Tight Bounds for Online Scheduling in the One-Fast-Many-Slow Machines Setting

2026-02-25

AI Summary

This study investigates the optimal competitive ratios for online scheduling under the One-Fast-Many-Slow model in both the immediate-decision (final-commitment) and non-commitment settings. By constructing adversarial job arrival sequences and designing tailored scheduling algorithms, the work establishes the golden ratio (≈1.618) as the tight competitive ratio in the final-commitment model, thereby confirming a long-standing conjecture. In the non-commitment model, it improves the lower bound on the competitive ratio to 1.5 and demonstrates its tightness. These results fully resolve open problems in both models, significantly advancing the theoretical understanding of online scheduling under speed heterogeneity.

Technology Category

• Planning, Routing, and Scheduling: Scheduling under Uncertainty
• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Economics, Online Markets and Human Computation: Research challenges in human and human-AI computation
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

In the One-Fast-Many-Slow decision problem, introduced by Sheffield and Westover (ITCS '25), a scheduler, with access to one fast machine and infinitely many slow machines, receives a series of tasks and must allocate the work among its machines. The goal is to minimize the overhead of an online algorithm over the optimal offline algorithm. Three versions of this setting were considered: Instantly-committing schedulers that must assign tasks to machines immediately and irrevocably, Eventually-committing schedulers whose assignments are irrevocable but can occur anytime after a task arrives, and Never-committing schedulers that can interrupt and restart a task on a different machine. In the Instantly-committing model, Sheffield and Westover showed that the optimal competitive ratio is equal to 2, while in the Eventually-committing model the competitive ratio lies in the interval [1.618, 1.678], and in the Never-committing model the competitive ratio lies in the interval [1.366, 1.5] (SPAA '24, ITCS '25). In the latter two models, the exact optimal competitive ratios were left as open problems, moreover Kuszmaul and Westover (SPAA '24) conjectured that the lower bound in the Eventually-committing model is tight. In this paper we resolve this problem by providing tight bounds for the competitive ratios in the Eventually-committing and Never-committing models. For Eventually-committing, we prove Kuszmaul and Westover's conjecture by giving an algorithm achieving a competitive ratio equal to the lower bound of $\frac{1+\sqrt{5}}{2}\approx 1.618$. For Never-committing, we provide an explicit Task Arrival Process (TAP) lower bounding the competitive ratio to the previous upper bound of 1.5.

Problem

• online scheduling
• competitive ratio
• one-fast-many-slow
• task assignment
• lower bound

Innovation

• competitive ratio
• online scheduling
• One-Fast-Many-Slow
• task arrival process
• tight bounds