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Efficient Diagonalization of Symmetric Matrices Associated with Graphs of Small Treewidth

Authors Martin Fürer , Carlos Hoppen , Vilmar Trevisan

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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

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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.

Cite As Get 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

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.

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