Scheduling on a Constant Number of Machines

We consider the problem of scheduling independent jobs on a constant number of machines. We illustrate two important approaches for obtaining polynomial time approximation schemes for two different variants of the problem, more precisely the multiprocessor-job and the unrelated-machines models, and two different optimization criteria: the makespan and the sum of weigthed completion times. In the past few years, there have been significant developments in the area of approximation algorithms for AfP-hard scheduling problems, see e.g. [S] and the references at the end of this paper. Our current, admittedly optimistic, opinion is the following: for any scheduling problem where the schedules can be stretched without unduly affecting the cost function, and in which each job is specified by a constant number of parameters, there should be a way to construct a PTAS by a suitable combination of known algorithmic and approximation techniques. We present here two approaches for obtaining efficient PTAS that we illustrate by two examples dealing eachone with a different optimization criterion. In the first case, we consider the problem of minimizing the makespan for a multiprocessor job system [12]. In this model, we are given a set of jobs J such that each job requires to be processed simultaneously by several processors. In the dedicated variant of this model, each job requires the simultaneous use of a prespecified set of processors fixj. Since each processor can process at most one job at a time, jobs that share at least one resource cannot be scheduled at the same time step and are said to be incompatdb2e. Hence, jobs are subject to compatibility constraints. Thus, every job is specified by its execution time pj and the prespecified subset of processors f ixj on which it must be executed. By tj we denote the starting time of job j and the completion time of j is equal to Cj = tj +pj. The objective is to find a feasible schedule minimizing the makespan Cma, i.e. the maximum conipletion time of any job. Using the standard three field notation of Graham et al. [14], this problem is classified as PmlfkjICmm. In the second case, we are given n independent jobs that have to be executed on m unrelated machines. Each job j is specified by its execution times p(ii) on D. Hochbaum et al.