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

This paper studies the online scheduling problem of minimizing total flow time for $n$ jobs on $m$ identical machines. A classical $\Omega(n)$ lower bound shows that no deterministic single-machine algorithm can beat the trivial greedy, even when $n$ is known in advance. However, this barrier is specific to deterministic algorithms on a single machine, leaving open what randomization, multiple machines, or the kill-and-restart capability can achieve. We give a nearly complete answer. For randomized non-preemptive algorithms, we establish a tight $\Theta(\sqrt{n/m})$ competitive ratio, which also improves the best offline approximation to $O(\sqrt{n/m})$. For deterministic non-preemptive algorithms on multiple machines, we prove an $O(n/m^2 + \sqrt{n/m}\log m)$ upper bound and an $\Omega(n/m^2 + \sqrt{n/m})$ lower bound. In the kill-and-restart model, we reveal a sharp transition for deterministic algorithms: $\Omega(n/\log n)$ for $m = 1$ versus $\Theta(\sqrt{n/m})$ for $m \ge 2$; the latter matches the optimal randomized ratio, and we further show that randomization provides no additional power in this model. We also investigate the setting where $n$ is unknown. We prove that no randomized non-preemptive algorithm achieves $o(n)$ on one machine or $o(n/m^2 + \sqrt{n/m})$ on $m$ machines. In contrast, our kill-and-restart algorithm achieves $O(n^{\alpha}/\sqrt{m})$ for $m \ge 2$, where $\alpha = (\sqrt{5}-1)/2$, breaking the trivial bound without knowledge of $n$.