License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
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DOI: 10.4230/LIPIcs.ICALP.2018.80
URN: urn:nbn:de:0030-drops-90849
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2018/9084/
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Jeong, Jisu ; Kim, Eun Jung ; Oum, Sang-il

Finding Branch-Decompositions of Matroids, Hypergraphs, and More

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Abstract

Given n subspaces of a finite-dimensional vector space over a fixed finite field F, we wish to find a "branch-decomposition" of these subspaces of width at most k, that is a subcubic tree T with n leaves mapped bijectively to the subspaces such that for every edge e of T, the sum of subspaces associated to the leaves in one component of T-e and the sum of subspaces associated to the leaves in the other component have the intersection of dimension at most k. This problem includes the problems of computing branch-width of F-represented matroids, rank-width of graphs, branch-width of hypergraphs, and carving-width of graphs.
We present a fixed-parameter algorithm to construct such a branch-decomposition of width at most k, if it exists, for input subspaces of a finite-dimensional vector space over F. Our algorithm is analogous to the algorithm of Bodlaender and Kloks (1996) on tree-width of graphs. To extend their framework to branch-decompositions of vector spaces, we developed highly generic tools for branch-decompositions on vector spaces. For this problem, a fixed-parameter algorithm was known due to Hlinený and Oum (2008). But their method is highly indirect. Their algorithm uses the non-trivial fact by Geelen et al. (2003) that the number of forbidden minors is finite and uses the algorithm of Hlinený (2006) on checking monadic second-order formulas on F-represented matroids of small branch-width. Our result does not depend on such a fact and is completely self-contained, and yet matches their asymptotic running time for each fixed k.

BibTeX - Entry

@InProceedings{jeong_et_al:LIPIcs:2018:9084,
  author =	{Jisu Jeong and Eun Jung Kim and Sang-il Oum},
  title =	{{Finding Branch-Decompositions of Matroids, Hypergraphs, and More}},
  booktitle =	{45th International Colloquium on Automata, Languages, and  Programming (ICALP 2018)},
  pages =	{80:1--80:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-076-7},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{107},
  editor =	{Ioannis Chatzigiannakis and Christos Kaklamanis and D{\'a}niel Marx and Donald Sannella},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2018/9084},
  URN =		{urn:nbn:de:0030-drops-90849},
  doi =		{10.4230/LIPIcs.ICALP.2018.80},
  annote =	{Keywords: branch-width, rank-width, carving-width, fixed-parameter tractability}
}

Keywords: branch-width, rank-width, carving-width, fixed-parameter tractability
Collection: 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)
Issue Date: 2018
Date of publication: 04.07.2018

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