Eppstein, David ;
Löffler, Maarten ;
Strash, Darren
Listing all maximal cliques in sparse graphs in near-optimal time
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:DagSemProc.10441.2,
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 = {https://drops.dagstuhl.de/opus/volltexte/2011/2935},
URN = {urn:nbn:de:0030-drops-29356},
doi = {10.4230/DagSemProc.10441.2},
annote = {Keywords: Clique, backtracking, degeneracy, worst-case optimality}
}
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Keywords: |
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Clique, backtracking, degeneracy, worst-case optimality |
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Seminar: |
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10441 - Exact Complexity of NP-hard Problems
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Issue date: |
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2011 |
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Date of publication: |
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27.01.2011 |
27.01.2011
