Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2202.05011

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

Ming Ding·Rasmus Kyng·Maximilian Probst Gutenberg·Peng Zhang
View original source ↗Read source document ↗Export metadata ↓

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.

Explore related subjects

cs.CCcs.DScs.NAmath.ATmath.NA

Keep this discovery

BibTeXRIS

Ming Ding, Rasmus Kyng, Maximilian Probst Gutenberg, Peng Zhang. 2022-02-10. Hardness Results for Laplacians of Simplicial Complexes via Sparse-Linear Equation Complete Gadgets. https://arxiv.org/abs/2202.05011

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

The Computational Complexity of Holant Problems on 4-regular Graphs from the Stable Subgroup Sequence of $SL(2,\mathbb{C})$

The Holant framework provides a general setting for studying counting problems and includes graph homomorphisms (\#GH) and counting constraint satisfaction problems (\#CSP) as special cases. Over the past twenty years, a series of computational complexity dichotomies have been established for Holant problems, but the classification for complex-valued signatures is still open. The main obstacle is the case in which all signatures have even arity. In this paper, we establish a dichotomy for Holant problems with a complex-valued 4-ary signature, which is a key base case for the full classification of Holant problems. We present a new strategy by introducing Schur's theorem, the classification of finite subgroups of $\mathrm{SL}(2,\mathbb{C})$ and stable subgroup sequences into the proof. These new techniques are of independent interest.

cs.CC↗
arXiv2026

Topology inside NC$^1$

We show that ACC$^0$ is precisely what can be computed with constant-width circuits of polynomial size and polylogarithmic genus. This extends a characterization given by Hansen, showing that planar constant-width circuits also characterize ACC$^0$. Thus polylogarithmic genus provides no additional computational power in this model. We consider other generalizations of planarity, including crossing number and thickness. We show that constant-width circuits of polynomial size and thickness two already suffice to capture all of NC$^1$.

cs.CC↗