Document Open Access Logo

Efficient Diagonalization of Symmetric Matrices Associated with Graphs of Small Treewidth

Authors Martin Fürer , Carlos Hoppen , Vilmar Trevisan

PDF
Thumbnail PDF

File

LIPIcs.ICALP.2020.52.pdf
  • Filesize: 0.52 MB
  • 18 pages

Document Identifiers

Author Details

Martin Fürer
  • Pennsylvania State University, University Park, PA, USA
Carlos Hoppen
  • Universidade Federal do Rio Grande do Sul, Porto Alegre, Brazil
Vilmar Trevisan
  • Universidade Federal do Rio Grande do Sul, Porto Alegre, Brazil

Funding

  • Hoppen, Carlos: CNPq 308054/2018-0 and FAPERGS 19/2551-0001727-8.
  • Trevisan, Vilmar: CNPq 409746/2016-9 and 303334/2016-9, CAPES-PRINT 88887.467572/2019-00, and FAPERGS 17/2551-0001.

Cite AsGet BibTex

Martin Fürer, Carlos Hoppen, and Vilmar Trevisan. Efficient Diagonalization of Symmetric Matrices Associated with Graphs of Small Treewidth. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 52:1-52:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.ICALP.2020.52

Abstract

Let M = (m_{ij}) be a symmetric matrix of order n and let G be the graph with vertex set {1,…,n} such that distinct vertices i and j are adjacent if and only if m_{ij} ≠ 0. We introduce a dynamic programming algorithm that finds a diagonal matrix that is congruent to M. If G is given with a tree decomposition 𝒯 of width k, then this can be done in time O(k|𝒯| + k² n), where |𝒯| denotes the number of nodes in 𝒯.

Subject Classification

ACM Subject Classification
  • Computing methodologies → Linear algebra algorithms
  • Theory of computation → Fixed parameter tractability
  • Mathematics of computing → Graph theory
Keywords
  • Treewidth
  • Diagonalization
  • Eigenvalues

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

References

  1. Hans L. Bodlaender, Paul S. Bonsma, and Daniel Lokshtanov. The fine details of fast dynamic programming over tree decompositions. In Gregory Z. Gutin and Stefan Szeider, editors, IPEC, volume 8246 of Lecture Notes in Computer Science, pages 41-53. Springer, 2013. Google Scholar
  2. Hans L. Bodlaender, John R. Gilbert, Hjálmtýr Hafsteinsson, and Ton Kloks. Approximating treewidth, pathwidth, frontsize, and shortest elimination tree. J. Algorithms, 18(2):238-255, 1995. Google Scholar
  3. Derek G. Corneil and Udi Rotics. On the relationship between clique-width and treewidth. SIAM J. Comput, 34(4):825-847, 2005. Google Scholar
  4. Bruno Courcelle and Stephan Olariu. Upper bounds to the clique width of graphs. Discrete Appl. Math., 101(1-3):77-114, 2000. Google Scholar
  5. Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015. Google Scholar
  6. Marek Cygan, Jesper Nederlof, Marcin Pilipczuk, Michal Pilipczuk, Joham M. M. van Rooij, and Jakub Onufry Wojtaszczyk. Solving connectivity problems parameterized by treewidth in single exponential time. In Proceedings of the 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science, pages 150-159, 2011. Google Scholar
  7. Rodney G. Downey and Michael R. Fellows. Fundamentals of Parameterized Complexity. Texts in Computer Science. Springer, 2013. Google Scholar
  8. Jörg Flum and Martin Grohe. Parameterized Complexity Theory. Texts in Theoretical Computer Science. An EATCS Series. Springer, 2006. Google Scholar
  9. 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. Google Scholar
  10. Martin Fürer. A natural generalization of bounded tree-width and bounded clique-width. In Alberto Pardo and Alfredo Viola, editors, LATIN 2014: Theoretical Informatics - 11th Latin American Symposium, Montevideo, Uruguay, March 31 - April 4, 2014. Proceedings, volume 8392 of Lecture Notes in Computer Science, pages 72-83. Springer, 2014. URL: https://doi.org/10.1007/978-3-642-54423-1_7.
  11. Martin Fürer. Multi-clique-width. In Christos H. Papadimitriou, editor, 8th Innovations in Theoretical Computer Science Conference, ITCS 2017, January 9-11, 2017, Berkeley, CA, USA, volume 67 of LIPIcs, pages 14:1-14:13. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2017. URL: https://doi.org/10.4230/LIPIcs.ITCS.2017.14.
  12. Martin Fürer, Carlos Hoppen, David P. Jacobs, and Vilmar Trevisan. Eigenvalue location in graphs of small clique-width. Linear Algebra and its Applications, 560:56-85, 2019. Google Scholar
  13. Martin Fürer and Huiwen Yu. Space saving by dynamic algebraization based on tree-depth. Theory of Computing Systems, 61(2):283-304, 2017. Google Scholar
  14. Archontia C. Giannopoulou, George B. Mertzios, and Rolf Niedermeier. Polynomial fixed-parameter algorithms: A case study for longest path on interval graphs. Theor. Comput. Sci., 689:67-95, 2017. URL: https://doi.org/10.1016/j.tcs.2017.05.017.
  15. T. Kloks. Treewidth: Computations and Approximations, volume 842 of Lecture Notes in Computer Science. Springer Verlag, 1994. Google Scholar
  16. Carl Meyer. Matrix analysis and applied linear algebra. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2000. Google Scholar
  17. Rolf Niedermeier. Invitation to Fixed-Parameter Algorithms. Oxford University Press, 2006. Google Scholar
  18. Neil Robertson and Paul D. Seymour. Graph minors II. Algorithmic aspects of tree-width. J. Algorithms, 7(3):309-322, 1986. Google Scholar
  19. Donald J. Rose, R. Endre. Tarjan, and George S. Lueker. Algorithmic aspects of vertex elimination on graphs. SIAM Journal on Computing, 5(2):266-283, 1976. Google Scholar
  20. Donald J. Rose and Robert Endre Tarjan. Algorithmic aspects of vertex elimination on directed graphs. SIAM Journal on Applied Mathematics, 34(1):176-197, 1978. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact