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Hodge Decomposition and General Laplacian Solvers for Embedded Simplicial Complexes

Authors Mitchell Black, Amir Nayyeri

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Author Details

Mitchell Black
  • School of Electrical Engineering and Computer Science, Oregon State University, Corvallis, OR, USA
Amir Nayyeri
  • School of Electrical Engineering and Computer Science, Oregon State University, Corvallis, OR, USA

Funding

The authors were supported in part by NSF grants CCF-1941086 and CCF-1816442.

Acknowledgements

The authors would like to thank the reviewers for their helpful comments, especially for an observation that improved the dependence on β in the runtime.

Cite As Get BibTex

Mitchell Black and Amir Nayyeri. Hodge Decomposition and General Laplacian Solvers for Embedded Simplicial Complexes. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 23:1-23:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.ICALP.2022.23

Abstract

We describe a nearly-linear time algorithm to solve the linear system L₁x = b parameterized by the first Betti number of the complex, where L₁ is the 1-Laplacian of a simplicial complex K that is a subcomplex of a collapsible complex X linearly embedded in ℝ³. Our algorithm generalizes the work of Black et al. [SODA2022] that solved the same problem but required that K have trivial first homology. Our algorithm works for complexes K with arbitrary first homology with running time that is nearly-linear with respect to the size of the complex and polynomial with respect to the first Betti number. The key to our solver is a new algorithm for computing the Hodge decomposition of 1-chains of K in nearly-linear time. Additionally, our algorithm implies a nearly quadratic solver and nearly quadratic Hodge decomposition for the 1-Laplacian of any simplicial complex K embedded in ℝ³, as K can always be expanded to a collapsible embedded complex of quadratic complexity.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Mathematics of computing → Algebraic topology
  • Theory of computation → Design and analysis of algorithms
  • Mathematics of computing → Numerical analysis
Keywords
  • Computational Topology
  • Laplacian solvers
  • Combinatorial Laplacian
  • Hodge decomposition
  • Parameterized Complexity

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