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.
@InProceedings{furer_et_al:LIPIcs.ICALP.2020.52, author = {F\"{u}rer, Martin and Hoppen, Carlos and Trevisan, Vilmar}, title = {{Efficient Diagonalization of Symmetric Matrices Associated with Graphs of Small Treewidth}}, booktitle = {47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)}, pages = {52:1--52:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-138-2}, ISSN = {1868-8969}, year = {2020}, volume = {168}, editor = {Czumaj, Artur and Dawar, Anuj and Merelli, Emanuela}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2020.52}, URN = {urn:nbn:de:0030-drops-124590}, doi = {10.4230/LIPIcs.ICALP.2020.52}, annote = {Keywords: Treewidth, Diagonalization, Eigenvalues} }
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