Malleable scheduling beyond identical machines

Abstract

In malleable job scheduling, jobs can be executed simultaneously on multiple machines with the processing time depending on the number of allocated machines. Jobs are required to be executed non-preemptively and in unison, in the sense that they occupy, during their execution, the same time interval over all the machines of the allocated set. In this work, we study generalizations of malleable job scheduling inspired by standard scheduling on unrelated machines. Specifically, we introduce a general model of malleable job scheduling, where each machine has a (possibly different) speed for each job, and the processing time of a job j on a set of allocated machines S depends on the total speed of S for j. For machines with unrelated speeds, we show that the optimal makespan cannot be approximated within a factor less than e e−1 , unless P = N P . On the positive side, we present polynomial-time algorithms with approximation ratios 2e e−1 for machines with unrelated speeds, 3 for machines with uniform speeds, and 7/3 for restricted assignments on identical machines. Our algorithms are based on deterministic LP rounding and result in sparse schedules, in the sense that each machine shares at most one job with other machines. We also prove lower bounds on the integrality gap of 1 + ϕ for unrelated speeds (ϕ is the golden ratio) and 2 for uniform speeds and restricted assignments. To indicate the generality of our approach, we show that it also yields constant factor approximation algorithms (i) for minimizing the sum of weighted completion times; and (ii) a variant where we determine the effective speed of a set of allocated machines based on the Lp norm of their speeds.

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

“…If, on the other hand, max i∈M j∈J:i∈Sj 1/g j (S j ) > α/V , then we conclude that the MMFA instance does not allow for a solution of value V (because such a solution would have load 1/V in the Assignment instance). ⊓ ⊔ This black-box reduction, together with the 3.16-approximation for speedimplementable functions [9] implies a 3-augmented 3.16-approximation for the well-known Santa Claus problem [2] (the special case of MMFA with linear utilities). A more careful analysis delivers the following stronger result: Corollary 1.…”

“…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).…”

“…On the other hand, scheduling malleable jobs on non-identical machines has received much less attention. As a natural first step, building on previous work by Correa et al [7] on the closely related splittable job model, Fotakis et al [9] introduced the setting of speed-implementable processing-time functions, where each machine i has an unrelated "speed" s ij for each job j and a job's processing time is a non-decreasing function of the total allocated speed fulfilling a natural generalization of the non-decreasing work assumption. They devised an LP-based 3.16-approximation for this setting.…”