On Solving Asymmetric Diagonally Dominant Linear Systems in Sublinear Time

TL;DR

This work introduces a sublinear-time framework for solving asymmetric row/column diagonally dominant systems $M x=b$ by focusing on estimating $t^{ op}x^{ ast}$ for a fixed $t$ and a fixed solution $x^{ ast}$ with $M x^{ ast}=b$. Central to the approach is a Neumann-series representation of $x^{ ast}$, complemented by a novel maximum $p$-norm gap $ ilde{eta}_{ ext{max}}(M)$ that generalizes the spectral gap to directed/asymmetric matrices and governs truncation error. The authors develop a toolkit combining random-walk sampling, local push, and a bidirectional method to obtain a range of upper bounds under various access models and error measures, with concrete applications to PageRank and single-pair effective resistance. The results unify and broaden prior SDD sublinear-time solvers, reveal intrinsic connections between ForwardPush and BackwardPush, and establish both algorithmic upper bounds and lower bounds that highlight the computational role of the $p$-norm gaps in directed spectral graph theory. The framework thus advances sublinear solvers for directed graphs, local graph algorithms, and directed spectral theory, while suggesting practical impact for ranking, resistance computations, and graph-based inference in sublinear-time regimes.

Abstract

We initiate a study of solving a row/column diagonally dominant (RDD/CDD) linear system $Mx=b$ in sublinear time, with the goal of estimating $t^{\top}x^*$ for a given vector $t\in R^n$ and a specific solution $x^*$. This setting naturally generalizes the study of sublinear-time solvers for symmetric diagonally dominant (SDD) systems [AKP19] to the asymmetric case. Our first contributions are characterizations of the problem's mathematical structure. We express a solution $x^*$ via a Neumann series, prove its convergence, and upper bound the truncation error on this series through a novel quantity of $M$, termed the maximum $p$-norm gap. This quantity generalizes the spectral gap of symmetric matrices and captures how the structure of $M$ governs the problem's computational difficulty. For systems with bounded maximum $p$-norm gap, we develop a collection of algorithmic results for locally approximating $t^{\top}x^*$ under various scenarios and error measures. We derive these results by adapting the techniques of random-walk sampling, local push, and their bidirectional combination, which have proved powerful for special cases of solving RDD/CDD systems, particularly estimating PageRank and effective resistance on graphs. Our general framework yields deeper insights, extended results, and improved complexity bounds for these problems. Notably, our perspective provides a unified understanding of Forward Push and Backward Push, two fundamental approaches for estimating random-walk probabilities on graphs. Our framework also inherits the hardness results for sublinear-time SDD solvers and local PageRank computation, establishing lower bounds on the maximum $p$-norm gap or the accuracy parameter. We hope that our work opens the door for further study into sublinear solvers, local graph algorithms, and directed spectral graph theory.

Theorems & Definitions (79)

• Theorem 1.1
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• Corollary 1.10