Randomized approximation schemes for minimizing the weighted makespan on identical parallel machines

In this paper, we discuss scheduling problems with m identical machines and n jobs where each job has to be assigned to some machine. The objective is to minimize the weighted makespan of jobs, i.e., the maximum weighted completion time of jobs. This scheduling problem is a generalization of minimizing the makespan on parallel machine scheduling problem. We present a ( \(2-\frac{1}{m}\) $2-\frac{1}{m}$ )-approximation algorithm and a randomized efficient polynomial time approximation scheme (EPTAS) for the problem. We also design a randomized fully polynomial time approximation scheme (FPTAS) for the special case when the number of machines is fixed.