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DOI: 10.4230/LIPIcs.ITCS.2020.38
URN: urn:nbn:de:0030-drops-117239
URL: https://drops.dagstuhl.de/opus/volltexte/2020/11723/
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Bera, Suman K. ; Pashanasangi, Noujan ; Seshadhri, C.

Linear Time Subgraph Counting, Graph Degeneracy, and the Chasm at Size Six

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Abstract

We consider the problem of counting all k-vertex subgraphs in an input graph, for any constant k. This problem (denoted SUB-CNT_k) has been studied extensively in both theory and practice. In a classic result, Chiba and Nishizeki (SICOMP 85) gave linear time algorithms for clique and 4-cycle counting for bounded degeneracy graphs. This is a rich class of sparse graphs that contains, for example, all minor-free families and preferential attachment graphs. The techniques from this result have inspired a number of recent practical algorithms for SUB-CNT_k. Towards a better understanding of the limits of these techniques, we ask: for what values of k can SUB_CNT_k be solved in linear time? We discover a chasm at k=6. Specifically, we prove that for k < 6, SUB_CNT_k can be solved in linear time. Assuming a standard conjecture in fine-grained complexity, we prove that for all k ⩾ 6, SUB-CNT_k cannot be solved even in near-linear time.

BibTeX - Entry

@InProceedings{bera_et_al:LIPIcs:2020:11723,
  author =	{Suman K. Bera and Noujan Pashanasangi and C. Seshadhri},
  title =	{{Linear Time Subgraph Counting, Graph Degeneracy, and the Chasm at Size Six}},
  booktitle =	{11th Innovations in Theoretical Computer Science Conference (ITCS 2020)},
  pages =	{38:1--38:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-134-4},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{151},
  editor =	{Thomas Vidick},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2020/11723},
  URN =		{urn:nbn:de:0030-drops-117239},
  doi =		{10.4230/LIPIcs.ITCS.2020.38},
  annote =	{Keywords: Subgraph counting, bounded degeneracy graphs, fine-grained complexity}
}

Keywords: Subgraph counting, bounded degeneracy graphs, fine-grained complexity
Seminar: 11th Innovations in Theoretical Computer Science Conference (ITCS 2020)
Issue Date: 2020
Date of publication: 10.01.2020


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