Single machine scheduling with deadlines and increasing rates of processing times

The makespan, flow time and maximum lateness problems of scheduling a set of tasks with deadlines and increasing rates of processing times on a single machine are considered in this paper. We first show that, when the increasing rates of processing time are identical, the makespan problem is equivalent to the corresponding flow time problem. Both problems are solvable in $O(n^5)$ time by a dynamic programming algorithm. As an application of the dynamic programming algorithm, we demonstrate that the corresponding maximum lateness problem can be solved in $O(n^6\log n)$ time. We then show that the general makespan problem is strongly NP-complete. Thus, both the corresponding flow time problem and maximum lateness problem are also strongly NP-complete.