The maximum clique enumeration problem: algorithms, applications, and implementations.

BACKGROUND: The maximum clique enumeration (MCE) problem 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. RESULTS: An extensive testbed of benchmark graphs are created using publicly available transcriptomic datasets from the Gene Expression Omnibus (GEO). 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. CONCLUSIONS: Several algorithmic improvements to MCE are made which progressively decrease the run time on graphs in the testbed. Frequently the final runtime improvement is several orders of magnitude. As a result, instances which were once prohibitively time-consuming to solve are brought into the domain of realistic feasibility.

A systematic comparison of genome-scale clustering algorithms.

BACKGROUND: A wealth of clustering algorithms has been applied to gene co-expression experiments. These algorithms cover a broad range of approaches, from conventional techniques such as k-means and hierarchical clustering, to graphical approaches such as k-clique communities, weighted gene co-expression networks (WGCNA) and paraclique. Comparison of these methods to evaluate their relative effectiveness provides guidance to algorithm selection, development and implementation. Most prior work on comparative clustering evaluation has focused on parametric methods. Graph theoretical methods are recent additions to the tool set for the global analysis and decomposition of microarray co-expression matrices that have not generally been included in earlier methodological comparisons. In the present study, a variety of parametric and graph theoretical clustering algorithms are compared using well-characterized transcriptomic data at a genome scale from Saccharomyces cerevisiae. METHODS: For each clustering method under study, a variety of parameters were tested. Jaccard similarity was used to measure each cluster's agreement with every GO and KEGG annotation set, and the highest Jaccard score was assigned to the cluster. Clusters were grouped into small, medium, and large bins, and the Jaccard score of the top five scoring clusters in each bin were averaged and reported as the best average top 5 (BAT5) score for the particular method. RESULTS: Clusters produced by each method were evaluated based upon the positive match to known pathways. This produces a readily interpretable ranking of the relative effectiveness of clustering on the genes. Methods were also tested to determine whether they were able to identify clusters consistent with those identified by other clustering methods. CONCLUSIONS: Validation of clusters against known gene classifications demonstrate that for this data, graph-based techniques outperform conventional clustering approaches, suggesting that further development and application of combinatorial strategies is warranted.