We study two \(\mathcal {NP}\)-hard single-machine scheduling problems with generalized due-dates. In such problems, due-dates are associated with positions in the job sequence rather than with jobs. Accordingly, the job that is assigned to position j in the job processing order (job sequence), is assigned with a predefined due-date, \(\delta _{j}\). In the first problem, the objective consists of finding a job schedule that minimizes the maximal absolute lateness, while in the second problem, we aim to maximize the weighted number of jobs completed exactly at their due-date. Both problems are known to be strongly \(\mathcal {NP}\)-hard when the instance includes an arbitrary number of different due-dates. Our objective is to study the tractability of both problems with respect to the number of different due-dates in the instance, \(\nu _{d}\). We show that both problems remain \( \mathcal {NP}\)-hard even when \(\nu _{d}=2\), and are solvable in pseudo-polynomial time when the value of \(\nu _{d}\) is upper bounded by a constant. To complement our results, we show that both problems are fixed parameterized tractable (FPT) when we combine the two parameters of number of different due-dates (\(\nu _{d}\)) and number of different processing times (\(\nu _{p}\)).

我们研究了两个具有广义截止日期的\(\mathcal {NP}\) -hard 单机调度问题。在此类问题中，截止日期与工作序列中的职位相关联，而不是与工作相关联。因此，分配给作业处理订单（作业序列）中位置j的作业被分配了预定义的到期日期\(\delta _{j}\)。在第一个问题中，目标包括找到一个最小化最大绝对迟到的工作计划，而在第二个问题中，我们的目标是最大化在到期日准确完成的工作的加权数量。众所周知，这两个问题都是强\(\mathcal {NP}\)-hard 当实例包含任意数量的不同截止日期时。我们的目标是研究这两个问题相对于实例中不同截止日期的数量\(\nu _{d}\)的可处理性。我们证明这两个问题即使在\(\nu _{d}=2\)时仍然是\( \mathcal {NP}\) -hard ，并且当\(\nu _{ d}\)的上限为常数。为了补充我们的结果，我们表明，当我们结合不同到期日期的数量（\(\nu _{d}\) ）和不同处理时间的数量（\ (\nu _{p}\) )。