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

TL;DR

This paper demonstrates that solving linear equations in combinatorial Laplacians of 2-complexes is computationally as hard as solving general linear equations, indicating significant complexity challenges in this area.

Contribution

The paper establishes a nearly-linear time reduction from general linear equations to those involving combinatorial Laplacians of 2-complexes, showing their computational hardness.

Findings

Linear equations in 2-complex Laplacians are as hard as general linear equations.

Nearly-linear time solvers for 2-complex Laplacians would imply solutions for all linear equations.

The reduction preserves sparsity up to poly-logarithmic factors.

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 $c \geq 1$, if we can solve linear equations in combinatorial Laplacians of 2-complexes up to high accuracy in time $\tilde{O}((\# \text{ of nonzero coefficients})^c)$, then we can solve general linear equations with polynomially bounded integer coefficients and condition numbers up to high accuracy in time $\tilde{O}((\# \text{ of nonzero coefficients})^c)$. 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.