On Scheduling Independent Tasks with Restricted Execution Times

We consider the problem of nonpreemptively scheduling n independent tasks on m identical and parallel machines with the objective of minimizing the overall finishing time. The problem has been shown to be NP-complete in the strong sense and hence there probably will not be any pseudopolynomial time algorithm for solving this problem. We show, however, that if the execution times are restricted to a fixed number, say k, of different values, then it can be solved in polynomial time. Our algorithm can be implemented to run in time 0(log₂p * log₂m * n^2(k−1)) and space 0(log₂m * n^k−1) in the worst case, where p is the largest execution time.