Minimizing Completion Times for Stochastic Jobs via Batched Free Times

We study the classic problem of minimizing the expected total completion time of jobs on $m$ identical machines in the setting where the sizes of the jobs are stochastic. Specifically, the size of each job is a random variable whose distribution is known to the algorithm, but whose realization is revealed only after the job is scheduled. While minimizing the total completion time is easy in the deterministic setting, the stochastic problem has long been notorious: all known algorithms have approximation ratios that either depend on the variances, or depend linearly on the number of machines. We give an $\widetilde{O}(\sqrt{m})$-approximation for stochastic jobs which have Bernoulli processing times. This is the first approximation for this problem that is both independent of the variance in the job sizes, and is sublinear in the number of machines $m$. Our algorithm is based on a novel reduction from minimizing the total completion time to a natural makespan-like objective, which we call the weighted free time. We hope this free time objective will be useful in further improvements to this problem, as well as other stochastic scheduling problems.