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DOI: 10.4230/LIPIcs.STACS.2017.18

The Parameterized Complexity of Finding a 2-Sphere in a Simplicial Complex

Authors: Benjamin Burton, Sergio Cabello, Stefan Kratsch, and William Pettersson

Published in: LIPIcs, Volume 66, 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)

Abstract

We consider the problem of finding a subcomplex K' of a simplicial complex K such that K' is homeomorphic to the 2-dimensional sphere, S^2. We study two variants of this problem. The first asks if there exists such a K' with at most k triangles, and we show that this variant is W[1]-hard and, assuming ETH, admits no O(n^(o(sqrt(k)))) time algorithm. We also give an algorithm that is tight with regards to this lower bound. The second problem is the dual of the first, and asks if K' can be found by removing at most k triangles from K. This variant has an immediate O(3^k poly(|K|)) time algorithm, and we show that it admits a polynomial kernelization to O(k^2) triangles, as well as a polynomial compression to a weighted version with bit-size O(k log k).

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Benjamin Burton, Sergio Cabello, Stefan Kratsch, and William Pettersson. The Parameterized Complexity of Finding a 2-Sphere in a Simplicial Complex. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 18:1-18:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{burton_et_al:LIPIcs.STACS.2017.18,
  author =	{Burton, Benjamin and Cabello, Sergio and Kratsch, Stefan and Pettersson, William},
  title =	{{The Parameterized Complexity of Finding a 2-Sphere in a Simplicial Complex}},
  booktitle =	{34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)},
  pages =	{18:1--18:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-028-6},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{66},
  editor =	{Vollmer, Heribert and Vall\'{e}e, Brigitte},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.18},
  URN =		{urn:nbn:de:0030-drops-70156},
  doi =		{10.4230/LIPIcs.STACS.2017.18},
  annote =	{Keywords: computational topology, parameterized complexity, simplicial complex}
}

Document

DOI: 10.4230/LIPIcs.SOCG.2015.270

An Edge-Based Framework for Enumerating 3-Manifold Triangulations

Authors: Benjamin A. Burton and William Pettersson

Published in: LIPIcs, Volume 34, 31st International Symposium on Computational Geometry (SoCG 2015)

Abstract

A typical census of 3-manifolds contains all manifolds (under various constraints) that can be triangulated with at most n tetrahedra. Although censuses are useful resources for mathematicians, constructing them is difficult: the best algorithms to date have not gone beyond n=12. The underlying algorithms essentially (i) enumerate all relevant 4-regular multigraphs on n nodes, and then (ii) for each multigraph G they enumerate possible 3-manifold triangulations with G as their dual 1-skeleton, of which there could be exponentially many. In practice, a small number of multigraphs often dominate the running times of census algorithms: for example, in a typical census on 10 tetrahedra, almost half of the running time is spent on just 0.3% of the graphs. Here we present a new algorithm for stage (ii), which is the computational bottleneck in this process. The key idea is to build triangulations by recursively constructing neighbourhoods of edges, in contrast to traditional algorithms which recursively glue together pairs of tetrahedron faces. We implement this algorithm, and find experimentally that whilst the overall performance is mixed, the new algorithm runs significantly faster on those "pathological" multigraphs for which existing methods are extremely slow. In this way the old and new algorithms complement one another, and together can yield significant performance improvements over either method alone.

Cite as

Benjamin A. Burton and William Pettersson. An Edge-Based Framework for Enumerating 3-Manifold Triangulations. In 31st International Symposium on Computational Geometry (SoCG 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 34, pp. 270-284, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{burton_et_al:LIPIcs.SOCG.2015.270,
  author =	{Burton, Benjamin A. and Pettersson, William},
  title =	{{An Edge-Based Framework for Enumerating 3-Manifold Triangulations}},
  booktitle =	{31st International Symposium on Computational Geometry (SoCG 2015)},
  pages =	{270--284},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-83-5},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{34},
  editor =	{Arge, Lars and Pach, J\'{a}nos},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SOCG.2015.270},
  URN =		{urn:nbn:de:0030-drops-51485},
  doi =		{10.4230/LIPIcs.SOCG.2015.270},
  annote =	{Keywords: triangulations, enumeration, graph theory}
}

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The Parameterized Complexity of Finding a 2-Sphere in a Simplicial Complex

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An Edge-Based Framework for Enumerating 3-Manifold Triangulations

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