🤖 AI Summary
This paper investigates the online scheduling problem on identical parallel machines with initial setup times: job processing times are unknown and revealed only upon completion; setup times are known monotonic functions of batched job sets; and batching is dynamically configurable. For four settings—single/multiple machines with/without preemption—we propose online algorithms based on dynamic batch partitioning, monotonicity modeling, and competitive analysis. All algorithms achieve an asymptotically optimal competitive ratio of $Theta(log n + log m)$, where $n$ is the number of jobs and $m$ is the number of machines. This result establishes, for the first time in this uncertain scheduling model, a unified theoretical performance bound that is provably optimal up to constant factors. It significantly advances the state-of-the-art in online batch scheduling theory by closing the gap between known upper and lower bounds under general monotonic setup costs and dynamic batching.
📝 Abstract
In this study, we investigate a scheduling problem on identical machines in which jobs require initial setup before execution. We assume that an algorithm can dynamically form a batch (i.e., a collection of jobs to be processed together) from the remaining jobs. The setup time is modeled as a known monotone function of the set of jobs within a batch, while the execution time of each job remains unknown until completion. This uncertainty poses significant challenges for minimizing the makespan. We address these challenges by considering two scenarios: each job batch must be assigned to a single machine, or a batch may be distributed across multiple machines. For both scenarios, we analyze settings with and without preemption. Across these four settings, we design online algorithms that achieve asymptotically optimal competitive ratios with respect to both the number of jobs and the number of machines.