The Power of Amortization on Scheduling with Explorable Uncertainty

In this work, we study a scheduling problem with explorable uncertainty. Each job comes with an upper limit of its processing time, which could be potentially reduced by testing the job, which also takes time. The objective is to schedule all jobs on a single machine with a minimum total completion time. The challenge lies in deciding which jobs to test and the order of testing/processing jobs. The online problem was first introduced with unit testing time and later generalized to variable testing times. For this general setting, the upper bounds of the competitive ratio are shown to be $4$ and $3.3794$ for deterministic and randomized online algorithms; while the lower bounds for unit testing time stands, which are $1.8546$ (deterministic) and $1.6257$ (randomized). We continue the study on variable testing times setting. We first enhance the analysis framework and improve the competitive ratio of the deterministic algorithm from $4$ to $1+\sqrt{2} \approx 2.4143$. Using the new analysis framework, we propose a new deterministic algorithm that further improves the competitive ratio to $2.316513$. The new framework also enables us to develop a randomized algorithm improving the expected competitive ratio from $3.3794$ to $2.152271$.