Explorable Uncertainty in Scheduling with Non-Uniform Testing Times

TL;DR

This paper extends scheduling with testing under explorable uncertainty to non-uniform testing times, providing the first competitive algorithms and bounds for sum of completion times and makespan objectives.

Contribution

It introduces the first competitive algorithms for scheduling with non-uniform testing times, expanding previous unit-time models and analyzing both deterministic and randomized approaches.

Findings

Constant competitive ratios for sum of completion times in deterministic case.

First lower bounds for preemptive setting with non-uniform testing times.

Tight competitive ratios for makespan minimization in both deterministic and randomized cases.

Abstract

The problem of scheduling with testing in the framework of explorable uncertainty models environments where some preliminary action can influence the duration of a task. In the model, each job has an unknown processing time that can be revealed by running a test. Alternatively, jobs may be run untested for the duration of a given upper limit. Recently, D\"urr et al. [5] have studied the setting where all testing times are of unit size and have given lower and upper bounds for the objectives of minimizing the sum of completion times and the makespan on a single machine. In this paper, we extend the problem to non-uniform testing times and present the first competitive algorithms. The general setting is motivated for example by online user surveys for market prediction or querying centralized databases in distributed computing. Introducing general testing times gives the problem a new flavor and requires updated methods with new techniques in the analysis. We present constant competitive ratios for the objective of minimizing the sum of completion times in the deterministic case, both in the non-preemptive and preemptive setting. For the preemptive setting, we additionally give a first lower bound. We also present a randomized algorithm with improved competitive ratio. Furthermore, we give tight competitive ratios for the objective of minimizing the makespan, both in the deterministic and the randomized setting.