Efficient Algorithms for Finding Maximum and Maximal Cliques and Their Applications

The problem of finding a maximum clique or enumerating all maximal cliques is very important and has been explored in several excellent survey papers. Here, we focus our attention on the step-by-step examination of a series of branch-and-bound depth-first search algorithms: Basics, MCQ, MCR, MCS, and MCT. Subsequently, as with the depth-first search as above, we present our algorithm, CLIQUES, for enumerating all maximal cliques. Finally, we describe some of the applications of the algorithms and their variants in bioinformatics, data mining, and other fields.

[1]  Robert E. Tarjan,et al.  Finding a Maximum Independent Set , 1976, SIAM J. Comput..

[2]  T. Akutsu,et al.  Multiple methods for protein side chain packing using maximum weight cliques. , 2006, Genome informatics. International Conference on Genome Informatics.

[3]  Shinya Takahashi,et al.  A Simple and Faster Branch-and-Bound Algorithm for Finding a Maximum Clique , 2010, WALCOM.

[4]  Makoto Haraguchi,et al.  A fast and complete algorithm for enumerating pseudo-cliques in large graphs , 2016, International Journal of Data Science and Analytics.

[5]  Akira Tanaka,et al.  The Worst-Case Time Complexity for Finding All the Cliques , 2016 .

[6]  Etsuji Tomita,et al.  A Much Faster Branch-and-Bound Algorithm for Finding a Maximum Clique , 2016, FAW.

[7]  Qinghua Wu,et al.  A review on algorithms for maximum clique problems , 2015, Eur. J. Oper. Res..

[8]  Shuji Tsukiyama,et al.  A New Algorithm for Generating All the Maximal Independent Sets , 1977, SIAM J. Comput..

[9]  Makoto Haraguchi,et al.  Structural Change Pattern Mining Based on Constrained Maximal k-Plex Search , 2012, Discovery Science.

[10]  Makoto Haraguchi,et al.  Enumerating Maximal Isolated Cliques Based on Vertex-Dependent Connection Lower Bound , 2016, MLDM.

[11]  Wilbert E. Wilhelm,et al.  Clique-detection models in computational biochemistry and genomics , 2006, Eur. J. Oper. Res..

[12]  Makoto Haraguchi,et al.  Relational Change Pattern Mining Based on Modularity Difference , 2013, MIWAI.

[13]  Sergiy Butenko,et al.  Clique Relaxation Models in Social Network Analysis , 2012 .

[14]  Atsuhiro Takasu,et al.  A clique-based method for the edit distance between unordered trees and its application to analysis of glycan structures , 2011, BMC Bioinformatics.

[15]  Panos M. Pardalos,et al.  Improvements to MCS algorithm for the maximum clique problem , 2014, J. Comb. Optim..

[16]  Akira Tanaka,et al.  The worst-case time complexity for generating all maximal cliques and computational experiments , 2006, Theor. Comput. Sci..

[17]  Tatsuya Akutsu,et al.  Dynamic Programming and Clique Based Approaches for Protein Threading with Profiles and Constraints , 2006, IEICE Trans. Fundam. Electron. Commun. Comput. Sci..

[18]  Atsuhiro Takasu,et al.  A Clique-Based Method Using Dynamic Programming for Computing Edit Distance Between Unordered Trees , 2012, J. Comput. Biol..

[19]  David S. Johnson,et al.  Cliques, Coloring, and Satisfiability , 1996 .

[20]  Tatsuya Akutsu,et al.  Efficient Algorithms for Finding Maximum and Maximal Cliques: Effective Tools for Bioinformatics , 2011 .

[21]  Pablo San Segundo,et al.  Infra-chromatic bound for exact maximum clique search , 2015, Comput. Oper. Res..

[22]  Etsuji Tomita,et al.  Clique-based data mining for related genes in a biomedical database , 2009, BMC Bioinformatics.

[23]  Chu Min Li,et al.  Combining Graph Structure Exploitation and Propositional Reasoning for the Maximum Clique Problem , 2010, 2010 22nd IEEE International Conference on Tools with Artificial Intelligence.

[24]  Etsuji Tomita,et al.  An Efficient Branch-and-bound Algorithm for Finding a Maximum Clique with Computational Experiments , 2001, J. Glob. Optim..

[25]  Etsuji Tomita,et al.  A Simple Algorithm for Finding a Maximum Clique and Its Worst-Case Time Complexity , 1990, Systems and Computers in Japan.

[26]  Tatsuya Akutsu,et al.  Protein Threading with Profiles and Distance Constraints Using Clique Based Algorithms , 2006, J. Bioinform. Comput. Biol..

[27]  Diogo Vieira Andrade,et al.  Fast local search for the maximum independent set problem , 2012, J. Heuristics.

[28]  W. Alshehri,et al.  Clique relaxation models in social network analysis , 2011 .

[29]  Tatsuya Akutsu,et al.  Protein Side-chain Packing Problem: A Maximum Edge-weight Clique Algorithmic Approach , 2005, APBC.

[30]  Etsuji Tomita,et al.  A Simple and Faster Branch-and-Bound Algorithm for Finding a Maximum Clique with Computational Experiments , 2013, IEICE Trans. Inf. Syst..

[31]  J. Jeffry Howbert,et al.  The Maximum Clique Problem , 2007 .

[32]  Liu Zheng-lian,et al.  An improved branch and bound algorithm , 2011 .

[33]  Kazuhisa Makino,et al.  New Algorithms for Enumerating All Maximal Cliques , 2004, SWAT.

[34]  Mikhail Batsyn,et al.  Speeding up branch and bound algorithms for solving the maximum clique problem , 2013, Journal of Global Optimization.

[35]  P. Pardalos,et al.  An exact algorithm for the maximum clique problem , 1990 .

[36]  David Eppstein,et al.  Listing All Maximal Cliques in Large Sparse Real-World Graphs , 2011, JEAL.

[37]  Janez Konc,et al.  An improved branch and bound algorithm for the maximum clique problem , 2007 .

[38]  J. Moon,et al.  On cliques in graphs , 1965 .

[39]  C. Bron,et al.  Algorithm 457: finding all cliques of an undirected graph , 1973 .

[40]  Tatsuya Akutsu,et al.  Point matching under non-uniform distortions and protein side chain packing based on an efficient maximum clique algorithm. , 2002, Genome informatics. International Conference on Genome Informatics.

[41]  Kengo Katayama,et al.  An effective local search for the maximum clique problem , 2005, Inf. Process. Lett..

[42]  Makoto Haraguchi,et al.  Enumerating Maximal Clique Sets with Pseudo-Clique Constraint , 2015, Discovery Science.

[43]  Etsuji Tomita,et al.  An Efficient Branch-and-Bound Algorithm for Finding a Maximum Clique , 2003, DMTCS.

[44]  Panos M. Pardalos,et al.  The maximum clique problem , 1994, J. Glob. Optim..