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Fast Gaussian Elimination for Low Treewidth Matrices

Authors Martin Fürer , Carlos Hoppen , Vilmar Trevisan

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Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
Keywords
  • Gaussian elimination
  • FPT algorithms
  • treewidth

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Abstract

Let A = (a_{ij}) be an m× n matrix whose elements lie in an arbitrary field 𝔽, and let G be the bipartite graph with vertex set {v_1,…,v_m} ∪ {w_1,…,w_n} such that vertices v_i and w_j are adjacent if and only if a_{ij} ≠ 0. We introduce an algorithm that finds an m× n matrix U in row echelon form and a permutation matrix Q of order n, such that AQ is row equivalent to U. If a tree decomposition 𝒯 of G of width k and size O(k(m+n)) is part of the input, then Q and the columns of U that contain a pivot can be computed in time O(k²(m+n)). Among other things, this allows us to compute the rank and the determinant of A in time O(k²(m+n)). It also allows us to decide in time O(k²(m+n)) whether the linear system Ax = b has a solution and to compute a solution of the linear system in case it exists.

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Martin Fürer, Carlos Hoppen, and Vilmar Trevisan. Fast Gaussian Elimination for Low Treewidth Matrices. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 116:1-116:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ESA.2025.116

Author Details

Martin Fürer
  • Department of Computer Science and Engineering, Pennsylvania State University, University Park, PA, USA
Carlos Hoppen
  • Instituto de Matemática e Estatística, Universidade Federal do Rio Grande do Sul, Brazil
Vilmar Trevisan
  • Instituto de Matemática e Estatística, Universidade Federal do Rio Grande do Sul, Brazil

Funding

  • Hoppen, Carlos: C. Hoppen acknowledges the support of CNPq 315132/2021-3.
  • Trevisan, Vilmar: V. Trevisan acknowledges partial support of CNPq grants 409746/2016-9 and 303334/2016-9, CAPES under project MATHAMSUD 18-MATH-01 and FAPERGS under Project PqG 17/2551-0001. CNPq is Conselho Nacional de Desenvolvimento Científico e Tecnológico, CAPES is Coordenação de Aperfeiçoamento de Pessoal de Nível Superior and FAPERGS is Fundação de Amparo à Pesquisa do Estado do Rio Grande do Sul.

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