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DOI: 10.4230/LIPIcs.ESA.2025.116

Fast Gaussian Elimination for Low Treewidth Matrices

Authors: Martin Fürer, Carlos Hoppen, and Vilmar Trevisan

Published in: LIPIcs, Volume 351, 33rd Annual European Symposium on Algorithms (ESA 2025)

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)

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@InProceedings{furer_et_al:LIPIcs.ESA.2025.116,
  author =	{F\"{u}rer, Martin and Hoppen, Carlos and Trevisan, Vilmar},
  title =	{{Fast Gaussian Elimination for Low Treewidth Matrices}},
  booktitle =	{33rd Annual European Symposium on Algorithms (ESA 2025)},
  pages =	{116:1--116:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-395-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{351},
  editor =	{Benoit, Anne and Kaplan, Haim and Wild, Sebastian and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2025.116},
  URN =		{urn:nbn:de:0030-drops-245855},
  doi =		{10.4230/LIPIcs.ESA.2025.116},
  annote =	{Keywords: Gaussian elimination, FPT algorithms, treewidth}
}

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Track A: Algorithms, Complexity and Games

DOI: 10.4230/LIPIcs.ICALP.2020.52

Efficient Diagonalization of Symmetric Matrices Associated with Graphs of Small Treewidth

Authors: Martin Fürer, Carlos Hoppen, and Vilmar Trevisan

Published in: LIPIcs, Volume 168, 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)

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.

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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)

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@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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DOI: 10.4230/LIPIcs.ITCS.2017.14

Multi-Clique-Width

Authors: Martin Fürer

Published in: LIPIcs, Volume 67, 8th Innovations in Theoretical Computer Science Conference (ITCS 2017)

Abstract

Multi-clique-width is obtained by a simple modification in the definition of clique-width. It has the advantage of providing a natural extension of tree-width. Unlike clique-width, it does not explode exponentially compared to tree-width. Efficient algorithms based on multi-clique-width are still possible for interesting tasks like computing the independent set polynomial or testing c-colorability. In particular, c-colorability can be tested in time linear in n and singly exponential in c and the width k of a given multi-k-expression. For these tasks, the running time as a function of the multi-clique-width is the same as the running time of the fastest known algorithm as a function of the clique-width. This results in an exponential speed-up for some graphs, if the corresponding graph generating expressions are given. The reason is that the multi-clique-width is never bigger, but is exponentially smaller than the clique-width for many graphs. This gap shows up when the tree-width is basically equal to the multi-clique width as well as when the tree-width is not bounded by any function of the clique-width.

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Martin Fürer. Multi-Clique-Width. In 8th Innovations in Theoretical Computer Science Conference (ITCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 67, pp. 14:1-14:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{furer:LIPIcs.ITCS.2017.14,
  author =	{F\"{u}rer, Martin},
  title =	{{Multi-Clique-Width}},
  booktitle =	{8th Innovations in Theoretical Computer Science Conference (ITCS 2017)},
  pages =	{14:1--14:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-029-3},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{67},
  editor =	{Papadimitriou, Christos H.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2017.14},
  URN =		{urn:nbn:de:0030-drops-81775},
  doi =		{10.4230/LIPIcs.ITCS.2017.14},
  annote =	{Keywords: clique-width, parameterized complexity, tree-width, independent set polynomial, graph coloring}
}

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Documents authored by Fürer, Martin

Fast Gaussian Elimination for Low Treewidth Matrices

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Efficient Diagonalization of Symmetric Matrices Associated with Graphs of Small Treewidth

Abstract

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Multi-Clique-Width

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