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URN: urn:nbn:de:0030-drops-29356
URL: http://drops.dagstuhl.de/opus/volltexte/2011/2935/
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Eppstein, David ; Löffler, Maarten ; Strash, Darren

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

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Abstract

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

@InProceedings{eppstein_et_al:DSP:2011:2935,
  author =	{David Eppstein and Maarten L{\"o}ffler and Darren Strash},
  title =	{{Listing all maximal cliques in sparse graphs in near-optimal time}},
  booktitle =	{Exact Complexity of NP-hard Problems},
  year =	{2011},
  editor =	{Thore Husfeldt and Dieter Kratsch and Ramamohan Paturi and Gregory B. Sorkin},
  number =	{10441},
  series =	{Dagstuhl Seminar Proceedings},
  ISSN =	{1862-4405},
  publisher =	{Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, Germany},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2011/2935},
  annote =	{Keywords: Clique, backtracking, degeneracy, worst-case optimality}
}

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


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