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 Gamma. The objective is to minimize the maximum number of tardy jobs over all possible scenarios. Since the problem is NP-hard in general, we focus on analyzing its tractability under the assumption that some natural parameter of the problem is bounded by a constant. We consider three parameters: the robustness parameter Gamma, the number of distinct due dates in the instance, and the number of jobs with nonzero deviations. Using parametrized-complexity theory, we prove that the problem is W[1]-hard with respect to Gamma, but can be solved in XP time with respect to the same parameter. With respect to the number of different 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, which we prove to be solvable in polynomial time. Finally, we prove that the problem is solvable in FPT time with respect to the number of nonzero deviations.

翻译：本研究探讨了处理时间不确定条件下的鲁棒单机调度问题。不确定性采用预算化方法建模，其中每个作业具有标称处理时间和偏差处理时间，且偏差数量受Gamma值约束。目标是最小化所有可能场景下最大延迟作业数量。由于该问题在一般情况下属于NP难问题，我们重点分析在问题自然参数有界条件下的可解性。我们考虑三个参数：鲁棒性参数Gamma、实例中不同截止日期的数量，以及具有非零偏差的作业数量。运用参数化复杂性理论，我们证明了该问题关于Gamma是W[1]难的，但能以XP时间关于同一参数求解。关于不同截止日期数量，我们建立了更强的硬度结果：即使仅存在两种不同截止日期时问题仍保持NP难，而当截止日期数量以常数为上界时可在伪多项式时间内求解。作为补充，我们证明具有共同（单一）截止日期的特例可归约为具有相同物品收益的鲁棒二元背包问题，并证明该问题可在多项式时间内求解。最后，我们证明了该问题关于非零偏差数量可在FPT时间内求解。