On the tractability of hard scheduling problems with generalized due-dates with respect to the number of different due-dates

We study two 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, δ_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 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, ν_d. We show that both problems remain NP-hard even when ν_d= 2 , and are solvable in pseudo-polynomial time when the value of ν_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 (ν_d) and number of different processing times (ν_p).