Multi-agent scheduling on a single machine to minimize total weighted number of tardy jobs

We consider the feasibility model of multi-agent scheduling on a single machine, where each agent's objective function is to minimize the total weighted number of tardy jobs. We show that the problem is strongly NP-complete in general. When the number of agents is fixed, we first show that the problem can be solved in pseudo-polynomial time for integral weights, and can be solved in polynomial time for unit weights; then we present a fully polynomial-time approximation scheme for the problem.