some insight for the reduction in the other way, but have not succeeded.
Answering my question
I ended up using SUBSET-SUM for the reduction. Observing that if we set the cost function $c=1$ for all agents, this is essentially a subset-sum problem. If someone sees a direct reduction from one of the problems on the list in the question I'd still like to see it.