On the Minimization of the Makespan Subject to Flowtime Optimality

When scheduling n jobs on m identical machines in parallel, two performance criteria are of particular interest: the makespan the completion time of the last job and the flowtime the sum of the completion times of all n jobs. Whereas minimizing makespan is NP-hard, many schedules minimize flowtime, and they are easy to characterize. This paper considers the problem of minimizing the makespan among flowtime-optimal schedules. Heuristics have appeared in the literature that result in flowtime-optimal schedules with makespans which are always less than or equal to 5/4 times the minimum flowtime-optimal makespan. In this note, we present new heuristics for this problem, one of which yields a worst-case bound of 28/27 for the two-machine case. The new heuristics have a running time of On log n. Extensions are discussed for general m.