The worst-case time complexity for generating all maximal cliques and computational experiments

Authors:
Etsuji Tomita;Akira Tanaka;Haruhisa Takahashi
Affiliations:
The University of Electro-Communications, Department of Information and Communication Engineering, Chofu, Tokyo, Japan;The University of Electro-Communications, Department of Information and Communication Engineering, Chofu, Tokyo, Japan and Toyota Techno Service Corporation, Toyota, Aichi, Japan;The University of Electro-Communications, Department of Information and Communication Engineering, Chofu, Tokyo, Japan
Venue:
Theoretical Computer Science - Computing and combinatorics
Year:
2006

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Abstract

We present a depth-first search algorithm for generating all maximal cliques of an undirected graph, in which pruning methods are employed as in the Bron-Kerbosch algorithm. All the maximal cliques generated are output in a tree-like form. Subsequently, we prove that its worst-case time complexity is O(3n/3) for an n-vertex graph. This is optimal as a function of n, since there exist up to 3n/3 maximal cliques in an n-vertex graph. The algorithm is also demonstrated to run very fast in practice by computational experiments.