An FPTAS for the parallel two-stage flowshop problem

We consider the NP-hard m-parallel two-stage flowshop problem, abbreviated as the ( m , 2 ) -PFS problem, where we need to schedule n jobs to m parallel identical two-stage flowshops in order to minimize the makespan, i.e. the maximum completion time of all the jobs on the m flowshops. The ( m , 2 ) -PFS problem can be decomposed into two subproblems: to assign the n jobs to the m parallel flowshops, and for each flowshop to schedule the jobs assigned to the flowshop. We first present a pseudo-polynomial time dynamic programming algorithm to solve the ( m , 2 ) -PFS problem optimally, for any fixed m, based on an earlier idea for solving the ( 2 , 2 ) -PFS problem. Using the dynamic programming algorithm as a subroutine, we design a fully polynomial-time approximation scheme (FPTAS) for the ( m , 2 ) -PFS problem.