Minimizing mean flow time in two-machine open shops and flow shops

Preemptive scheduling in open shops and flow shops so as to minimize the mean flow time are both known to be strongly NP-hard on three or more machines. However, their complexity on two machines remained open for more than a decade. In this paper we show that open shop is NP-hard in the ordinary sense and flow shop is NP-hard in the strong sense. A linear time algorithm is also given to produce optimal schedules for two-machine open shops if the optimal order of completion times is known in advance. Finally, an O(N log N) time algorithm is given to produce optimal schedules for two-machine flow shops if the optimal order of completion times on the first machine is known in advance, where N is the number of jobs.