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

TL;DR

This paper tackles the makespan minimization problem for monotone moldable scheduling on $m$ identical machines, including the contiguous variant. It introduces a practically efficient algorithm that achieves an approximation ratio of about $1.4593$ (i.e., $1.4593+\varepsilon$) with runtime $O(nm\log(1/\varepsilon))$, by extending the classical three-shelf framework with a three-class MCKP partition step and a repair mechanism that ensures contiguity. A notable analytic feature is the appearance of the Lambert $W$ function in a critical bound, reflecting a careful geometric/work-area argument. Experimental results on randomly generated instances show the method performs significantly better in practice than the worst-case bound, often approaching $10/7$ of $OPT$, and demonstrate the practicality of this approach for large instances. Overall, the work breaks the long-standing barrier of $1.5$ for practically efficient algorithms in this domain and provides a foundation for future improvements and extensions to related scheduling and packing problems.

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 $Ω(n^{g(1/\varepsilon)})$, where $g(\cdot)$ is a super-exponential function [Jansen and Thöle '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.

Figures (4)

• Figure 1: Schedule before \ref{['lem:machinesS0S1']} (left) and after (right). Pink jobs are in $\mathcal{C}_1$, blue jobs in $\mathcal{C}_2$, and yellow jobs in $\mathcal{C}_3$.
• Figure 2: Infeasible 3-shelf schedule. Blue jobs are in $S_0$, pink jobs in $S_1$, and yellow jobs in $S_2$.
• Figure 4: Runtime of the algorithm for varying number of machines.
• Figure 5: Runtime of the algorithm for varying number of jobs.

Theorems & Definitions (16)

• Theorem 1
• Definition 2: Canonical Number of Machines
• Theorem 3
• Lemma 4
• Definition 5: MCKP
• Lemma 6
• Remark 7
• Definition 8: 3-Shelf Schedule
• Lemma 8
• Lemma 8