A PTAS for Scheduling Unrelated Machines of Few Different Types

Scheduling on Unrelated Machines is a classical optimization problem where [Formula: see text] jobs have to be distributed to [Formula: see text] machines. Each of the jobs [Formula: see text] has on machine [Formula: see text] a processing time [Formula: see text]. The goal is to minimize the makespan, i.e., the maximum completion time of the longest-running machine. Unless [Formula: see text], this problem does not allow for a polynomial-time approximation algorithm with a ratio better than [Formula: see text]. A natural scenario is however that many machines are of the same type, like a CPU and GPU cluster: for each of the [Formula: see text] machine types, the machines [Formula: see text] of the same type [Formula: see text] satisfy [Formula: see text] for all jobs [Formula: see text]. For the case where the number [Formula: see text] of machine types is constant, this paper presents an approximation scheme, i.e., an algorithm of approximation ratio [Formula: see text] for [Formula: see text], with an improved running time only single exponential in [Formula: see text].