A polynomial-time approximation scheme for an arbitrary number of parallel two-stage flow-shops

We investigate the approximability of the m parallel two-stage flow-shop (mP2FS) problem, where a set of jobs is scheduled on the multiple identical two-stage flow-shops to minimize the makespan, i.e., the finishing time of the last job. Each job needs to be processed non-preemptively on one flow-shop without switching to the other flow-shops. This problem is a hybrid of the classic parallel machine scheduling and two-stage flow-shop scheduling problems. Its strong NP-hardness follows from the parallel machine scheduling problem when the number of machines is part of the input. Our main contribution is a polynomial-time approximation scheme (PTAS) for the mP2FS problem when the number of shops is part of the input, which improves the previous best approximation algorithm of a ratio (2+ϵ). Owing to the strong NP-hardness, our PTAS achieves the best possible approximation ratio.