Listing All Maximal Cliques in Sparse Graphs in Near-Optimal Time
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- Eppstein D., Löffler M., Strash D. (2010) Listing All Maximal Cliques in Sparse Graphs in Near-Optimal Time. In: Cheong O., Chwa KY., Park K. (eds) Algorithms and Computation. ISAAC 2010. Lecture Notes in Computer Science, vol 6506. Springer, Berlin, Heidelberg
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(dn3d/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 n ≥ d + 3) is (n − d)3d/3. Therefore, our algorithm matches the Θ(d(n − d)3d/3) worst-case output size of the problem whenever n − d = Ω(n).
Keywordssparse graphs d-degenerate graphs maximal clique listing algorithms Bron–Kerbosch algorithm fixed-parameter tractability
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