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. As we show, when the processing speeds are fractionally subadditive, the problem of scheduling malleable jobs at minimum makespan can be approximated by a considerably simpler assignment problem. Moreover, we provide efficient approximation algorithms, with 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.

A Constant-Factor Approximation for Generalized Malleable Scheduling under $M^\natural$-Concave Processing Speeds

“…Our main contribution is an O(1)-approximation for the Assignment problem with M ♮ -concave processing speeds (see Section 1.2 for definitions of submodularity and M ♮ -concavity). Because all M ♮ -concave functions are submodular, our results, together with the aforementioned transformation [10], imply the following theorem.…”

“…Parallel execution of a job on multiple machines is often used to optimize the overall makespan in time-critical task scheduling systems. Practical applications are numerous and diverse, varying from task scheduling in production and logistics, such as quay crane allocation in naval logistics [4,13] and cleaning activities on trains [3], to optimizing the performance of computationally demanding tasks, such as web search index update [27] and training neural networks [11] (see also [9,10] for further references and examples).…”

“…Fotakis et al [10] left open whether there are processing speed functions more general than scaled matroid rank functions for which generalized malleable scheduling can be approximated within a constant factor. In this work, employing notions and techniques from the field of Discrete Convexity [21], we present a constant-factor approximation algorithm for job-dependent M ♮ -concave (a.k.a.…”

A Constant-Factor Approximation for Generalized Malleable Scheduling Under $$M^\natural $$-Concave Processing Speeds

“…Parallel execution of a job on multiple machines is often used to optimize the overall makespan in time-critical task scheduling systems. Practical applications are numerous and diverse, varying from task scheduling in production and logistics, such as quay crane allocation in naval logistics [4,15] and cleaning activities on trains [3], to optimizing the performance of computationally demanding tasks, such as web search index update [28] and training neural networks [12] (see also [9,10] for further references and examples).…”

“…However, as recently observed in [10], the aforementioned models, in which the processing power of a heterogeneous set of machines is expressed by a single scalar, cannot capture the (possibly complicated) combinatorial interaction effects arising among different machines processing the same job. Practical settings where such complicated interdependencies among machines may arise include modern heterogeneous parallel computing systems, typically consisting of CPUs, GPUs, and I/O nodes [5], and highly distributed processing systems, where massive parallelization is subject to constraints imposed by the underlying communication network [1]; see [10] for further references and examples. Having such practical settings in mind, Fotakis et al [10] introduced a generalized malleable scheduling model, where the processing time f j (S) = 1/g j (S) of a job j depends on a job-specific processing speed function g j (S) of the set of machines S allocated to j.…”