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 completed by its deadline, and the error is simply the execution time of the unfinished portion. In this paper we 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 tasks 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. Motivated by the complexity of the problem, we consider the case of a single processor and a set of tasks with oppositely ordered ready times and deadlines. We give a pseudo-polynomial time algorithm and a fast heuristic for this special case. The heuristic is shown to have an asymptotic worst-case ratio of 3. Finally, we give a polynomial time algorithm for a set of tasks with oppositely ordered ready times and deadlines, and similarly ordered mandatory execution times and total execution times.