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

Authors Ming Ding, Rasmus Kyng, Maximilian Probst Gutenberg, Peng Zhang

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Author Details

Ming Ding
  • ETH Zürich, Switzerland
Rasmus Kyng
  • ETH Zürich, Switzerland
Maximilian Probst Gutenberg
  • ETH Zürich, Switzerland
Peng Zhang
  • Rutgers University, Piscataway, NJ, USA

Funding

  • Kyng, Rasmus: The research leading to these results has received funding from grant no. 200021 204787 of the Swiss National Science Foundation.
  • Gutenberg, Maximilian Probst: The research leading to these results has received funding from grant no. 200021 204787 of the Swiss National Science Foundation.

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Ming Ding, Rasmus Kyng, Maximilian Probst Gutenberg, and Peng Zhang. Hardness Results for Laplacians of Simplicial Complexes via Sparse-Linear Equation Complete Gadgets. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 53:1-53:19, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.ICALP.2022.53

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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
  • Mathematics of computing → Computations on matrices
  • Mathematics of computing → Algebraic topology
Keywords
  • Simplicial Complexes
  • Combinatorial Laplacians
  • Linear Equations
  • Fine-Grained Complexity

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