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

TL;DR

This paper develops approximation algorithms for scheduling malleable jobs with complex processing speed functions, achieving strong theoretical guarantees and demonstrating practical efficiency across different concavity assumptions.

Contribution

It introduces approximation algorithms for scheduling with subadditive and submodular processing speeds, extending malleable scheduling theory to more general speed functions.

Findings

Algorithms achieve logarithmic approximation for submodular speeds.

Constant approximation for matroid rank functions.

Practical algorithms outperform worst-case guarantees.

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.