Online Flow Time Minimization: Tight Bounds for Non-Preemptive Algorithms
Yutong Geng, Enze Sun, Zonghan Yang, Yuhao Zhang
TL;DR
The paper addresses online flow-time minimization for $n$ jobs on $m$ identical machines under non-preemptive constraints, resolving long-standing questions about the power of randomness and the deterministic optimum in multi-machine settings. It introduces an online rank-based partitioning framework that separates jobs into a bounded set of large jobs and a large body of small jobs, managed via NSJF for the latter and carefully scheduled injections for the former, using proxy jobs to handle dynamic re-classification. The main results include tight randomized bounds $Θ(\sqrt{n/m})$ for the multi-machine non-preemptive setting, and a near-tight deterministic bound $O(\frac{n}{m^{2}}+\sqrt{\frac{n}{m}}\log m)$, with a kill-and-restart variant achieving $O(\sqrt{n/m})$ for $m\ge2$; in the unknown-$n$ regime, kill-and-restart breaks the $O(n)$ barrier with an $O(n^{\alpha}/\sqrt{m})$-competitive algorithm where $\alpha=(\sqrt{5}-1)/2$. The results substantially advance understanding of online non-preemptive flow-time minimization, establish tightness via matching lower bounds, and show that kill-and-restart can yield substantial gains in certain regimes while randomization cannot surpass these limits. The techniques also yield an improved offline approximation of $O(\sqrt{n/m})$ by derandomizing the randomized online approach or applying the kill-and-restart offline scheme.
Abstract
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 $Ω(n)$ 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 $Θ(\sqrt{n/m})$ 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 $Ω(n/\log n)$ 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.