The Power of Amortization on Scheduling with Explorable Uncertainty

TL;DR

The paper tackles the single-machine scheduling problem under explorable uncertainty (SUP), where testing a job costs $t_j$ and can reduce its processing from $u_j$ to $p_j \in [0,u_j]$; online decisions must choose which jobs to test and in what order. It introduces an amortized enhancement of the $(\alpha,\beta)$-SORT framework and develops PCP$_{\alpha,\beta}$ and Rand-PCP$_{\beta}$ variants to bound total completion time. Deterministic results improve the competitive ratio from $4$ to $1+\sqrt{2}$ and further to $2.316513$, while randomized results improve from $3.3794$ to $2.152271$, with preemption shown to be unhelpful in this setting. The work also discusses parallel-machine extensions and outlines open questions on deeper amortization and fully online arrivals. These results provide quantitative guidance on how information acquisition via testing benefits online scheduling under bounded uncertainty.

Abstract

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 $2.152271$.

Figures (2)

• Figure 1: An example where ${p_k^*}$ is charged four times. The light blue and dark blue segments represent $c(k,j)$ and $c(j,k)$, respectively. The red segment represents ${p_k^*}$.
• Figure 2: The red arrows illustrate how to charge $c(k,j)+c(j,k)$ to the cost of tasks regarding $k$. Each row in the sub-figures is a permutation of how the tasks are executed. The circles and rectangles are testing tasks and execution tasks after testing, respectively. The rectangles with curly tops are execution tasks without testing. The tasks in gray are from the job $k$, and the tasks in white are from the job $j$. The light blue and dark blue line segments under the tasks represent the contribution $c(k,j)$ and $c(j,k)$, respectively.

Theorems & Definitions (33)

• lemma thmcounterlemma
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• corollary thmcountercorollary
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• lemma thmcounterlemma
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