Assigning and Scheduling Generalized Malleable Jobs Under Subadditive or Submodular Processing Speeds

Abstract

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.

Funding: D. Fotakis received financial support from the Hellenic Foundation for Research and Innovation (H.F.R.I.) [“First Call for H.F.R.I. Research Projects to Support Faculty Members and Researchers and the Procurement of High-Cost Research Equipment Grant,” Project BALSAM, HFRI-FM17-1424]. J. Matuschke received financial support from the Fonds Wetenschappelijk Onderzoek-Vlanderen [Research Project G072520N “Optimization and Analytics for Stochastic and Robust Project Scheduling”]. O. Papadigenopoulos received financial support from the National Science Foundation Institute for Machine Learning [Award 2019844].

Supplemental Material: The online appendix is available at https://doi.org/10.1287/opre.2022.0168.