Delayed-Clairvoyant Flow Time Scheduling via a Borrow Graph Analysis

We study the problem of preemptively scheduling jobs online over time on a single machine to minimize the total flow time. In the traditional clairvoyant scheduling model, the scheduler learns about the processing time of a job at its arrival, and scheduling at any time the job with the shortest remaining processing time (SRPT) is optimal. In contrast, the practically relevant non-clairvoyant model assumes that the processing time of a job is unknown at its arrival, and is only revealed when it completes. Non-clairvoyant flow time minimization does not admit algorithms with a constant competitive ratio. Consequently, the problem has been studied under speed augmentation (JACM'00) or with predicted processing times (STOC'21, SODA'22) to attain constant guarantees. In this paper, we consider $\alpha$-clairvoyant scheduling, where the scheduler learns the processing time of a job once it completes an $\alpha$-fraction of its processing time. This naturally interpolates between clairvoyant scheduling ($\alpha=0$) and non-clairvoyant scheduling ($\alpha=1$). By elegantly fusing two traditional algorithms, we propose a scheduling rule with a competitive ratio of $\mathcal{O}(\frac{1}{1-\alpha})$ whenever $0 \leq \alpha < 1$. As $\alpha$ increases, our competitive guarantee transitions nicely (up to constants) between the previously established bounds for clairvoyant and non-clairvoyant flow time minimization. We complement this positive result with a tight randomized lower bound.