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The worst-case time complexity for generating all maximal cliques and computational experiments. (English) Zbl 1153.68398
Summary: 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 \(O(3^{n/3})\) for an \(n\)-vertex graph. This is optimal as a function of \(n\), since there exist up to \(3^{n/3}\) maximal cliques in an \(n\)-vertex graph. The algorithm is also demonstrated to run very fast in practice by computational experiments.
Reviewer: Reviewer (Berlin)

MSC:
68Q25 Analysis of algorithms and problem complexity
05C69 Vertex subsets with special properties (dominating sets, independent sets, cliques, etc.)
05C85 Graph algorithms (graph-theoretic aspects)
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[1] T. Akutsu, M. Hayashida, E. Tomita, J. Suzuki, K. Horimoto, Protein threading with profiles and constraints, Proc. IEEE Symp. Bioinform. Bioeng. (2004) 537-544;; T. Akutsu, M. Hayashida, D.K.C. Bahadur, E. Tomita, J. Suzuki, K. Horimoto, Dynamic programming and clique based approaches for protein threading with profiles and constraints, IEICE Trans. Fund. Electron. Commun. Comput. Sci. E89-A (2006) 1215-1222.
[2] Bahadur, D.K.C.; Akutsu, T.; Tomita, E.; Seki, T.; Fujiyama, A., Point matching under non-uniform distortions and protein side chain packing based on efficient maximum clique algorithms, Genome inform., 13, 143-152, (2002)
[3] Bahadur, D.K.C.; Tomita, E.; Suzuki, J.; Akutsu, T., Protein side-chain packing problem: a maximum edge-weight clique algorithmic approach, J. bioinform. comput. biol., 3, 103-126, (2005)
[4] Bahadur, D.K.C.; Tomita, E.; Suzuki, J.; Horimoto, K.; Akutsu, T., Protein threading with profiles and distance constraints using clique based algorithms, J. bioinform. comput. biol., 4, 19-42, (2006)
[5] I.M. Bomze, M. Budinich, P.M. Pardalos, M. Pelillo, The maximum clique problem, in: D.-Z. Du, P.M. Pardalos (Eds.), Handbook of Combinatorial Optimization, Supplement Vol. A, Kluwer Academic Publishers, Dordrecht, 1999, pp. 1-74. · Zbl 1253.90188
[6] Bron, C.; Kerbosch, J., Algorithm 457, finding all cliques of an undirected graph, Comm. ACM, 16, 575-577, (1973) · Zbl 0261.68018
[7] Chiba, N.; Nishizeki, T., Arboricity and subgraph listing algorithms, SIAM J. comput., 14, 210-223, (1985) · Zbl 0572.68053
[8] Hattori, M.; Okuno, Y.; Goto, S.; Kanehisa, M., Development of a chemical structure comparison method for integrated analysis of chemical and genomic information in the metabolic pathways, J. amer. chem. soc., 125, 11853-11865, (2003)
[9] Johnston, H.C., Cliques of a graph—variations on the bron – kerbosch algorithm, Int. J. comput. inform. sci., 5, 209-238, (1976) · Zbl 0401.68042
[10] D.S. Johnson, M.A. Trick (Eds.), Cliques, Coloring, and Satisfiability, DIMACS Series in Discrete Mathematics and Theoretical Computer Science, Vol. 26, American Mathematical Society, Providence, RI, 1996.
[11] Kobayashi, S.; Kondo, T.; Okuda, K.; Tomita, E., Extracting globally structure free sequences by local structure freeness, Proc. DNA, 9, 206, (2003)
[12] Koch, I., Enumerating all connected maximal common subgraphs in two graphs, Theoret. comput. sci., 250, 1-30, (2001) · Zbl 0952.68105
[13] Lawler, E.L.; Lenstra, J.K.; Rinnooy Kan, A.H.G., Generating all maximal independent sets: NP-hardness and polynomial-time algorithms, SIAM J. comput., 9, 558-565, (1980) · Zbl 0445.68054
[14] K. Makino, T. Uno, New algorithms for enumerating all maximal cliques, SWAT 2004, Lecture Notes in Computer Science, Vol. 3111, 2004, pp. 260-272. · Zbl 1095.68626
[15] Mohseni-Zadeh, S.; Brézellec, P.; Risler, J.-L., Cluster-C, an algorithm for the large-scale clustering of protein sequences based on the extraction of maximal cliques, Comput. biol. chem., 28, 211-218, (2004) · Zbl 1088.92024
[16] Mohseni-Zadeh, S.; Louis, A.; Brézellec, P.; Risler, J.-L., PHYTOPROT: a database of clusters of plant proteins, Nucleic acids res., 32, D351-D353, (2004)
[17] Moon, J.W.; Moser, L., On cliques in graphs, Israel J. math., 3, 23-28, (1965) · Zbl 0144.23205
[18] T. Nakagawa, E. Tomita, Experimental comparisons of algorithms for generating all maximal cliques, Technical Report of IPSJ, 2004-MPS-52, 2004, pp. 41-44.
[19] Pardalos, P.M.; Xue, J., The maximum clique problem, J. global optim., 4, 301-328, (1994) · Zbl 0797.90108
[20] Robson, J.M., Algorithms for maximum independent sets, J. algorithms, 7, 425-440, (1986) · Zbl 0637.68080
[21] Tarjan, R.E.; Trojanowski, A.E., Finding a maximum independent set, SIAM J. comput., 6, 537-546, (1977) · Zbl 0357.68035
[22] E. Tomita, T. Seki, An efficient branch-and-bound algorithm for finding a maximum clique, DMTCS 2003, Lecture Notes in Computer Science, Vol. 2731, 2003, pp. 278-289;; E. Tomita, T. Kameda, An efficient branch-and-bound algorithm for finding a maximum clique with computational experiments, J. Global Optim. (2006), in press;; Y. Sutani, E. Tomita, Computational experiments and analyses of a more efficient algorithm for finding a maximum clique, Technical Report of IPSJ, 2005-MPS-57, 2005, pp. 45-48, and others. · Zbl 1038.68565
[23] E. Tomita, A. Tanaka, H. Takahashi, The worst-case time complexity for finding all the cliques, Technical Report of the University of Electro-Communications, UEC-TR-C5(2), 1988. · Zbl 1091.68562
[24] E. Tomita, A. Tanaka, H. Takahashi, An optimal algorithm for finding all the cliques, Technical Report of IPSJ, 1989-AL-12, 1989, pp. 91-98.
[25] E. Tomita, A. Tanaka, H. Takahashi, The worst-case time complexity for generating all maximal cliques, COCOON 2004, Lecture Notes in Computer Science, Vol. 3106, 2004, pp. 161-170. · Zbl 1091.68562
[26] Tsukiyama, S.; Ide, M.; Ariyoshi, H.; Shirakawa, I., A new algorithm for generating all the maximal independent sets, SIAM J. comput., 6, 505-517, (1977) · Zbl 0364.05027
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