An approximation scheme for minimizing the makespan of the parallel identical multi-stage flow-shops

In the parallel k-stage flow-shops problem, we are given m identical k-stage flow-shops and a set of jobs. Each job can be processed by any one of the flow-shops but switching between flow-shops is not allowed. The objective is to minimize the makespan, which is the finishing time of the last job. This problem generalizes the classical parallel identical machine scheduling (where k=1) and the classical flow-shop scheduling (where m=1) problems, and thus it is NP-hard. We present a polynomial-time approximation scheme (PTAS) for the problem, when m and k are fixed constants. The key technique is to partition the jobs into big jobs and small jobs, enumerate over all feasible schedules for the big jobs, and handle the small jobs by solving a linear program and employing a “sliding” method. Such a technique has been used in the design of PTAS for several flow-shop scheduling variants. Our main contributions are the non-trivial application of this technique and a valid answer to the open question in the literature.