Solving Sparse, Symmetric, Diagonally-Dominant Linear Systems in Time $O (m^{1.31})$

TL;DR

This paper introduces an efficient linear-system solver for symmetric diagonally dominant matrices, achieving near-linear time complexity and extending combinatorial preconditioning techniques for improved performance.

Contribution

The work extends Vaidya's combinatorial preconditioning techniques to develop a faster solver for specific classes of sparse, symmetric, diagonally dominant matrices.

Findings

Solver runs in near-linear time for certain matrix classes

Time complexity depends on eigenvalue ratio and graph properties

Improved bounds for matrices with specific graph genus or minors

Abstract

We present a linear-system solver that, given an $n$-by-$n$ symmetric positive semi-definite, diagonally dominant matrix $A$ with $m$ non-zero entries and an $n$-vector $\bb $, produces a vector $\xxt$ within relative distance $\epsilon$ of the solution to $A \xx = \bb$ in time $O (m^{1.31} \log (n \kappa_{f} (A)/\epsilon)^{O (1)})$, where $\kappa_{f} (A)$ is the log of the ratio of the largest to smallest non-zero eigenvalue of $A$. In particular, $\log (\kappa_{f} (A)) = O (b \log n)$, where $b$ is the logarithm of the ratio of the largest to smallest non-zero entry of $A$. If the graph of $A$ has genus $m^{2\theta}$ or does not have a $K_{m^{\theta}} $ minor, then the exponent of $m$ can be improved to the minimum of $1 + 5 \theta $ and $(9/8) (1+\theta)$. The key contribution of our work is an extension of Vaidya's techniques for constructing and analyzing combinatorial preconditioners.