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

2026-01-06

arXiv.org

AI Summary

This work addresses the continuous monotonic moldable job scheduling problem—assigning $n$ jobs with variable parallelism to $m$ identical machines to minimize the makespan—and presents the first approximation algorithm that combines strong theoretical guarantees with practical efficiency. Exploiting the non-increasing nature of job processing time with respect to the number of allocated machines, the algorithm integrates combinatorial optimization techniques to achieve a $(73/50 + \varepsilon) \approx 1.4593 + \varepsilon$ approximation ratio in $O(nm \log(1/\varepsilon))$ time. Compared to existing approaches, the proposed method significantly reduces time complexity while empirical evaluations demonstrate that its real-world performance substantially exceeds the theoretical worst-case bound.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Planning, Routing, and Scheduling: Mixed Discrete/Continuous Planning
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Architectures and workflows that use LLMs for crowd work
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

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.

Problem

• moldable job scheduling
• monotone
• contiguous
• approximation algorithm
• scheduling

Innovation

• moldable job scheduling
• approximation algorithm
• contiguous scheduling
• monotone processing time
• practical efficiency