Linear-Time Multilevel Graph Partitioning via Edge Sparsification (2504.17615v1)

Published 24 Apr 2025 in cs.DS

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 $1.49\times$ average speedup (up to $4\times$ for some instances) with only 1\% loss in solution quality. Moreover, our algorithm clearly outperforms state-of-the-art single-level and streaming approaches.

Summary

The paper introduces a novel linear-time (O(n+m)) multilevel graph partitioning algorithm by integrating edge sparsification techniques with multilevel refinement capabilities.
Edge sparsification is used to construct coarser graph levels and maintain cut quality, demonstrating practical effectiveness with a 1.49x speedup and only a 1% cut increase when integrated into a state-of-the-art partitioner.
This method provides a scalable approach for partitioning large graphs, offering insights into the trade-off between computational cost and partition quality while identifying graph modularity as a diagnostic tool.

Linear-Time Multilevel Graph Partitioning via Edge Sparsification

The research presented in this paper offers a novel and efficient approach to the classical problem of balanced graph partitioning by introducing a linear-time multilevel graph partitioning algorithm. Historically, graph partitioning algorithms have been dichotomized into either high-quality multilevel algorithms, which often incur superlinear computational costs, or more computationally economical single-level and streaming algorithms, which typically sacrifice quality. The authors bridge this divide by leveraging edge sparsification techniques to construct a hierarchy of increasingly coarser graph representations, ensuring a linear running time without major detriments to partition quality.

Core Contributions

The paper highlights several significant contributions:

Linear-Time Multilevel Algorithm: By integrating controlled edge sparsification with multilevel refinement capabilities, the authors demonstrate that it is possible to achieve linear time complexity, O(n + m), where n is the number of nodes and m is the number of edges of the graph. This approach circumvents traditional performance bottlenecks inherent to large-scale multilevel algorithms.
Edge Sparsification: Edge sparsification is utilized to determine geometric reductions in graph size at successive hierarchy levels. Various sampling strategies, such as uniform sampling and weighted threshold sampling, are deployed to select key edges while preserving pertinent partition characteristics. The authors provide theoretical evidence underpinning these methods, proving their efficacy in maintaining cut integrity.
Integration with State-of-the-Art: The proposed algorithm has been incorporated into the KAMINPAR multilevel partitioner, where it maintains strong parallel scalability, achieving a 1.49x average speedup in total partitioning time with only a minimal 1% increase in edge cuts compared to traditional methods. The algorithm outperforms established single-level (PULP) and streaming partitioners (CUTTANA) in both solution quality and speed.
Worst-Case Running Time Analysis: The paper provides insights into scenarios where running time performance may degrade, particularly in graphs with low modularity. Such graphs are identified as potential worst-case instances, suggesting the necessity of sparsification to achieve the desired linear complexity.

Practical and Theoretical Implications

The implications of this work are multifaceted:

Scalability: By establishing a framework for linear-time multilevel partitioning, the paper proposes a scalable approach for handling large-scale graph datasets, which is crucial for real-world applications requiring high computational efficiency, such as distributed systems and parallel computing environments.
Balanced Trade-offs: The research emphasizes the critical balance between computational cost and partition quality, advocating for methodologies that do not compromise on either front. This is particularly relevant for applications in scientific computing, where high-quality partitions can significantly enhance computation and memory performance.
Modularity as a Diagnostic Tool: The insights into modularity provide a diagnostic tool for identifying graphs that may challenge the algorithm's efficiency, offering a basis for further exploration within both empirical and theoretical domains.

Future Directions

Future work may explore various enhancements to the sparsification techniques, potentially developing smarter or adaptive methodologies that respond dynamically to graph structure variations. Additionally, extending the theoretical bounds and performance guarantees in heterogeneous computing environments or across varied graph topologies could further cement the applicability of this algorithm in diverse fields of data analysis and computational science.