  In this paper, we consider semi-online scheduling with migration on two
hierarchical machines, with the purpose of minimizing the makespan. The meaning
of two hierarchical machines is that one of the machines can run any job, while
the other machine can only run specific jobs. Every instance also has a fixed
parameter M \geq 0, known as the migration factor. Jobs are presented one by
one. Each new job has to be assigned to a machine when it arrives, and at the
same time it is possible to modify the assignment of previously assigned jobs,
such that the moved jobs have a total size not exceeding M times the size of
the new job.
  The semi-online variant studied here is called bin stretching. In this
problem, the optimal makespan is provided to the scheduler in advance. This is
still a non-trivial variant for any migration factor M \geq 0. We prove tight
bounds on the competitive ratio for any migration factor M, where the design
and analysis is split into several cases, based on the value of M, and the
resulting competitive ratio. Unlike the online variant with migration for two
hierarchical machines, this case allows an online approximation scheme.

Data Structures and Algorithms;Discrete Mathematics;Combinatorics;Optimization and Control