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DOI: 10.4230/LIPIcs.ISAAC.2024.38

Basis Sequence Reconfiguration in the Union of Matroids

Authors: Tesshu Hanaka, Yuni Iwamasa, Yasuaki Kobayashi, Yuto Okada, and Rin Saito

Published in: LIPIcs, Volume 322, 35th International Symposium on Algorithms and Computation (ISAAC 2024)

Abstract

Given a graph G and two spanning trees T and T' in G, normalSpanning Tree Reconfiguration asks whether there is a step-by-step transformation from T to T' such that all intermediates are also spanning trees of G, by exchanging an edge in T with an edge outside T at a single step. This problem is naturally related to matroid theory, which shows that there always exists such a transformation for any pair of T and T'. Motivated by this example, we study the problem of transforming a sequence of spanning trees into another sequence of spanning trees. We formulate this problem in the language of matroid theory: Given two sequences of bases of matroids, the goal is to decide whether there is a transformation between these sequences. We design a polynomial-time algorithm for this problem, even if the matroids are given as basis oracles. To complement this algorithmic result, we show that the problem of finding a shortest transformation is NP-hard to approximate within a factor of c log n for some constant c > 0, where n is the total size of the ground sets of the input matroids.

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Tesshu Hanaka, Yuni Iwamasa, Yasuaki Kobayashi, Yuto Okada, and Rin Saito. Basis Sequence Reconfiguration in the Union of Matroids. In 35th International Symposium on Algorithms and Computation (ISAAC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 322, pp. 38:1-38:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{hanaka_et_al:LIPIcs.ISAAC.2024.38,
  author =	{Hanaka, Tesshu and Iwamasa, Yuni and Kobayashi, Yasuaki and Okada, Yuto and Saito, Rin},
  title =	{{Basis Sequence Reconfiguration in the Union of Matroids}},
  booktitle =	{35th International Symposium on Algorithms and Computation (ISAAC 2024)},
  pages =	{38:1--38:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-354-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{322},
  editor =	{Mestre, Juli\'{a}n and Wirth, Anthony},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2024.38},
  URN =		{urn:nbn:de:0030-drops-221658},
  doi =		{10.4230/LIPIcs.ISAAC.2024.38},
  annote =	{Keywords: Combinatorial reconfiguration, Matroids, Polynomial-time algorithm, Inapproximability}
}

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DOI: 10.4230/LIPIcs.GD.2024.14

Bounding the Treewidth of Outer k-Planar Graphs via Triangulations

Authors: Oksana Firman, Grzegorz Gutowski, Myroslav Kryven, Yuto Okada, and Alexander Wolff

Published in: LIPIcs, Volume 320, 32nd International Symposium on Graph Drawing and Network Visualization (GD 2024)

Abstract

The treewidth is a structural parameter that measures the tree-likeness of a graph. Many algorithmic and combinatorial results are expressed in terms of the treewidth. In this paper, we study the treewidth of outer k-planar graphs, that is, graphs that admit a straight-line drawing where all the vertices lie on a circle, and every edge is crossed by at most k other edges. Wood and Telle [New York J. Math., 2007] showed that every outer k-planar graph has treewidth at most 3k + 11 using so-called planar decompositions, and later, Auer et al. [Algorithmica, 2016] proved that the treewidth of outer 1-planar graphs is at most 3, which is tight. In this paper, we improve the general upper bound to 1.5k + 2 and give a tight bound of 4 for k = 2. We also establish a lower bound: we show that, for every even k, there is an outer k-planar graph with treewidth k+2. Our new bound immediately implies a better bound on the cop number, which answers an open question of Durocher et al. [GD 2023] in the affirmative. Our treewidth bound relies on a new and simple triangulation method for outer k-planar graphs that yields few crossings with graph edges per edge of the triangulation. Our method also enables us to obtain a tight upper bound of k + 2 for the separation number of outer k-planar graphs, improving an upper bound of 2k + 3 by Chaplick et al. [GD 2017]. We also consider outer min-k-planar graphs, a generalization of outer k-planar graphs, where we achieve smaller improvements.

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Oksana Firman, Grzegorz Gutowski, Myroslav Kryven, Yuto Okada, and Alexander Wolff. Bounding the Treewidth of Outer k-Planar Graphs via Triangulations. In 32nd International Symposium on Graph Drawing and Network Visualization (GD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 320, pp. 14:1-14:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{firman_et_al:LIPIcs.GD.2024.14,
  author =	{Firman, Oksana and Gutowski, Grzegorz and Kryven, Myroslav and Okada, Yuto and Wolff, Alexander},
  title =	{{Bounding the Treewidth of Outer k-Planar Graphs via Triangulations}},
  booktitle =	{32nd International Symposium on Graph Drawing and Network Visualization (GD 2024)},
  pages =	{14:1--14:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-343-0},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{320},
  editor =	{Felsner, Stefan and Klein, Karsten},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2024.14},
  URN =		{urn:nbn:de:0030-drops-212988},
  doi =		{10.4230/LIPIcs.GD.2024.14},
  annote =	{Keywords: treewidth, outerplanar graphs, outer k-planar graphs, outer min-k-planar graphs, cop number, separation number}
}

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Documents authored by Okada, Yuto

Basis Sequence Reconfiguration in the Union of Matroids

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Bounding the Treewidth of Outer k-Planar Graphs via Triangulations

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