Algorithm 457: finding all cliques of an undirected graph
Communications of the ACM
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
COCOON '02 Proceedings of the 8th Annual International Conference on Computing and Combinatorics
Genome-Scale Computational Approaches to Memory-Intensive Applications in Systems Biology
SC '05 Proceedings of the 2005 ACM/IEEE conference on Supercomputing
Scalable Parallel Algorithms for FPT Problems
Algorithmica
An Efficient Branch-and-bound Algorithm for Finding a Maximum Clique with Computational Experiments
Journal of Global Optimization
The worst-case time complexity for generating all maximal cliques and computational experiments
Theoretical Computer Science - Computing and combinatorics
Computational Linguistics
A game-theoretic approach to partial clique enumeration
Image and Vision Computing
Combinatorial genetic regulatory network analysis tools for high throughput transcriptomic data
RECOMB'05 Proceedings of the 2005 joint annual satellite conference on Systems biology and regulatory genomics
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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.