Online Flow Time Minimization: Tight Bounds for Non-Preemptive Algorithms

This paper studies the classical online scheduling problem of minimizing total flow time for $n$ jobs on $m$ identical machines. Prior work often cites the $\Omega(n)$ lower bound for non-preemptive algorithms to argue for the necessity of preemption or resource augmentation, which shows the trivial $O(n)$-competitive greedy algorithm is tight. However, this lower bound applies only to \emph{deterministic} algorithms in the \emph{single-machine} case, leaving several fundamental questions unanswered. Can randomness help in the non-preemptive setting, and what is the optimal online deterministic algorithm when $m \geq 2$? We resolve both questions. We present a polynomial-time randomized algorithm with competitive ratio $\Theta(\sqrt{n/m})$ and prove a matching randomized lower bound, settling the randomized non-preemptive setting for every $m$. This also improves the best-known offline approximation ratio from $O(\sqrt{n/m}\log(n/m))$ to $O(\sqrt{n/m})$. On the deterministic side, we present a non-preemptive algorithm with competitive ratio $O(n/m^{2}+\sqrt{n/m}\log m)$ and prove a nearly matching lower bound. Our framework also extends to the kill-and-restart model, where we reveal a sharp transition of deterministic algorithms: we design an asymptotically optimal algorithm with the competitive ratio $O(\sqrt{n/m})$ for $m\ge 2$, yet establish a strong $\Omega(n/\log n)$ lower bound for $m=1$. Moreover, we show that randomization provides no further advantage, as the lower bound coincides with that of the non-preemptive setting. While our main results assume prior knowledge of $n$, we also investigate the setting where $n$ is unknown. We show kill-and-restart is powerful enough to break the $O(n)$ barrier for $m \geq 2$ even without knowing $n$. Conversely, we prove randomization alone is insufficient, as no algorithm can achieve an $o(n)$ competitive ratio in this setting.