Bin stretching with migration on two hierarchical machines

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≥0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M \ge 0$$\end{document}, 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 offline makespan is provided to the scheduler in advance. This is still a non-trivial variant for any migration factor M>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M > 0$$\end{document}. We prove tight bounds on the competitive ratio for any migration factor M. The design and analysis is split into several cases, based on the value of M, and on the resulting competitive ratio. Unlike the online variant with migration for two hierarchical machines, this case allows an online fully polynomial time approximation scheme.