Two competitive ratio approximation schemes for a classical scheduling problem

For a long time, optimal competitive factors of online algorithms used to be approximated by manually finding and improving upper and lower bounds. The concept of a competitive ratio approximation scheme (CRAS), introduced by Günther et al. in 2012 [13], defines a method of reducing this process to a task that can be executed automatically by a computer up to an arbitrarily good precision: A CRAS for an online problem is an algorithm that, for any ε > 0, computes the optimal competitive ratio for this problem up to a multiplicative error of 1 + ε, and also computes an algorithm that achieves the computed competitive ratio. We describe a CRAS for the classical scheduling problem P ||Cmax for any fixed number m of identical machines: A sequence of jobs must be scheduled on the machines with the objective of minimizing the makespan. The jobs arrive over a list and the processing time of each job is known upon arrival, but the processing times of future jobs and the length of the list are unknown; the next job (or the end of the list) is only revealed once the current job has been assigned to a machine. This is called the online-list model. In contrast, in the online-time model, jobs arrive over time (possibly multiple jobs at once) and one can wait until a later time to assign a job to a machine to start processing.