A PTAS for the Multiple Parallel Identical Multi-stage Flow-Shops to Minimize the Makespan - Tong, Weitian; Miyano, Eiji; Goebel, Randy; Lin, Guohui (Frontiers in Algorithmics, 2016)
In the parallelk-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 $$\text {NP}$$ -hard. We present a polynomial-time approximation scheme for the problem, when m and k are fixed constants. The key technique is to enumerate over schedules for big jobs, solve a linear programming for small jobs, and add the fractional small jobs at the end. Such a technique has been used in the design of similar approximation schemes.