Shortest path to nonpreemptive schedules of unit-time jobs on two identical parallel machines with minimum total completion time

Ideal schedules reach both minimum maximum completion time and minimum total completion time of jobs. It is known that there exist computable in polynomial time ideal nonpreemptive two-machine schedules of unit-time operation jobs with equal release dates and arbitrary precedence constraints on identical parallel machines, in flow shops and open shops. In this paper, we study the possibility of extending these results to the case where release dates can be different. We establish the complexity status of P2|prec,rj,pj=1|∑Cj and F2|prec,rj,pij=1|∑Cj showing that optimal schedules for these problems can also be found in polynomial time and conjecture that all such schedules are ideal indeed. On the other hand, we show that the ideal schedules in open shops do not always exist.