Hardness of Obligatory-Test Scheduling on Multiple Machines

Abstract

We study online scheduling with obligatory testing on $m$ identical parallel machines, with the objective of minimizing the sum of completion times. Each job comprises a test of known length and a processing operation of initially unknown length. The processing time is revealed only when the test completes. Unlike in optional testing models, the scheduler does not choose whether to acquire information. Instead, it must decide how to allocate machine capacity between testing unrevealed jobs and processing jobs whose sizes are already known. Previous single-machine lower-bound constructions suggest a natural $\sqrt{2}$ benchmark [ESA 2024: 48:1-14]. However, these constructions cannot be directly transferred to identical parallel machines by a simple replication argument. An online algorithm may interleave jobs from different copies, and the test and processing operation of a job need not be scheduled on the same machine. We address this difficulty by introducing a completion-threshold framework that reasons directly about global progress under total machine capacity. For each $X$, let $T_X$ be the earliest time at which the algorithm has completed at least $X$ jobs. The identity $\sum_{X=1}^{N}T_X$ then converts pointwise progress bounds into lower bounds on the total completion time. Using this framework, we prove a three-type lower bound of $1.4811$ and a dyadic multi-type lower bound tending to $3/2$. The latter also improves the deterministic single-machine lower bound from $\sqrt{2}$ to $3/2$. On the algorithmic side, we give a parallel version of single-machine 1-SORT and prove that, if single-machine 1-SORT is $\rho$-competitive, then its parallel version is $\frac{2(m+\rho-1)}{m+1}$-competitive on $m$ identical machines.