Approximability of flow shop scheduling

Shop scheduling problems are notorious for their intractability, both in theory and practice. In this paper, we demonstrate the existence of a polynomial approximation scheme for the flow shop scheduling problem with an arbitrary fixed number of machines. For the three common shop models (open, flow, and job), this result is the only known approximation scheme. Since none of the three models can be approximated arbitrarily closely in the general case (unless P=NP), the result demonstrates the approximability gap between the models in which the number of machines is fixed, and those in which it is part of the input of the instance. The result can be extended to flow shops with job release dates and delivery times and to flow shops with a fixed number stages, where the number of machines at any stage is fixed. We also describe a related polynomial approximation scheme for the problem of scheduling an open shop with a single bottleneck machine and an arbitrary number of non-bottleneck machines