Semi-online Algorithms on Two Hierarchical Machines with Reassignment

This paper studies the reassignment schedule problem on two identical machines with hierarchy. When the first machine has a hierarchy of 1 and the second machine has a hierarchy of 2, under the limitation of hierarchy constraints, after all jobs are assigned, we discuss the three situations of reassigning a job, maximizing the minimum machine load. The first situation is reassigning the last job in the sequence equivalent to the online situation, and the competitive ratio of any online algorithm is boundless; the second situation is reassigning the last job on the machine M2, proposing the optimal algorithm with a competitive ratio of 2; and the third situation is that reassigning any job is equivalent to the buffer size of 1. Addition, any k jobs with hierarchy 2 are reassigned, under the limitation of hierarchy constraints, and the competitive ratio of any online algorithm is at least 2.