Scheduling with Testing: Competitive Algorithms for Minimizing the Total Weighted Completion Time in the Adversarial Model

2026-06-23

AI Summary

This study addresses the weighted task scheduling problem under an adversarial model with a testing mechanism, where each task may either be executed directly or first tested to reveal a potentially shorter processing time, with the objective of minimizing total weighted completion time. For both single-machine and identical parallel machine settings, the work presents the first constant-competitive online algorithms that handle task-dependent weights, significantly improving upon existing results—even advancing the known upper bounds for the unweighted case. Building on list-scheduling strategies, the authors design deterministic and randomized algorithms achieving competitive ratios of 2.3166 and 2.1523 on a single machine, and 2.7763 and 2.5110 in the parallel setting, respectively.

Abstract

We study scheduling with testing on a single machine and on identical parallel machines to minimize the total \emph{weighted} completion time in the adversarial model. In this setting, each job is equipped with a weight, an upper bound on its processing time, and a testing time. An algorithm can either execute a job for an amount of time equal to the upper bound or test it first to reveal a potentially lower processing time used to schedule the job later. We establish the first constant-competitive algorithms for this problem with job-dependent weights that reflect each job's relative importance. For single-machine scheduling, we present a deterministic algorithm with a competitive ratio of 2.3166 and show that a randomized variant has a competitive ratio of 2.1523. These guarantees match the best-known upper bounds in the unweighted setting. Combining these algorithms with list scheduling yields competitive ratios of 2.7763 and 2.5110 for identical-parallel-machine scheduling, improving the previously best-known bounds even in the unweighted case.

Problem

• scheduling with testing
• weighted completion time
• adversarial model
• competitive algorithms
• parallel machines

Innovation

• scheduling with testing
• competitive algorithms
• weighted completion time
• adversarial model
• parallel machines