An Optimal algorithm for scheduling interval ordered tasks with communication on N processors

The problem of scheduling task graphs on multiprocessor systems have received considerable attention in recent years. This problem is known to be NP-hard in its general form as well as many restricted cases. Few polynomial algorithms have been developed for solving special cases of the scheduling problem when the communication cost is not considered. For example, Papadimitriou and Yannakakis showed that unit execution time tasks in interval orders can be scheduled in linear time on N processors when communication cost is assumed to be zero. They have also shown that the generalization of this problem to arbitrary execution times is NP-complete. The complexity of the problem arises even more when communication among tasks is considered. Papadimitriou and Yannakakis also showed that the problem of optimally scheduling unit-time task graphs with communication on an unlimited number of processors is NP-complete. They provided a simple way to approximate the optimal schedule length within a factor of two. In this paper, we study the problem of scheduling task graphs with communication on a given number of processors when the task graph is an interval order. We introduce an optimal algorithm for solving the scheduling problem when the execution cost of the system tasks is identical and equal to the communication cost between any pair of processors.