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

Authors David Eppstein, Maarten Löffler, Darren Strash

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David Eppstein
Maarten Löffler
Darren Strash

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David Eppstein, Maarten Löffler, and Darren Strash. Listing all maximal cliques in sparse graphs in near-optimal time. In Exact Complexity of NP-hard Problems. Dagstuhl Seminar Proceedings, Volume 10441, pp. 1-14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2011)


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 $n-d=Omega(n)$.
  • Clique
  • backtracking
  • degeneracy
  • worst-case optimality


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