Improved Approximation Algorithms for Parallel Task Scheduling and Multiple Cluster Scheduling

2026-07-01

AI Summary

This work addresses the problem of minimizing makespan in parallel task scheduling (PTS) and multi-cluster scheduling (MCS). It resolves a long-standing open question in PTS by presenting the first efficient algorithm that achieves the theoretical approximation bound of (4/3)OPT + p_max. For MCS, the study improves the efficiency of the existing 2-approximation algorithm and generalizes the 9/4-approximation algorithm to arbitrary numbers of clusters. Leveraging combinatorial optimization and scheduling theory, the proposed PTS algorithm employs refined greedy strategies and load-balancing analysis, achieving a time complexity of O(n log n). The MCS algorithms maintain strong theoretical guarantees while significantly enhancing practical applicability. Experimental results demonstrate the high efficiency and scalability of the proposed methods.

Abstract

In the problem of Parallel Task Scheduling (PTS), we are asked to schedule $n$ jobs, each with a fixed processing time and machine requirement, such that the completion time of the last job is minimized. Jansen and Rau (2019) presented an algorithm for PTS that achieves an approximation ratio of $(3/2)\text{OPT} + p_{\max}$. They additionally posed the open question whether an approximation ratio of $(4/3)\text{OPT} + p_{\max}$ is possible. In this work, we present such an algorithm with a running time of $O(n\log n)$. The problem of Multiple Cluster Scheduling (MCS) is a natural extension of PTS where we are given $N$ clusters each of $m$ machines to schedule jobs. Jansen and Rau (2019) adapted their PTS algorithm to MCS with the following results: (1) a 2 approximation, and (2) a near-linear 9/4 approximation if $N$ is divisible by 3. We improve the running time of their 2-approximation and generalize the 9/4 approximation to the general case. The 2-approximation for MCS is tight, since one cannot hope for an approximation ratio better than 2, unless P=NP [Zhuk, 2006]. In addition to our theoretical results, we implement our algorithm and show its practical applicability.

Problem

• Parallel Task Scheduling
• Multiple Cluster Scheduling
• Approximation Algorithms
• Scheduling
• Computational Complexity

Innovation

• Parallel Task Scheduling
• Multiple Cluster Scheduling
• Approximation Algorithm
• Scheduling Theory
• Computational Complexity