Complexity Results for Multiprocessor Scheduling under Resource Constraints

We examine the computational complexity of scheduling problems associated with a certain abstract model of a multiprocessing system. The essential elements of the model are a finite number of identical processors, a finite set of tasks to be executed, a partial order constraining the sequence in which tasks may be executed, a finite set of limited resources, and, for each task, the time required for its execution and the amount of each resource which it requires. We focus on the complexity of algorithms for determining a schedule which satisfies the partial order and resource usage constraints and which completes all required processing before a given fixed deadline. For certain special cases, it is possible to give such a scheduling algorithm which runs in low order polynomial time. However, the main results of this paper imply that almost all cases of this scheduling problem, even with only one resource, are NP-complete and hence are as difficult as the notorious traveling salesman problem.