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DOI: 10.4230/LIPIcs.SoCG.2018.13
URN: urn:nbn:de:0030-drops-87268
URL: http://drops.dagstuhl.de/opus/volltexte/2018/8726/
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Bjerkevik, Håvard Bakke ; Botnan, Magnus Bakke

Computational Complexity of the Interleaving Distance

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LIPIcs-SoCG-2018-13.pdf (0.5 MB)


Abstract

The interleaving distance is arguably the most prominent distance measure in topological data analysis. In this paper, we provide bounds on the computational complexity of determining the interleaving distance in several settings. We show that the interleaving distance is NP-hard to compute for persistence modules valued in the category of vector spaces. In the specific setting of multidimensional persistent homology we show that the problem is at least as hard as a matrix invertibility problem. Furthermore, this allows us to conclude that the interleaving distance of interval decomposable modules depends on the characteristic of the field. Persistence modules valued in the category of sets are also studied. As a corollary, we obtain that the isomorphism problem for Reeb graphs is graph isomorphism complete.

BibTeX - Entry

@InProceedings{bjerkevik_et_al:LIPIcs:2018:8726,
  author =	{Håvard Bakke Bjerkevik and Magnus Bakke Botnan},
  title =	{{Computational Complexity of the Interleaving Distance}},
  booktitle =	{34th International Symposium on Computational Geometry (SoCG 2018)},
  pages =	{13:1--13:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-066-8},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{99},
  editor =	{Bettina Speckmann and Csaba D. T{\'o}th},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2018/8726},
  URN =		{urn:nbn:de:0030-drops-87268},
  doi =		{10.4230/LIPIcs.SoCG.2018.13},
  annote =	{Keywords: Persistent Homology, Interleavings, NP-hard}
}

Keywords: Persistent Homology, Interleavings, NP-hard
Seminar: 34th International Symposium on Computational Geometry (SoCG 2018)
Issue Date: 2018
Date of publication: 24.05.2018


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