Robust algorithms for total completion time

We revisit the problem of scheduling or assigning jobs non-preemptively so as to minimize the total completion time on m identical machines and on m uniformly related machines. This problem is polynomially solvable if all jobs are presented at once, even for unrelated machines. An online algorithm receives jobs one by one, such that every job is scheduled before the next job is presented. A robust algorithm with migration factor γ>0 also receives jobs one at a time to be assigned immediately, but when a job of size p arrives, the algorithm can re-assign a subset of jobs of total size γ⋅p. That is, it can remove the jobs of one such subset from their positions and schedule them again in an arbitrary way. The goal is to obtain optimal or almost optimal solutions. We use the term schedule for a solution where every job is scheduled to a time slot on a machine, and the term assignment is used for a solution where a job is assigned to a machine and an optimal ordering of jobs (by SPT) is always used for every machine. We show that a nearly optimal schedule cannot be obtained for m≥1 machines for any constant migration factor. For the variant of creating assignments, for one machine already the online problem is trivial, and we prove that an optimal assignment cannot be obtained for any constant migration factor γ for any m≥2 and identical machines. Then, we deal with the problem of finding almost optimal assignments. We provide a fully polynomial time approximation scheme (FPTAS) for identical machines with constant migration factor, show how its running time can be reduced, and extend our result to the case of uniformly related machines. Our approximation schemes work even for the case where job departures may occur.