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License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SoCG.2016.12
URN: urn:nbn:de:0030-drops-59040
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2016/5904/
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Bagchi, Bhaskar ; Datta, Basudeb ; Burton, Benjamin A. ; Singh, Nitin ; Spreer, Jonathan

Efficient Algorithms to Decide Tightness

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LIPIcs-SoCG-2016-12.pdf (0.6 MB)

Abstract

Tightness is a generalisation of the notion of convexity: a space is tight if and only if it is "as convex as possible", given its topological constraints. For a simplicial complex, deciding tightness has a straightforward exponential time algorithm, but more efficient methods to decide tightness are only known in the trivial setting of triangulated surfaces.

In this article, we present a new polynomial time procedure to decide tightness for triangulations of 3-manifolds - a problem which previously was thought to be hard. In addition, for the more difficult problem of deciding tightness of 4-dimensional combinatorial manifolds, we describe an algorithm that is fixed parameter tractable in the treewidth of the 1-skeletons of the vertex links. Finally, we show that simpler treewidth parameters are not viable: for all non-trivial inputs, we show that the treewidths of both the 1-skeleton and the dual graph must grow too quickly for a standard treewidth-based algorithm to remain tractable.

BibTeX - Entry

@InProceedings{bagchi_et_al:LIPIcs:2016:5904,
  author =	{Bhaskar Bagchi and Basudeb Datta and Benjamin A. Burton and Nitin Singh and Jonathan Spreer},
  title =	{{Efficient Algorithms to Decide Tightness}},
  booktitle =	{32nd International Symposium on Computational Geometry (SoCG 2016)},
  pages =	{12:1--12:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-009-5},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{51},
  editor =	{S{\'a}ndor Fekete and Anna Lubiw},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2016/5904},
  URN =		{urn:nbn:de:0030-drops-59040},
  doi =		{10.4230/LIPIcs.SoCG.2016.12},
  annote =	{Keywords: discrete geometry and topology,  polynomial time algorithms,  fixed parameter tractability,  tight triangulations,  simplicial complexes}
}

Keywords: discrete geometry and topology, polynomial time algorithms, fixed parameter tractability, tight triangulations, simplicial complexes
Collection: 32nd International Symposium on Computational Geometry (SoCG 2016)
Issue Date: 2016
Date of publication: 10.06.2016

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