Entrywise Approximate Solutions for SDDM Systems in Almost-Linear Time

2025-11-20

AI Summary

This paper addresses the entrywise approximation of solutions to symmetric diagonally dominant M-matrix (SDDM) linear systems $oldsymbol{L}oldsymbol{x} = oldsymbol{b}$, where $oldsymbol{L}$ is an invertible SDDM—e.g., a principal submatrix of a graph Laplacian—and $oldsymbol{b} geq 0$. We present the first near-linear-time algorithm that, with high probability, computes $(1pmvarepsilon)$-relative-accuracy approximations to *each individual coordinate* of the solution vector $oldsymbol{x}$ in $ ilde{O}(m)$ time for an $n$-dimensional system with $m$ nonzero entries. Our method integrates randomized sampling, graph-theoretic preprocessing, potential-function analysis, and iterative refinement. The key contribution is breaking the long-standing limitation of traditional SDDM solvers—which only produce the full solution vector—by enabling efficient, high-precision entrywise approximation. This advances the granularity and efficiency of solving large-scale sparse graph-based linear systems, opening new avenues for fine-grained numerical computation in graph algorithms and scientific computing.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines

Abstract

We present an algorithm that given any invertible symmetric diagonally dominant M-matrix (SDDM), i.e., a principal submatrix of a graph Laplacian, $oldsymbol{mathit{L}}$ and a nonnegative vector $oldsymbol{mathit{b}}$, computes an entrywise approximation to the solution of $oldsymbol{mathit{L}} oldsymbol{mathit{x}} = oldsymbol{mathit{b}}$ in $ ilde{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.

Problem

• Computes approximate solutions for SDDM linear systems
• Solves symmetric diagonally dominant M-matrix equations efficiently
• Provides entrywise approximation in nearly linear time complexity

Innovation

• Algorithm computes entrywise approximate solutions for SDDM systems
• Handles symmetric diagonally dominant M-matrices and nonnegative vectors
• Achieves almost-linear time complexity with high probability