License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2023.48
URN: urn:nbn:de:0030-drops-188739
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2023/18873/
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Bishnu, Arijit ; Ghosh, Arijit ; Mishra, Gopinath

On the Complexity of Triangle Counting Using Emptiness Queries

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Abstract

Beame et al. [ITCS'18 & TALG'20] introduced and used the Bipartite Independent Set (BIS) and Independent Set (IS) oracle access to an unknown, simple, unweighted and undirected graph and solved the edge estimation problem. The introduction of this oracle set forth a series of works in a short time that either solved open questions mentioned by Beame et al. or were generalizations of their work as in Dell and Lapinskas [STOC'18 and TOCT'21], Dell, Lapinskas, and Meeks [SODA'20 and SICOMP'22], Bhattacharya et al. [ISAAC'19 & TOCS'21], and Chen et al. [SODA'20]. Edge estimation using BIS can be done using polylogarithmic queries, while IS queries need sub-linear but more than polylogarithmic queries. Chen et al. improved Beame et al.’s upper bound result for edge estimation using IS and also showed an almost matching lower bound. Beame et al. in their introductory work asked a few open questions out of which one was on estimating structures of higher order than edges, like triangles and cliques, using BIS queries.
In this work, we almost resolve the query complexity of estimating triangles using BIS oracle. While doing so, we prove a lower bound for an even stronger query oracle called Edge Emptiness (EE) oracle, recently introduced by Assadi, Chakrabarty, and Khanna [ESA'21] to test graph connectivity.

BibTeX - Entry

@InProceedings{bishnu_et_al:LIPIcs.APPROX/RANDOM.2023.48,
  author =	{Bishnu, Arijit and Ghosh, Arijit and Mishra, Gopinath},
  title =	{{On the Complexity of Triangle Counting Using Emptiness Queries}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)},
  pages =	{48:1--48:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-296-9},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{275},
  editor =	{Megow, Nicole and Smith, Adam},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2023/18873},
  URN =		{urn:nbn:de:0030-drops-188739},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2023.48},
  annote =	{Keywords: Triangle Counting, Emptiness Queries, Bipartite Independent Set Query}
}

Keywords: Triangle Counting, Emptiness Queries, Bipartite Independent Set Query
Collection: Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)
Issue Date: 2023
Date of publication: 04.09.2023

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