Preemptive scheduling of duoprocessor tasks on dedicated processors to minimize schedule length

In this paper we have analyzed the algorithmic complexity of preemptive scheduling of duoprocessor tasks on dedicated processors in the case where the number of machines is arbitrary. We have proved this problem to be NP-hard even if all tasks have the same processing time. In other words, this means that it is practically impossible to give an efficient polynomially bounded algorithm for finding optimal schedules. However, if the number of machines is fixed, we can solve our problem in polynomial time. Again, this means that we can use a practically efficient algorithm simplex to solve this problem to optimality. The general NP-hardness of a scheduling problem does not exclude the existence of efficient algorithms for some special cases allowing arbitrarily many machines. Therefore, in the second part of this paper we have identified a number of such special cases involving a simplified structure of scheduling graphs. Finally, we have pointed out some important equivalence between finding a minimum makespan schedule and establishing a fractional chromatic coloring of the edges of associated graph.