Upper bound on the number of processors for scheduling with interprocessor communication delays

The problem of scheduling a task system with communication delays on multiprocessor systems is known to be NP-hard in its general form as well as many restricted cases even on an unlimited number of processors. In this paper, we study the problem of determining an upper bound on the minimum number of processors achieved by a schedule that minimizes the makespan for scheduling problems with communication delays. We prove that the minimum number of partitioning paths of the precedence graph is an upper bound on the minimum number of processors for UET-UCT (Unit Execution Time-Unit Communication Time) task systems. Then we propose an algorithm of O(n) (n designates the number of tasks) to compute an upper bound, which is valid independently of task processing times and communication delays, in the special case when the precedence graph is an out-tree or an in-tree.