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Submodular Hypergraph Partitioning: Metric Relaxations and Fast Algorithms via an Improved Cut-Matching Game

Authors Antares Chen , Lorenzo Orecchia , Erasmo Tani

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ACM Subject Classification
  • Theory of computation → Facility location and clustering
Keywords
  • Hypergraph Partitioning
  • Cut Improvement
  • Cut-Matching Game

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Abstract

Despite there being significant work on developing spectral- [Chan et al., 2018; Lau et al., 2023; Kwok et al., 2022], and metric-embedding-based [Louis and Makarychev, 2016] approximation algorithms for hypergraph conductance, little is known regarding the approximability of other hypergraph partitioning objectives.
This work proposes algorithms for a general model of hypergraph partitioning that unifies both undirected and directed versions of many well-studied partitioning objectives. The first contribution of this paper introduces polymatroidal cut functions, a large class of cut functions amenable to approximation algorithms via metric embeddings and routing multicommodity flows. We demonstrate a simple O(√{log n})-approximation, where n is the number of vertices in the hypergraph, for these problems by rounding relaxations to metrics of negative-type.
The second contribution of this paper generalizes the cut-matching game framework of Khandekar et al. [Khandekar et al., 2007] to tackle polymatroidal cut functions. This yields an almost-linear time O(log n)-approximation algorithm for standard versions of undirected and directed hypergraph partitioning [Kwok et al., 2022]. A technical contribution of our construction is a novel cut-matching game, which greatly relaxes the set of allowed actions by the cut player and allows for the use of approximate s-t maximum flows by the matching player. We believe this to be of independent interest.

Cite As Get BibTex

Antares Chen, Lorenzo Orecchia, and Erasmo Tani. Submodular Hypergraph Partitioning: Metric Relaxations and Fast Algorithms via an Improved Cut-Matching Game. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 49:1-49:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ICALP.2025.49

Author Details

Antares Chen
  • University of Chicago, IL, USA
Lorenzo Orecchia
  • University of Chicago, IL, USA
Erasmo Tani
  • University of Chicago, IL, USA
  • Sapienza University, Rome, Italy

Funding

  • Chen, Antares: NSF DGE 2140001.
  • Orecchia, Lorenzo: NSF CAREER 1943510.

Acknowledgements

The authors would like to thank anonymous reviewers for suggesting related work on polymatroidal networks, Thatchaphol Saranurak for suggesting related work on the use of cut-matching games beyond partitioning graphs, and Jeffrey Negrea for assisting on topics related to online learning. AC would like to dedicate this submission to Luca Trevisan. A portion of this work was done while AC visited Bocconi; among many other fond memories, AC recalls long discussions with Luca on this project and other topics of research. AC would also like to thank Max Ovsiankin and Marco Pirazzini for their steadfast support in finding the sparsest of cuts.

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