Approximation Schemes for Scheduling on Parallel Machines with GoS Levels∗

We consider the offline scheduling problem of minimizing the makespan on m parallel and identical machines with certain feature. Every job and machine are labeled with the grade of service (GoS) levels, and each job can only be processed by the machine whose GoS level is no more than that of the job. In this paper, we present a polynomial time approximation scheme (PTAS) with running time O(n logn) for the special case where the GoS level is either 1 or 2, where the hidden constant depends exponentially on the reciprocal value of the desired precision. This solves an open problem left in [11] partially. We also present a new full polynomial time approximation scheme (FPTAS) with running time O(n) for the case where the number of machines is fixed.