2 Search Results for "Pretto, Luca"

Document
DOI: 10.4230/LIPIcs.ITCS.2026.89
On Solving Asymmetric Diagonally Dominant Linear Systems in Sublinear Time

Authors: Tsz Chiu Kwok, Zhewei Wei, and Mingji Yang

Published in: LIPIcs, Volume 362, 17th Innovations in Theoretical Computer Science Conference (ITCS 2026)

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^{⊤}x^{∗} for a given vector t ∈ ℝⁿ and a specific solution x^{∗}. This setting naturally generalizes the study of sublinear-time solvers for symmetric diagonally dominant (SDD) systems [Andoni-Krauthgamer-Pogrow, ITCS 2019] to the asymmetric case, which has remained underexplored despite extensive work on nearly-linear-time solvers for RDD/CDD systems. 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^{⊤}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.
Cite as

Tsz Chiu Kwok, Zhewei Wei, and Mingji Yang. On Solving Asymmetric Diagonally Dominant Linear Systems in Sublinear Time. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 89:1-89:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

Copy BibTex To Clipboard

@InProceedings{kwok_et_al:LIPIcs.ITCS.2026.89,
  author =	{Kwok, Tsz Chiu and Wei, Zhewei and Yang, Mingji},
  title =	{{On Solving Asymmetric Diagonally Dominant Linear Systems in Sublinear Time}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{89:1--89:25},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-410-9},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{362},
  editor =	{Saraf, Shubhangi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2026.89},
  URN =		{urn:nbn:de:0030-drops-253768},
  doi =		{10.4230/LIPIcs.ITCS.2026.89},
  annote =	{Keywords: Spectral Graph Theory, Linear Systems, Sublinear Algorithms}
}
Document
DOI: 10.4230/LIPIcs.STACS.2018.18
On Approximating the Stationary Distribution of Time-reversible Markov Chains

Authors: Marco Bressan, Enoch Peserico, and Luca Pretto

Published in: LIPIcs, Volume 96, 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)

Abstract
Approximating the stationary probability of a state in a Markov chain through Markov chain Monte Carlo techniques is, in general, inefficient. Standard random walk approaches require tilde{O}(tau/pi(v)) operations to approximate the probability pi(v) of a state v in a chain with mixing time tau, and even the best available techniques still have complexity tilde{O}(tau^1.5 / pi(v)^0.5); and since these complexities depend inversely on pi(v), they can grow beyond any bound in the size of the chain or in its mixing time. In this paper we show that, for time-reversible Markov chains, there exists a simple randomized approximation algorithm that breaks this "small-pi(v) barrier".
Cite as

Marco Bressan, Enoch Peserico, and Luca Pretto. On Approximating the Stationary Distribution of Time-reversible Markov Chains. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 18:1-18:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

Copy BibTex To Clipboard

@InProceedings{bressan_et_al:LIPIcs.STACS.2018.18,
  author =	{Bressan, Marco and Peserico, Enoch and Pretto, Luca},
  title =	{{On Approximating the Stationary Distribution of Time-reversible Markov Chains}},
  booktitle =	{35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)},
  pages =	{18:1--18:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-062-0},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{96},
  editor =	{Niedermeier, Rolf and Vall\'{e}e, Brigitte},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2018.18},
  URN =		{urn:nbn:de:0030-drops-84949},
  doi =		{10.4230/LIPIcs.STACS.2018.18},
  annote =	{Keywords: Markov chains, MCMC sampling, large graph algorithms, randomized algorithms, sublinear algorithms}
}
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