Minimizing the number of late tasks with error constraint

The problem of minimizing the number of late tasks in the imprecise computation model is considered. Each task consists of two subtasks, mandatory and optional. A task is said to be on-time if its mandatory part is completed by its deadline; otherwise, it is said to be late. An on-time task incurs an error if its optional part is not computed by the deadline, and the error is simply the execution time of the unfinished portion. The authors consider the problem of finding a preemptive schedule for a set of tasks on p ≥ 1 identical processors, such that the number of on-time tasks is maximized, (or equivalently, the number of late task is minimized), and the total error of the on-time tasks is no more than a given threshold K. Such a schedule is called an optimal schedule. It is shown that the problem of finding an optimal schedule is NP-hard for each fixed p ≥ 1, even if all tasks have the same ready time and the same deadline.