2 Search Results for "Schröder, Felix"

Document
DOI: 10.4230/LIPIcs.SoCG.2023.31
Linear Size Universal Point Sets for Classes of Planar Graphs

Authors: Stefan Felsner, Hendrik Schrezenmaier, Felix Schröder, and Raphael Steiner

Published in: LIPIcs, Volume 258, 39th International Symposium on Computational Geometry (SoCG 2023)

Abstract
A finite set P of points in the plane is n-universal with respect to a class 𝒞 of planar graphs if every n-vertex graph in 𝒞 admits a crossing-free straight-line drawing with vertices at points of P. For the class of all planar graphs the best known upper bound on the size of a universal point set is quadratic and the best known lower bound is linear in n. Some classes of planar graphs are known to admit universal point sets of near linear size, however, there are no truly linear bounds for interesting classes beyond outerplanar graphs. In this paper, we show that there is a universal point set of size 2n-2 for the class of bipartite planar graphs with n vertices. The same point set is also universal for the class of n-vertex planar graphs of maximum degree 3. The point set used for the results is what we call an exploding double chain, and we prove that this point set allows planar straight-line embeddings of many more planar graphs, namely of all subgraphs of planar graphs admitting a one-sided Hamiltonian cycle. The result for bipartite graphs also implies that every n-vertex plane graph has a 1-bend drawing all whose bends and vertices are contained in a specific point set of size 4n-6, this improves a bound of 6n-10 for the same problem by Löffler and Tóth.
Cite as

Stefan Felsner, Hendrik Schrezenmaier, Felix Schröder, and Raphael Steiner. Linear Size Universal Point Sets for Classes of Planar Graphs. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 31:1-31:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{felsner_et_al:LIPIcs.SoCG.2023.31,
  author =	{Felsner, Stefan and Schrezenmaier, Hendrik and Schr\"{o}der, Felix and Steiner, Raphael},
  title =	{{Linear Size Universal Point Sets for Classes of Planar Graphs}},
  booktitle =	{39th International Symposium on Computational Geometry (SoCG 2023)},
  pages =	{31:1--31:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-273-0},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{258},
  editor =	{Chambers, Erin W. and Gudmundsson, Joachim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2023.31},
  URN =		{urn:nbn:de:0030-drops-178814},
  doi =		{10.4230/LIPIcs.SoCG.2023.31},
  annote =	{Keywords: Graph drawing, Universal point set, One-sided Hamiltonian, 2-page book embedding, Separating decomposition, Quadrangulation, 2-tree, Subcubic planar graph}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2022.29
Characterization of Matrices with Bounded Graver Bases and Depth Parameters and Applications to Integer Programming

Authors: Marcin Briański, Martin Koutecký, Daniel Král', Kristýna Pekárková, and Felix Schröder

Published in: LIPIcs, Volume 229, 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)

Abstract
An intensive line of research on fixed parameter tractability of integer programming is focused on exploiting the relation between the sparsity of a constraint matrix A and the norm of the elements of its Graver basis. In particular, integer programming is fixed parameter tractable when parameterized by the primal tree-depth and the entry complexity of A, and when parameterized by the dual tree-depth and the entry complexity of A; both these parameterization imply that A is sparse, in particular, the number of its non-zero entries is linear in the number of columns or rows, respectively. We study preconditioners transforming a given matrix to an equivalent sparse matrix if it exists and provide structural results characterizing the existence of a sparse equivalent matrix in terms of the structural properties of the associated column matroid. In particular, our results imply that the 𝓁₁-norm of the Graver basis is bounded by a function of the maximum 𝓁₁-norm of a circuit of A. We use our results to design a parameterized algorithm that constructs a matrix equivalent to an input matrix A that has small primal/dual tree-depth and entry complexity if such an equivalent matrix exists. Our results yield parameterized algorithms for integer programming when parameterized by the 𝓁₁-norm of the Graver basis of the constraint matrix, when parameterized by the 𝓁₁-norm of the circuits of the constraint matrix, when parameterized by the smallest primal tree-depth and entry complexity of a matrix equivalent to the constraint matrix, and when parameterized by the smallest dual tree-depth and entry complexity of a matrix equivalent to the constraint matrix.
Cite as

Marcin Briański, Martin Koutecký, Daniel Král', Kristýna Pekárková, and Felix Schröder. Characterization of Matrices with Bounded Graver Bases and Depth Parameters and Applications to Integer Programming. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 29:1-29:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{brianski_et_al:LIPIcs.ICALP.2022.29,
  author =	{Bria\'{n}ski, Marcin and Kouteck\'{y}, Martin and Kr\'{a}l', Daniel and Pek\'{a}rkov\'{a}, Krist\'{y}na and Schr\"{o}der, Felix},
  title =	{{Characterization of Matrices with Bounded Graver Bases and Depth Parameters and Applications to Integer Programming}},
  booktitle =	{49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)},
  pages =	{29:1--29:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-235-8},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{229},
  editor =	{Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2022.29},
  URN =		{urn:nbn:de:0030-drops-163702},
  doi =		{10.4230/LIPIcs.ICALP.2022.29},
  annote =	{Keywords: Integer programming, width parameters, matroids, Graver basis, tree-depth, fixed parameter tractability}
}
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