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

2025-11-05

AI Summary

This paper studies the non-preemptive online scheduling of $n$ jobs on $m$ identical machines to minimize total flow time. To address the long-standing open questions regarding the power of randomness in online scheduling and the optimal competitive ratio of deterministic algorithms for multi-machine settings, we establish the first tight competitive ratio bound $Theta(sqrt{n/m})$ for non-preemptive randomized algorithms. We propose the first asymptotically optimal deterministic algorithm and uncover a phase-transition phenomenon in the kill-and-restart model: machine count critically affects performance. Through probabilistic analysis, competitive ratio analysis, and lower-bound construction, we achieve an $O(sqrt{n/m})$-competitive randomized algorithm with a matching lower bound. Furthermore, we improve the offline approximation ratio to $O(sqrt{n/m})$, and—crucially—when $n$ is unknown, we prove that kill-and-restart breaks the classical $Omega(n)$ competitive barrier, attaining $O(sqrt{n/m})$.

Technology Category

• Planning, Routing, and Scheduling: Scheduling under Uncertainty
• Machine Learning: Online Learning & Bandits
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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 $Omega(n)$ 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 $mge 2$, 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.

Problem

• Determining optimal non-preemptive online algorithms for flow time minimization
• Establishing randomized and deterministic competitive ratios for multi-machine scheduling
• Investigating kill-and-restart models and unknown job count scenarios

Innovation

• Randomized algorithm achieving competitive ratio O(sqrt(n/m))
• Deterministic non-preemptive algorithm with near-optimal bounds
• Kill-and-restart model enabling breakthrough for unknown n