License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
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DOI: 10.4230/LIPIcs.MFCS.2022.41
URN: urn:nbn:de:0030-drops-168392
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2022/16839/
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Deng, Mingyang ; Vassilevska Williams, Virginia ; Zhong, Ziqian

New Lower Bounds and Upper Bounds for Listing Avoidable Vertices

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Abstract

We consider the problem of listing all avoidable vertices in a given n vertex graph. A vertex is avoidable if every pair of its neighbors is connected by a path whose internal vertices are not neighbors of the vertex or the vertex itself. Recently, Papadopolous and Zisis showed that one can list all avoidable vertices in O(n^{ω+1}) time, where ω < 2.373 is the square matrix multiplication exponent, and conjectured that a faster algorithm is not possible.
In this paper we show that under the 3-OV Hypothesis, and thus the Strong Exponential Time Hypothesis, n^{3-o(1)} time is needed to list all avoidable vertices, and thus the current best algorithm is conditionally optimal if ω = 2. We then show that if ω > 2, one can obtain an improved algorithm that for the current value of ω runs in O(n^3.32) time. We also show that our conditional lower bound is actually higher and supercubic, under a natural High Dimensional 3-OV hypothesis, implying that for our current knowledge of rectangular matrix multiplication, the avoidable vertex listing problem likely requires Ω(n^3.25) time. We obtain further algorithmic improvements for sparse graphs and bounded degree graphs.

BibTeX - Entry

@InProceedings{deng_et_al:LIPIcs.MFCS.2022.41,
  author =	{Deng, Mingyang and Vassilevska Williams, Virginia and Zhong, Ziqian},
  title =	{{New Lower Bounds and Upper Bounds for Listing Avoidable Vertices}},
  booktitle =	{47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)},
  pages =	{41:1--41:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-256-3},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{241},
  editor =	{Szeider, Stefan and Ganian, Robert and Silva, Alexandra},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2022/16839},
  URN =		{urn:nbn:de:0030-drops-168392},
  doi =		{10.4230/LIPIcs.MFCS.2022.41},
  annote =	{Keywords: Avoidable Vertex, Fine-Grained Complexity}
}

Keywords: Avoidable Vertex, Fine-Grained Complexity
Collection: 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)
Issue Date: 2022
Date of publication: 22.08.2022

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