Approximation Algorithms for Multiprocessor Scheduling with Testing to Minimize the Total Job Completion Time

Abstract

In offline scheduling models, jobs are given with their exact processing times. In their online counterparts, jobs arrive in sequence together with their processing times and the scheduler makes irrevocable decisions on how to execute each of them upon its arrival. We consider a semi-online variant which has equally rich application background, called scheduling with testing, where the exact processing time of a job is revealed only after a required testing operation is finished, or otherwise the job has to be executed for a given possibly over-estimated length of time. For multiprocessor scheduling with testing to minimize the total job completion time, we present several first approximation algorithms with constant competitive ratios for various settings, including a \(2 \varphi \)-competitive algorithm for the non-preemptive general testing case and a \((0.0382 + 2.7925 (1 - \frac{1}{2\,m}))\)-competitive randomized algorithm, when the number of machines \(m \ge 37\) or otherwise 2.7925-competitive, where \(\varphi = (1 + \sqrt{5}) / 2 < 1.6181\) is the golden ratio and m is the number of machines, a \((3.5 - \frac{3}{2\,m})\)-competitive algorithm allowing job preemption when \(m \ge 3\) or otherwise 3-competitive, and a \((\varphi + \frac{\varphi + 1}{2} (1 - \frac{1}{\,}m))\)-competitive algorithm for the non-preemptive uniform testing case when \(m \ge 5\) or otherwise \((\varphi + 1)\)-competitive. Our results improve three previous best approximation algorithms for the single machine scheduling with testing problems, respectively.