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Linear-Time Multilevel Graph Partitioning via Edge Sparsification

Authors Lars Gottesbüren , Nikolai Maas , Dominik Rosch , Peter Sanders , Daniel Seemaier

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  • Theory of computation → Design and analysis of algorithms
Keywords
  • Graph Partitioning
  • Graph Algorithms
  • Linear Time Algorithms
  • Graph Sparsification

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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 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 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× average speedup (up to 4× for some instances) with only 1% loss in solution quality. Moreover, our algorithm clearly outperforms state-of-the-art single-level and streaming approaches.

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Lars Gottesbüren, Nikolai Maas, Dominik Rosch, Peter Sanders, and Daniel Seemaier. Linear-Time Multilevel Graph Partitioning via Edge Sparsification. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 32:1-32:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ESA.2025.32

Author Details

Lars Gottesbüren
  • Google Research, Zürich, Switzerland
Nikolai Maas
  • Karlsruhe Institute of Technology, Germany
Dominik Rosch
  • Karlsruhe Institute of Technology, Germany
Peter Sanders
  • Karlsruhe Institute of Technology, Germany
Daniel Seemaier
  • Karlsruhe Institute of Technology, Germany

Funding

This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement No. 882500).

Supplementary Materials

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