Some Local Cases for Online Scheduling Algorithm with Optimal Time Prediction

We studied the online scheduling problem with optimal prediction. The online scheduling problem means that there are several machines to execute the tasks one following one. Every time the decision maker gets a new task, he must immediately decide the machine to execute it. When the next task arrives, previous decision cannot be reversed. After all tasks are assigned to their own machines, the machines begin to execute them, and the time of the last machine which finishes all the tasks is the total time of the system.In our paper, we focus on identical machines, which means the machines are totally the same. The main goal of our research is approximation algorithm design, that is, to design an online algorithm to arrange the matching of these tasks and machines. We are concerned about the problem with a prediction, that is, the total time of the system after all tasks are optimally allocated is known before the arrival of the first task. With additional information, the performance of the approximation algorithm may be better than the original.In this problem, we find that the performance of the approximation algorithm has been greatly improved. We take the comparison of the upper and lower bounds of algorithms without prediction and with prediction respectively. The upper bound is the algorithm that can do the best in the worst case, and the lower bound is a situation that no algorithm can does better. When the two overlap, a tight bound is obtained. We prove that for 2 machines, the tight bound of the algorithm without prediction is 1.5, and the tight bound of the algorithm with prediction is 1.33. When the number of machines is 3, the lower bound of the algorithm without prediction is 1.5, and the upper bound of the algorithm with prediction is 1.5, and last we propose an algorithm that may improve the upper bound to 1.4 for 3 machines.