Minimizing the number of tardy jobs with precedence constraints and agreeable due dates Original Research Article

Minimizing the number of precedence constrained, unit-time tardy jobs is strongly NP-hard on a single machine. We study a special case of the problem where a job is tardy if it is finished more than a fixed K time units after its earliest possible completion time under the precedence constraints. We prove that the problem remains strongly NP-hard even with these special due dates. We also present polynomial time solutions for the weighted version of the problem if the precedence constraints are out-forests or interval orders. In the process, we also present a polynomial time solution for a special case of the minimum weight hitting set problem.