Scheduling Unit Execution Time Tasks on Two Parallel Machines with the Criteria of Makespan and Total Completion Time

Two extensions of the classical scheduling model with two parallel identical machines and a partially ordered set of unit execution time tasks are considered. It is well known that the Coffman—Graham algorithm constructs for this model a so-called ideal schedule: that is, a schedule which is optimal for both makespan and total completion time criteria simultaneously. The question of the existence of such a schedule for the extension of this model, where each task has a release time, has remained open over several decades. The paper gives a positive answer to this question and presents the corresponding polynomial-time algorithm. Another straightforward generalization of the considered classical model is obtained by the introduction of multiprocessor tasks. It is shown that, despite the fact that a slightly modified Coffman-Graham algorithm solves the makespan problem with multiprocessor tasks for arbitrary precedence constraints, generally an ideal schedule does not exist and the problem with the criterion of total completion time turns out to be NP-hard in the strong sense even for in-trees.