Solving Sparse, Symmetric, Diagonally-Dominant Linear Systems in Time $O (m^{1.31})$
Daniel A. Spielman, Shang-Hua Teng
TL;DR
The paper develops fast solvers for symmetric positive semidefinite diagonally dominant (PSDDD) systems by extending Vaidya’s combinatorial preconditioning into a support-theoretic framework built around Alon–Karp–Peleg–West trees. It introduces a provably efficient preconditioner B with a quantified bound on $\kappa_f(A,B)$ and constructs both one-shot and recursive Chebyshev-based solvers, achieving a near $O(m^{1.31+o(1)})$ runtime under broad topological conditions. The results include a non-recursive $O(m^{18/13+o(1)})$ solver and a recursive scheme whose exponent approaches $1.309$, with further improvements for planar, minor-free, or Gremban-cover graphs. These methods enable substantially faster structure-exploiting solvers for elliptic, resistive-network, and related PSDDD problems, especially when the sparsity graph has favorable topology. All mathematical notation is maintained with $...$ delimiters.
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 $ε$ of the solution to $A \xx = \bb$ in time $O (m^{1.31} \log (n κ_{f} (A)/ε)^{O (1)})$, where $κ_{f} (A)$ is the log of the ratio of the largest to smallest non-zero eigenvalue of $A$. In particular, $\log (κ_{f} (A)) = O (b \log n)$, 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 $m^{2θ}$ or does not have a $K_{m^θ} $ minor, then the exponent of $m$ can be improved to the minimum of $1 + 5 θ$ and $(9/8) (1+θ)$. The key contribution of our work is an extension of Vaidya's techniques for constructing and analyzing combinatorial preconditioners.