The Maximum Clique Enumeration Problem: Algorithms, Applications and Implementations

Algorithms are designed, analyzed and implemented for the maximum clique enumeration (MCE) problem, which asks that we identify all maximum cliques in a finite, simple graph. MCE is closely related to two other well-known and widely-studied problems: the maximum clique optimization problem, which asks us to determine the size of a largest clique, and the maximal clique enumeration problem, which asks that we compile a listing of all maximal cliques. Naturally, these three problems are NP-hard, given that they subsume the classic version of the NP-complete clique decision problem. MCE can be solved in principle with standard enumeration methods due to Bron, Kerbosch, Kose and others. Unfortunately, these techniques are ill-suited to graphs encountered in our applications. We must solve MCE on instances deeply seeded in data mining and computational biology, where high-throughput data capture often creates graphs of extreme size and density. MCE can also be solved in principle using more modern algorithms based in part on vertex cover and the theory of fixed-parameter tractability (FPT). While FPT is an improvement, these algorithms too can fail to scale sufficiently well as the sizes and densities of our datasets grow. An extensive testbed of benchmark MCE instances is devised, based on applications in transcriptomic data analysis. Empirical testing reveals crucial but latent features of such high-throughput biological data. In turn, it is shown that these features distinguish real data from random data intended to reproduce salient topological features. In particular, with real data there tends to be an unusually high degree of maximum clique overlap. Armed with this knowledge, novel decomposition strategies are tuned to the data and coupled with the best FPT MCE implementations. It is demonstrated that the resultant run times are frequently reduced by several orders of magnitude, and that instances once prohibitively time-consuming to solve are now often brought into the domain of realistic feasibility.

[1]  Eric R. Harley.  Comparison of Clique-Listing Algorithms , 2004, MSV/AMCS.

[2]  S. Ross.  A random graph , 1981 .

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

[4]  Nagiza F. Samatova,et al.  Genome-Scale Computational Approaches to Memory-Intensive Applications in Systems Biology , 2005, ACM/IEEE SC 2005 Conference (SC'05).

[5]  David Fernández-Baca,et al.  The Perfect Phylogeny Problem , 2001 .

[6]  Michael A. Langston,et al.  Combinatorial Genetic Regulatory Network Analysis Tools for High Throughput Transcriptomic Data , 2005, Systems Biology and Regulatory Genomics.

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

[8]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[9]  Faisal N. Abu-Khzam,et al.  Scalable Parallel Algorithms for FPT Problems , 2006, Algorithmica.

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

[11]  Alex E. Lash,et al.  Gene Expression Omnibus: NCBI gene expression and hybridization array data repository , 2002, Nucleic Acids Res..

[12]  Robert W. Williams,et al.  Complex trait analysis of gene expression uncovers polygenic and pleiotropic networks that modulate nervous system function , 2005, Nature Genetics.

[13]  Michael A. Langston,et al.  High performance computational tools for Motif discovery , 2004, 18th International Parallel and Distributed Processing Symposium, 2004. Proceedings..

[14]  Michael A. Langston,et al.  Threshold selection in gene co-expression networks using spectral graph theory techniques , 2009, BMC Bioinformatics.

[15]  Arnold M Saxton,et al.  Comparison of threshold selection methods for microarray gene co-expression matrices , 2009, BMC Research Notes.

[16]  Béla Bollobás,et al.  Random Graphs , 1985 .

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

[18]  John David Eblen.  The M aximum Clique Problem: Algorithms, Applications, and Implementations , 2010 .

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

[20]  Andrea Torsello,et al.  A game-theoretic approach to partial clique enumeration , 2009, Image Vis. Comput..

[21]  K. Kwast,et al.  Metabolic-State-Dependent Remodeling of the Transcriptome in Response to Anoxia and Subsequent Reoxygenation in Saccharomyces cerevisiae , 2006, Eukaryotic Cell.

[22]  Michael A. Langston,et al.  Graph algorithms for integrated biological analysis, with applications to type 1 diabetes data , 2009 .

[23]  Henning Fernau,et al.  On Parameterized Enumeration , 2002, COCOON.

[24]  Albert,et al.  Emergence of scaling in random networks , 1999, Science.

[25]  T. Vicsek,et al.  Uncovering the overlapping community structure of complex networks in nature and society , 2005, Nature.

[26]  Thomas Linke,et al.  Visualizing plant metabolomic correlation networks using clique-metabolite matrices , 2001, Bioinform..

[27]  Faisal N. Abu-Khzam,et al.  Using out-of-core techniques to produce exact solutions to the maximum clique problem on extremely large graphs , 2009, 2009 IEEE/ACS International Conference on Computer Systems and Applications.

[28]  Michael R. Fellows,et al.  Parameterized Complexity , 1998 .

[29]  Walter Willinger,et al.  Towards a Theory of Scale-Free Graphs: Definition, Properties, and Implications , 2005, Internet Math..

[30]  Michael A. Langston,et al.  Computational, Integrative, and Comparative Methods for the Elucidation of Genetic Coexpression Networks , 2005, Journal of biomedicine & biotechnology.

[31]  Rob Malouf,et al.  Maximal Consistent Subsets , 2007, Computational Linguistics.