Approximation algorithms for the parallel flow shop problem

We consider the NP-hard problem of scheduling n jobs in m two-stage parallel flow shops so as to minimize the makespan. This problem decomposes into two subproblems: assigning the jobs to parallel flow shops; and scheduling the jobs assigned to the same flow shop by use of Johnson’s rule. For m=2, we present a 32-approximation algorithm, and for m=3, we present a 127-approximation algorithm. Both these algorithms run in O(nlogn) time. These are the first approximation algorithms with fixed worst-case performance guarantees for the parallel flow shop problem.