A Practical 73/50 Approximation for Contiguous Monotone Moldable Job Scheduling

TL;DR

This paper introduces a new practical algorithm for monotone moldable job scheduling with an approximation ratio of about 1.4593 + ε, improving efficiency and performance over previous methods, and demonstrates its effectiveness through implementation.

Contribution

The paper presents a practically efficient algorithm with a near 1.46 approximation ratio for contiguous monotone moldable job scheduling, with improved time complexity and empirical performance.

Findings

Algorithm achieves approx. ratio of 1.4593 + ε.

Practical implementation outperforms theoretical worst-case.

Applicable to both standard and contiguous variants.

Abstract

In moldable job scheduling, we are provided $m$ identical machines and $n$ jobs that can be executed on a variable number of machines. The execution time of each job depends on the number of machines assigned to execute that job. For the specific problem of monotone moldable job scheduling, jobs are assumed to have a processing time that is non-increasing in the number of machines. The previous best-known algorithms are: (1) a polynomial-time approximation scheme with time complexity $\Omega(n^{g(1/\varepsilon)})$, where $g(\cdot)$ is a super-exponential function [Jansen and Th\"ole '08; Jansen and Land '18], (2) a fully polynomial approximation scheme for the case of $m \geq 8\frac{n}{\varepsilon}$ [Jansen and Land '18], and (3) a $\frac{3}{2}$ approximation with time complexity $O(nm\log(mn))$ [Wu, Zhang, and Chen '23]. We present a new practically efficient algorithm with an approximation ratio of $\approx (1.4593 + \varepsilon)$ and a time complexity of $O(nm \log \frac{1}{\varepsilon})$. Our result also applies to the contiguous variant of the problem. In addition to our theoretical results, we implement the presented algorithm and show that the practical performance is significantly better than the theoretical worst-case approximation ratio.