Precedence-constrained Scheduling of Malleable Jobs with Preemption

TL;DR

This paper develops a near-optimal approximation algorithm for scheduling malleable jobs with precedence constraints on identical machines, focusing on concave processing functions common in cloud computing, improving efficiency over previous methods.

Contribution

It introduces a $(2+ ext{epsilon})$-approximation algorithm for general concave processing functions and a $(1+ ext{epsilon})$-approximation for power functions, advancing scheduling theory.

Findings

Achieves a $(2+ ext{epsilon})$-approximation for concave processing functions.

Improves to a $(1+ ext{epsilon})$-approximation for power functions.

Establishes the best possible approximation ratio under complexity assumptions.

Abstract

Scheduling jobs with precedence constraints on a set of identical machines to minimize the total processing time (makespan) is a fundamental problem in combinatorial optimization. In practical settings such as cloud computing, jobs are often malleable, i.e., can be processed on multiple machines simultaneously. The instantaneous processing rate of a job is a non-decreasing function of the number of machines assigned to it (we call it the processing function). Previous research has focused on practically relevant concave processing functions, which obey the law of diminishing utility and generalize the classical (non-malleable) problem. Our main result is a $(2+\epsilon)$-approximation algorithm for concave processing functions (for any $\epsilon > 0$), which is the best possible under complexity theoretic assumptions. The approximation ratio improves to $(1 + \epsilon)$ for the interesting and practically relevant special case of power functions, i.e., $p_j(z) = c_j \cdot z^{\gamma}$.