A Parametrized Complexity View on Robust Scheduling with Budgeted Uncertainty
In this study, we investigate a robust single-machine scheduling problem under processing time uncertainty. The uncertainty is modeled using the budgeted approach, where each job has a nominal and deviation processing time, and the number of deviations is bounded by Γ. The objective is to minimize the number of tardy jobs where a job is considered tardy if there is some scenario in which it is completed after its due date. Since the problem is NP-hard in general, we focus on analyzing its tractability under the assumption that certain natural parameters of the problem are each bounded by a constant. We consider three parameters: the robustness parameter Γ, the number of distinct due dates in the instance, and the number of jobs with nonzero deviations. Using parameterized-complexity theory, we prove that the problem is W[1]-hard with respect to Γ, but can be solved in XP time with respect to the same parameter. With respect to the number of distinct due dates, we establish a stronger hardness result by showing that the problem remains NP-hard even when there are only two different due dates and is solvable in pseudo-polynomial time when the number of due dates is upper bounded by a constant. To complement these results, we show that the case of a common (single) due date reduces to a robust binary knapsack problem with equal item profits, a problem we prove to be solvable in polynomial time. Finally, we prove that the problem is fixed-parameter tractable with respect to the number of jobs with nonzero deviations.