Speed-Robust Scheduling -- Sand, Bricks, and Rocks

TL;DR

This paper introduces algorithms for speed-robust scheduling that guarantee near-optimal performance despite unknown machine speeds, with improved robustness factors for various job and machine scenarios.

Contribution

It presents new algorithms with improved robustness factors for speed-robust scheduling, including special cases like equal-size and infinitesimal jobs, and specific machine environments.

Findings

Robustness factor of 2-1/m for general setting

Robustness factor of 1.8 for equal-size jobs

Optimal robustness factor of e/(e-1) for infinitesimal jobs

Abstract

The speed-robust scheduling problem is a two-stage problem where given $m$ machines, jobs must be grouped into at most $m$ bags while the processing speeds of the given $m$ machines are unknown. After the speeds are revealed, the grouped jobs must be assigned to the machines without being separated. To evaluate the performance of algorithms, we determine upper bounds on the worst-case ratio of the algorithm's makespan and the optimal makespan given full information. We refer to this ratio as the robustness factor. We give an algorithm with a robustness factor $2-1/m$ for the most general setting and improve this to $1.8$ for equal-size jobs. For the special case of infinitesimal jobs, we give an algorithm with an optimal robustness factor equal to $e/(e-1) \approx 1.58$. The particular machine environment in which all machines have either speed $0$ or $1$ was studied before by Stein and Zhong (SODA 2019). For this setting, we provide an algorithm for scheduling infinitesimal jobs with an optimal robustness factor of $(1+\sqrt{2})/2 \approx 1.207$. It lays the foundation for an algorithm matching the lower bound of $4/3$ for equal-size jobs.