License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ICALP.2022.53
URN: urn:nbn:de:0030-drops-163945
URL: https://drops.dagstuhl.de/opus/volltexte/2022/16394/
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Ding, Ming ; Kyng, Rasmus ; Gutenberg, Maximilian Probst ; Zhang, Peng

Hardness Results for Laplacians of Simplicial Complexes via Sparse-Linear Equation Complete Gadgets

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Abstract

We study linear equations in combinatorial Laplacians of k-dimensional simplicial complexes (k-complexes), a natural generalization of graph Laplacians. Combinatorial Laplacians play a crucial role in homology and are a central tool in topology. Beyond this, they have various applications in data analysis and physical modeling problems. It is known that nearly-linear time solvers exist for graph Laplacians. However, nearly-linear time solvers for combinatorial Laplacians are only known for restricted classes of complexes.
This paper shows that linear equations in combinatorial Laplacians of 2-complexes are as hard to solve as general linear equations. More precisely, for any constant c ≥ 1, if we can solve linear equations in combinatorial Laplacians of 2-complexes up to high accuracy in time Õ((# of nonzero coefficients)^c), then we can solve general linear equations with polynomially bounded integer coefficients and condition numbers up to high accuracy in time Õ((# of nonzero coefficients)^c). We prove this by a nearly-linear time reduction from general linear equations to combinatorial Laplacians of 2-complexes. Our reduction preserves the sparsity of the problem instances up to poly-logarithmic factors.

BibTeX - Entry

@InProceedings{ding_et_al:LIPIcs.ICALP.2022.53,
  author =	{Ding, Ming and Kyng, Rasmus and Gutenberg, Maximilian Probst and Zhang, Peng},
  title =	{{Hardness Results for Laplacians of Simplicial Complexes via Sparse-Linear Equation Complete Gadgets}},
  booktitle =	{49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)},
  pages =	{53:1--53:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-235-8},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{229},
  editor =	{Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2022/16394},
  URN =		{urn:nbn:de:0030-drops-163945},
  doi =		{10.4230/LIPIcs.ICALP.2022.53},
  annote =	{Keywords: Simplicial Complexes, Combinatorial Laplacians, Linear Equations, Fine-Grained Complexity}
}

Keywords: Simplicial Complexes, Combinatorial Laplacians, Linear Equations, Fine-Grained Complexity
Collection: 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)
Issue Date: 2022
Date of publication: 28.06.2022

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