Document Open Access Logo

Fast Gaussian Elimination for Low Treewidth Matrices

Authors Martin Fürer , Carlos Hoppen , Vilmar Trevisan

PDF HTML 5

Files

Thumbnail PDF
PDF
LIPIcs.ESA.2025.116.pdf
  • Filesize: 0.79 MB
  • 15 pages
HTML5 Logo HTML (experimental)
LIPIcs.ESA.2025.116.html

Document Identifiers

Subject Classification

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

Metrics

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.

Cite As Get BibTex

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.

References

  1. U. Bertelè and F. Brioschi. Nonserial Dynamic Programming. Elsevier, 1972. Google Scholar
  2. Hans L. Bodlaender. A linear-time algorithm for finding tree-decompositions of small treewidth. SIAM J. Comput., 25(6):1305-1317, 1996. URL: https://doi.org/10.1137/S0097539793251219.
  3. Hans L. Bodlaender, Pål Grønås Drange, Markus S. Dregi, Fedor V. Fomin, Daniel Lokshtanov, and Michał Pilipczuk. A c^k n 5-approximation algorithm for treewidth. SIAM Journal on Computing, 45(2):317-378, 2016. URL: https://doi.org/10.1137/130947374.
  4. Sally Dong, Yin Tat Lee, and Guanghao Ye. A nearly-linear time algorithm for linear programs with small treewidth: a multiscale representation of robust central path. In Samir Khuller and Virginia Vassilevska Williams, editors, STOC '21: 53rd Annual ACM SIGACT Symposium on Theory of Computing, Virtual Event, Italy, June 21-25, 2021, pages 1784-1797. ACM, 2021. URL: https://doi.org/10.1145/3406325.3451056.
  5. Fedor V. Fomin, Daniel Lokshtanov, Saket Saurabh, Michal Pilipczuk, and Marcin Wrochna. Fully polynomial-time parameterized computations for graphs and matrices of low treewidth. ACM Trans. Algorithms, 14(3):34:1-34:45, 2018. URL: https://doi.org/10.1145/3186898.
  6. Martin Fürer, Carlos Hoppen, and Vilmar Trevisan. Efficient diagonalization of symmetric matrices associated with graphs of small treewidth. Theoretical Computer Science, 1040:115187, 2025. URL: https://doi.org/10.1016/j.tcs.2025.115187.
  7. Martin C. Golumbic and Clinton F. Goss. Perfect elimination and chordal bipartite graphs. Journal of Graph Theory, 2(2):155-163, 1978. URL: https://doi.org/10.1002/jgt.3190020209.
  8. Rudolf Halin. S-functions for graphs. Journal of Geometry, 8(1):171-186, 1976. URL: https://doi.org/10.1007/BF01917434.
  9. Ton Kloks. Treewidth, Computations and Approximations, volume 842 of Lecture Notes in Computer Science. Springer, 1994. URL: https://doi.org/10.1007/BFB0045375.
  10. Tuukka Korhonen. A single-exponential time 2-approximation algorithm for treewidth. SIAM Journal on Computing, pages FOCS21-174, November 2023. URL: https://doi.org/10.1137/22M147551X.
  11. Seymour V. Parter. The use of linear graphs in Gauss elimination. SIAM Review, 3(2):119-130, 1961. URL: https://doi.org/10.1137/1003021.
  12. Richard Peng and Santosh S. Vempala. Solving sparse linear systems faster than matrix multiplication. Commun. ACM, 67(7):79-86, 2024. URL: https://doi.org/10.1145/3615679.
  13. Venkatesh Radhakrishnan, Harry B. Hunt III, and Richard E. Stearns. Efficient algorithms for solving systems of linear equations and path problems. In Alain Finkel and Matthias Jantzen, editors, STACS 92, 9th Annual Symposium on Theoretical Aspects of Computer Science, Cachan, France, February 13-15, 1992, Proceedings, volume 577 of Lecture Notes in Computer Science, pages 109-119. Springer, 1992. URL: https://doi.org/10.1007/3-540-55210-3_177.
  14. Neil Robertson and Paul D. Seymour. Graph minors. I. excluding a forest. J. Comb. Theory B, 35(1):39-61, 1983. URL: https://doi.org/10.1016/0095-8956(83)90079-5.
  15. Neil Robertson and Paul D. Seymour. Graph minors. II. algorithmic aspects of tree-width. J. Algorithms, 7(3):309-322, 1986. URL: https://doi.org/10.1016/0196-6774(86)90023-4.
  16. Donald J. Rose. Triangulated graphs and the elimination process. Journal of Mathematical Analysis and Applications, 32(3):597-609, 1970. URL: https://doi.org/10.1016/0022-247X(70)90282-9.
  17. Donald J. Rose and Robert E. Tarjan. Algorithmic aspects of vertex elimination on directed graphs. SIAM Journal on Applied Mathematics, 34(1):176-197, 1978. Google Scholar
  18. Donald J. Rose, Robert E. Tarjan, and George S. Lueker. Algorithmic aspects of vertex elimination on graphs. SIAM Journal on Computing, 5(2):266-283, 1976. URL: https://doi.org/10.1137/0205021.
  19. David R. Wood. On tree-partition-width. European Journal of Combinatorics, 30(5):1245-1253, 2009. Part Special Issue on Metric Graph Theory. URL: https://doi.org/10.1016/j.ejc.2008.11.010.
Any Issues?
X

Feedback on the Current Page

CAPTCHA

Thanks for your feedback!

Feedback submitted to Dagstuhl Publishing

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact