License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
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DOI: 10.4230/DagSemProc.10441.2
URN: urn:nbn:de:0030-drops-29356
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Eppstein, David ; Löffler, Maarten ; Strash, Darren

Listing all maximal cliques in sparse graphs in near-optimal time

10441.EppsteinDavid.Paper.2935.pdf (0.4 MB)


The degeneracy of an $n$-vertex graph $G$ is the smallest number $d$ such that every subgraph of $G$ contains a vertex of degree at most $d$. We show that there exists a nearly-optimal fixed-parameter tractable algorithm for enumerating all maximal cliques, parametrized by degeneracy. To achieve this result, we modify the classic Bron--Kerbosch algorithm and show that it runs in time $O(dn3^{d/3})$. We also provide matching upper and lower bounds showing that the largest possible number of maximal cliques in an $n$-vertex graph with degeneracy $d$ (when $d$ is a multiple of 3 and $nge d+3$) is $(n-d)3^{d/3}$. Therefore, our algorithm matches the $Theta(d(n-d)3^{d/3})$ worst-case output size of the problem whenever $n-d=Omega(n)$.

BibTeX - Entry

  author =	{Eppstein, David and L\"{o}ffler, Maarten and Strash, Darren},
  title =	{{Listing all maximal cliques in sparse graphs in near-optimal time}},
  booktitle =	{Exact Complexity of NP-hard Problems},
  pages =	{1--14},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2011},
  volume =	{10441},
  editor =	{Thore Husfeldt and Dieter Kratsch and Ramamohan Paturi and Gregory B. Sorkin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{},
  URN =		{urn:nbn:de:0030-drops-29356},
  doi =		{10.4230/DagSemProc.10441.2},
  annote =	{Keywords: Clique, backtracking, degeneracy, worst-case optimality}

Keywords: Clique, backtracking, degeneracy, worst-case optimality
Collection: 10441 - Exact Complexity of NP-hard Problems
Issue Date: 2011
Date of publication: 27.01.2011

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