Hardness of Graph-Structured Algebraic and Symbolic Problems

TL;DR

This paper investigates the computational hardness of solving graph-structured algebraic linear systems over finite fields and polynomial rings, establishing reductions from general systems and analyzing symbolic matrix problems.

Contribution

It introduces reductions from general linear systems to graph Laplacians and walk matrices over finite fields, and formalizes the complexity of symbolic matrix identity testing.

Findings

Reducing general linear systems to graph Laplacians over finite fields shows their computational equivalence.

Symbolic matrix identity testing with degree 1 and variable multiplicity 3 is as hard as over the reals.

The results suggest limitations in extending Laplacian solvers from real to finite-field settings.

Abstract

In this paper, we study the hardness of solving graph-structured linear systems with coefficients over a finite field $\mathbb{Z}_p$ and over a polynomial ring $\mathbb{F}[x_1,\ldots,x_t]$. We reduce solving general linear systems in $\mathbb{Z}_p$ to solving unit-weight low-degree graph Laplacians over $\mathbb{Z}_p$ with a polylogarithmic overhead on the number of non-zeros. Given the hardness of solving general linear systems in $\mathbb{Z}_p$ [Casacuberta-Kyng 2022], this result shows that it is unlikely that we can generalize Laplacian solvers over $\mathbb{R}$, or finite-element based methods over $\mathbb{R}$ in general, to a finite-field setting. We also reduce solving general linear systems over $\mathbb{Z}_p$ to solving linear systems whose coefficient matrices are walk matrices (matrices with all ones on the diagonal) and normalized Laplacians (Laplacians that are also walk matrices) over $\mathbb{Z}_p$. We often need to apply linear system solvers to random linear systems, in which case the worst case analysis above might be less relevant. For example, we often need to substitute variables in a symbolic matrix with random values. Here, a symbolic matrix is simply a matrix whose entries are in a polynomial ring $\mathbb{F}[x_1, \ldots, x_t]$. We formally define the reducibility between symbolic matrix classes, which are classified in terms of the degrees of the entries and the number of occurrences of the variables. We show that the determinant identity testing problem for symbolic matrices with polynomial degree $1$ and variable multiplicity at most $3$ is at least as hard as the same problem for general matrices over $\mathbb{R}$.