arXiv:2606.25166v1 Announce Type: new 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.