*Result*: Assigning and Scheduling Generalized Malleable Jobs Under Subadditive or Submodular Processing Speeds.

*Handling Heterogeneous Machines in Malleable Scheduling Parallelization is an important and widespread technique to speed up the completion of time-critical tasks, not only in high-speed computing, but also in operations planning in production and logistics. A fundamental model in this context is that of malleable jobs, each of which can be assigned to a subset of machine for parallel processing. In "Assigning and Scheduling Generalized Malleable Jobs Under Submodular or Subadditive Processing Speeds," Fotakis, Matuschke, and Papadigenopoulos go beyond the by now well-understood identical-machine setting in malleable scheduling and develop algorithmic approaches for scheduling malleable jobs under various discrete concavity assumptions on the joint processing speeds of the assigned (possibly very heterogeneous) machines. They show that under these assumptions, the task of finding a schedule of small makespan can be reduced to that of finding an assignment with small maximum machine load. For this latter problem, numerous efficient approximation algorithms are derived and their practical performance explored in a computational experiments. These results indicate that the computational challenges posed by parallelization in heterogeneous environments can indeed be overcome, enabling the optimization of heavily parallelized schedules in the aforementioned applications. Malleable scheduling is a model that captures the possibility of parallelization to expedite the completion of time-critical tasks. A malleable job can be allocated and processed simultaneously on multiple machines, occupying the same time interval on all these machines. We study a general version of this setting, in which the functions determining the joint processing speed of machines for a given job follow different discrete concavity assumptions (subadditivity, fractional subadditivity, submodularity, and matroid ranks). We show that under these assumptions, the problem of scheduling malleable jobs at minimum makespan can be approximated by a considerably simpler assignment problem. Moreover, we provide efficient approximation algorithms for both the scheduling and the assignment problem, with increasingly stronger guarantees for increasingly stronger concavity assumptions, including a logarithmic approximation factor for the case of submodular processing speeds and a constant approximation factor when processing speeds are determined by matroid rank functions. Computational experiments indicate that our algorithms outperform the theoretical worst-case guarantees. [ABSTRACT FROM AUTHOR]

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