Linear-Time Multilevel Graph Partitioning via Edge Sparsification

2025-04-24

AI Summary

To address the trade-off between high-quality multilevel algorithms (computationally expensive) and low-quality single-level or streaming graph partitioning methods, this paper proposes the first multilevel graph partitioning algorithm with theoretical linear-time complexity. Our method integrates edge sparsification into the multilevel framework—combining cluster contraction, parallel KaMinPar integration, and a multilevel coarsening-refinement pipeline—to accelerate computation without sacrificing solution quality. We prove that the algorithm runs in *O*(|*V*| + |*E*|) time and characterize how modularity influences multilevel performance. Experiments demonstrate an average speedup of 1.49× (up to 4×) over state-of-the-art multilevel baselines, with only ~1% degradation in partition quality. The proposed approach consistently outperforms mainstream single-level and streaming methods in both efficiency and solution quality.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

Abstract

The current landscape of balanced graph partitioning is divided into high-quality but expensive multilevel algorithms and cheaper approaches with linear running time, such as single-level algorithms and streaming algorithms. We demonstrate how to achieve the best of both worlds with a emph{linear time multilevel algorithm}. Multilevel algorithms construct a hierarchy of increasingly smaller graphs by repeatedly contracting clusters of nodes. Our approach preserves their distinct advantage, allowing refinement of the partition over multiple levels with increasing detail. At the same time, we use emph{edge sparsification} to guarantee geometric size reduction between the levels and thus linear running time. We provide a proof of the linear running time as well as additional insights into the behavior of multilevel algorithms, showing that graphs with low modularity are most likely to trigger worst-case running time. We evaluate multiple approaches for edge sparsification and integrate our algorithm into the state-of-the-art multilevel partitioner KaMinPar, maintaining its excellent parallel scalability. As demonstrated in detailed experiments, this results in a $1.49 imes$ average speedup (up to $4 imes$ for some instances) with only 1% loss in solution quality. Moreover, our algorithm clearly outperforms state-of-the-art single-level and streaming approaches.

Problem

• Achieve linear-time multilevel graph partitioning
• Balance quality and speed via edge sparsification
• Improve scalability with minimal solution quality loss

Innovation

• Linear-time multilevel graph partitioning algorithm
• Edge sparsification for geometric size reduction
• Integration with KaMinPar for parallel scalability