Online Makespan Minimization: The Power of Restart

TL;DR

This paper introduces a new online algorithm for makespan minimization on identical machines that uses job restarts, achieving better competitive ratios than previous algorithms for both general and two-machine cases.

Contribution

It resolves the open problem of whether restart-based algorithms can surpass existing competitive ratios, providing a novel algorithm with improved performance guarantees.

Findings

Improves the 1.5 competitive ratio for general machines

Achieves a competitive ratio better than 1.382 for two machines

Demonstrates the effectiveness of restart strategies in online scheduling

Abstract

We consider the online makespan minimization problem on identical machines. Chen and Vestjens (ORL 1997) show that the largest processing time first (LPT) algorithm is 1.5-competitive. For the special case of two machines, Noga and Seiden (TCS 2001) introduce the SLEEPY algorithm that achieves a competitive ratio of $(5 - \sqrt{5})/2 \approx 1.382$, matching the lower bound by Chen and Vestjens (ORL 1997). Furthermore, Noga and Seiden note that in many applications one can kill a job and restart it later, and they leave an open problem whether algorithms with restart can obtain better competitive ratios. We resolve this long-standing open problem on the positive end. Our algorithm has a natural rule for killing a processing job: a newly-arrived job replaces the smallest processing job if 1) the new job is larger than other pending jobs, 2) the new job is much larger than the processing one, and 3) the processed portion is small relative to the size of the new job. With appropriate choice of parameters, we show that our algorithm improves the 1.5 competitive ratio for the general case, and the 1.382 competitive ratio for the two-machine case.