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Entrywise Approximate Solutions for SDDM Systems in Almost-Linear Time

Angelo Farfan, Mehrdad Ghadiri, Junzhao Yang

TL;DR

The paper tackles the problem of solving $\boldsymbol{L}\boldsymbol{x}=\boldsymbol{b}$ for invertible SDDM matrices with strong entrywise guarantees. It introduces a novel combination of a low-diameter (LD) cover and a threshold-decay (TD) framework, augmented with predictive insights, to drive only a small, targeted active set in each iteration. The main contribution is an entrywise $\underline{\approx}_{\epsilon}$ solution achieved in time $\widetilde{O}(m 2^{O(\sqrt{\log n})} \log^{2}(U\epsilon^{-1}\delta^{-1}))$, with high probability and modest bit complexity, by solving a sequence of normwise partial systems on strategically chosen subgraphs. This approach advances practical solvers for Laplacian-like systems by enabling accurate recovery of large coordinates while efficiently handling many small coordinates, with implications for numerical linear algebra and graph-based computations in near-linear time.

Abstract

We present an algorithm that given any invertible symmetric diagonally dominant M-matrix (SDDM), i.e., a principal submatrix of a graph Laplacian, $\boldsymbol{\mathit{L}}$ and a nonnegative vector $\boldsymbol{\mathit{b}}$, computes an entrywise approximation to the solution of $\boldsymbol{\mathit{L}} \boldsymbol{\mathit{x}} = \boldsymbol{\mathit{b}}$ in $\tilde{O}(m n^{o(1)})$ time with high probability, where $m$ is the number of nonzero entries and $n$ is the dimension of the system.

Entrywise Approximate Solutions for SDDM Systems in Almost-Linear Time

TL;DR

The paper tackles the problem of solving for invertible SDDM matrices with strong entrywise guarantees. It introduces a novel combination of a low-diameter (LD) cover and a threshold-decay (TD) framework, augmented with predictive insights, to drive only a small, targeted active set in each iteration. The main contribution is an entrywise solution achieved in time , with high probability and modest bit complexity, by solving a sequence of normwise partial systems on strategically chosen subgraphs. This approach advances practical solvers for Laplacian-like systems by enabling accurate recovery of large coordinates while efficiently handling many small coordinates, with implications for numerical linear algebra and graph-based computations in near-linear time.

Abstract

We present an algorithm that given any invertible symmetric diagonally dominant M-matrix (SDDM), i.e., a principal submatrix of a graph Laplacian, and a nonnegative vector , computes an entrywise approximation to the solution of in time with high probability, where is the number of nonzero entries and is the dimension of the system.

Paper Structure

This paper contains 32 sections, 16 theorems, 78 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Random walks.
  4. Solving SDDM linear systems with normwise error bounds.
  5. Technical Overview
  6. Threshold decay (TD) framework.
  7. TD with prediction.
  8. Low-diameter (LD) cover.
  9. Almost linear-time solver using LD cover.
  10. Constructing the LD cover.
  11. Related Work
  12. Low-Diameter Cover
  13. Probability Distance
  14. Constructing a Cover with Low Probability Distance Diameter
  15. The Almost-Linear-Time Algorithm
  16. ...and 17 more sections

Key Result

Theorem 1.1

There exists a randomized algorithm alg:SDDMSolve such that, for any $\delta \in (0, 1)$ and any $\epsilon > (nU)^{-2^{\sqrt{\log n}}}$ (i.e., not exponentially small), any invertible SDDM matrix $\boldsymbol{\mathit{L}} \in \mathbb{Z}^{n \times n}$ with $m$ nonzero integer entries in $[-U, U]$ and whose entries are represented by $O(\log (nU/\epsilon))$-bit floating points, using$\widetilde{O}$

Figures (4)

  • Figure 1: low-diameter construction
  • Figure 2: The Threshold Decay framework.
  • Figure 3: Solve the linear system restricted on a subset of vertices using the low-diameter cover
  • Figure 4: The entrywise approximate linear system solver for SDDM matrices in almost-linear-time

Theorems & Definitions (38)

  • Theorem 1.1
  • Definition 1.1: RDDL and SDDM Matrices
  • Definition 1.2: Associated Graph of an RDDL Matrix
  • Definition 1.3: Escape Probability
  • Lemma 1.4: Lemma 2.7 of GNY25
  • Lemma 1.5: Theorem 5.5 of ST04:journal
  • Lemma 1.6: Corollary 2.5 of GNY25
  • Corollary 1.7: Corollary 3.2 of GNY25
  • Definition 2.1: Probability distance
  • Claim 2.2: Properties of the probability distance
  • ...and 28 more