Scheduling start time dependent tasks with deadlines and identical initial processing times on a single machine

In this paper, we study the feasibility problem of scheduling a set of start time dependent tasks on a single machine with deadlines, processing rates and identical initial processing times. First, we show that the cases with arbitrary deadlines are strongly NP-complete. Second, we show that the cases with two distinct deadlines are NP-complete in the ordinary sense. Finally, we give an optimal polynomial algorithm for the makespan problem with two distinct processing rates. We solve a series of open problems in the literature and give a sharp boundary delineating the complexity of the problems.