Minimizing the Weighted Makespan with Restarts on a Single Machine

We consider the problem of minimizing the weighted makespan on a single machine with restarts. Restarts are similar to preemptions but weaker: a job can be interrupted, but then it has to be run again from the start instead of resuming at the point of interruption later. The objective is to minimize the weighted makespan, defined as the maximum weighted completion time of jobs. We establish a lower bound of 1.4656 on the competitive ratio achievable by deterministic online algorithms. For the case where all jobs have identical processing times, we design and analyze a deterministic online algorithm that improves the competitive ratio to better than 1.3098. Finally, we prove a lower bound of 1.2344 for this case.