A note on scheduling multiprocessor tasks with identical processing times

We study the scheduling situation where n tasks with identical processing times have to be scheduled on m parallel processors. Each task is subjected to a release date and requires simultaneously a fixed number of processors. We show that, for each fixed value of m, the problem of minimizing total completion time can be solved in polynomial time. The complexity status of the corresponding problem Pm|r_i,p_i=p,size_i|∑C_i was unknown before.Scope and purposeThere has been increasing interest in multiprocessor scheduling, i.e., in scheduling models where tasks require several processors (machines) simultaneously. Many scheduling problems fit in this model and a large amount of research has been carried on theoretical multiprocessor scheduling. In this paper we study the situation where tasks, subjected to release dates, have identical processing time and we introduce a dynamic programming algorithm that can compute the minimum total completion time. Although this scheduling problem has been open in the literature for several years, our algorithm is simple and easy to understand.