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

Abstract

BackgroundThe 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 scriptNscriptP-hard, given that they subsume the classic version of the scriptNscriptP-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.ResultsAn 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.ConclusionsSeveral 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.

Task graph scheduling in the presence of performance fluctuations ofcomputational resources

“…, t i,s } represents a set of tasks that are likely to be executed concurrently. The set of maximal cliques in a graph can be computed using the Bron-Kerbosch algorithm [25]. For example, the set of maximal cliques in the concurrency graph in Figure 2b is given below the figure.…”

Discrete Optimization with Decision Diagrams

“…The MISP (which is equivalent to the maximum clique problem) has found applications in many areas, including data mining (Edachery et al 1999), bioinformatics (Eblen et al 2011), and social network analysis (Balasundaram et al 2011).…”

Centrality Metrics for Identifying Network Fragility in Protein-Protein Interaction Networks and Synthesized NK Systems

“…A number of biological problems [6,7,23] can be posed in terms of graph and combinatorics [4]. While some of these can be analyzed using closed form mathematics, many interesting biological data sets are available in the form of simple or weighted [5,69] graph or network structures and are too large and complex to be analyzed except using numerical or enumerative graph methods [20]. Some examples include genetic transcription networks and other experimentally obtained bio-net data such as: interaction networks [9]; conformation space networks for protein folding [59,60,78]; peptide folding applications [10,65]; and metabolic graphs [61,77].…”