Maximal Clique Enumeration Problem Research Articles

Maximal clique enumeration problem on graphs: status and challenges

随着大数据时代的到来,图数据挖掘成为了一个热门的研究方向.极大团枚举(Maximal Clique Enumeration, MCE)作为图论中的一个基本问题, 在很多领域都有着广泛的应用. 然而, 鉴于极大团枚举问题本身的复杂性以及现实图数据规模的飞速增长, 在现实图数据上进行极大团枚举是很耗时的. 目前已经有大量的工作对该问题的求解算法进行改进, 或采用各种计算优化方法减少算法的运行时间. 本文就极大团枚举问题做了如下工作: 对现有的极大团枚举问题的研究工作进行了分类归纳; 对极大团枚举问题的研究现状进行了详细介绍; 对该问题进一步发展所面临的挑战和发展方向进行了讨论和展望.

Accelerating the Bron-Kerbosch Algorithm for Maximal Clique Enumeration Using GPUs

Maximal clique enumeration (MCE) is a classic problem in graph theory to identify all complete subgraphs in a graph. In prior MCE work, the Bron-Kerbosch algorithm is one of the most popular solutions, and there are several improved algorithms proposed on CPU platforms. However, while few studies have focused on the related issue of parallel implementation, recently, there have been numerous explorations of the acceleration of general purpose applications using a graphics processing unit (GPU) to reduce the computing power consumption. In this article, we develop a GPU-based Bron-Kerbosch algorithm that efficiently solves the MCE problem in parallel by optimizing the process of subproblem decomposition and computing resource usage. To speed up the computations, we use coalesced memory accesses and warp reductions to increase bandwidth and reduce memory latency. Our experimental results show that the proposed algorithm can fully exploit the resources of GPU architectures, allowing for the vast acceleration of operations to solve the MCE problem.

Listing all maximal cliques in large graphs on vertex-centric model

Maximal Clique Enumeration (MCE), which consists to enumerate all maximal complete subgraphs in a given graph, is a fundamental problem in graph theory, and it is used in several applications. It is one of Karp’s 21 NP-complete problems. In the literature, this problem has been widely studied. One of the most notable, efficient, successful and extensively used solutions is the Bron–Kerbosch (BK) algorithm. The latter is a sequential algorithm which is able to enumerate all maximal cliques in an undirected graph without duplication. Furthermore, it is used to solve large problems such as maximum clique problem, community detection and graph clustering. However, for large graphs, sequential algorithms are slow and do not scale well. Thus, processing efficiently this kind of graphs needs to develop distributed algorithms under parallel and distributed platforms or large graph mining frameworks. In this setting, we propose new efficient distributed algorithms for maximal clique enumerating based on the vertex-centric model. These algorithms use the BK algorithm principle to deal with the MCE problem on large cluster. The proposed algorithms are implemented in Giraph and evaluated by using real-world graphs and computer-generated benchmark networks. Our experiments on a Hadoop cluster show that the proposed algorithms can effectively process a variety of large real-world and computer-generated graphs and scale well with increasing the dataset size and the number of nodes in the cluster. Furthermore, the proposed algorithms are provably work-efficient compared with other algorithms including BK algorithm.

A new decomposition technique for maximal clique enumeration for sparse graphs

Given a graph, we are interested in the question of finding all its maximal cliques. This question models the community detection problem and has been extensively studied. Here, we approach it under the light of an important graph parameter. The degeneracy of a graph G is the smallest integer k such that every subgraph of G contains a vertex of degree at most k. Using a new decomposition technique, we present two output sensitive algorithms for the maximal clique enumeration problem. The first one has enumeration time depending only on the degeneracy of the graph. This is the first such result in the literature. This algorithm requires that the cliques are stored in memory. Thus, we propose a second one, which has enumeration time depending on the degeneracy and the maximum degree, and which only requires memory polynomial in the degeneracy of the graph (besides the space to store the graph itself). Then we show that this algorithm can be easily parallelized. As a by-product of our decomposition technique, we propose new algorithms for the maximum clique and p-clique problems as well as an approximation algorithm counting maximal cliques, with approximation ratio the maximum clique size.

ModuleDiscoverer: Identification of regulatory modules in protein-protein interaction networks

The identification of disease-associated modules based on protein-protein interaction networks (PPINs) and gene expression data has provided new insights into the mechanistic nature of diverse diseases. However, their identification is hampered by the detection of protein communities within large-scale, whole-genome PPINs. A presented successful strategy detects a PPIN’s community structure based on the maximal clique enumeration problem (MCE), which is a non-deterministic polynomial time-hard problem. This renders the approach computationally challenging for large PPINs implying the need for new strategies. We present ModuleDiscoverer, a novel approach for the identification of regulatory modules from PPINs and gene expression data. Following the MCE-based approach, ModuleDiscoverer uses a randomization heuristic-based approximation of the community structure. Given a PPIN of Rattus norvegicus and public gene expression data, we identify the regulatory module underlying a rodent model of non-alcoholic steatohepatitis (NASH), a severe form of non-alcoholic fatty liver disease (NAFLD). The module is validated using single-nucleotide polymorphism (SNP) data from independent genome-wide association studies and gene enrichment tests. Based on gene enrichment tests, we find that ModuleDiscoverer performs comparably to three existing module-detecting algorithms. However, only our NASH-module is significantly enriched with genes linked to NAFLD-associated SNPs. ModuleDiscoverer is available at http://www.hki-jena.de/index.php/0/2/490 (Others/ModuleDiscoverer).

A linear time algorithm for maximal clique enumeration in large sparse graphs

A maximal clique is one of the most fundamental dense substructures in an undirected graph, and maximal clique enumeration (MCE) plays an essential role in densely connected subgraphs discovering. Existing algorithms of maximal clique enumeration employ recursive iteration of adjacent nodes as guiding thought, which incurs high time complexity. In this paper, we propose a linear time algorithm, CM-Constructor (Candidate Map Constructor), for maximal clique enumeration in large sparse graphs which generates a novel data structure called candidate map as result. A candidate map holds not only all maximal cliques of an undirected graph but also some non-maximal cliques that can be easily discarded via an inverted clique tree. To the best of our knowledge, CM-Constructor is the first algorithm to tackle maximal clique enumeration problem utilizing linear procedure. It generates all maximal cliques without duplications for an undirected graph G=(V,E) within O(|E|) time.

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

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 -hard, given that they subsume the classic version of the -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.

A scalable, parallel algorithm for maximal clique enumeration

The problem of maximal clique enumeration (MCE) is to enumerate all of the maximal cliques in a graph. Once enumerated, maximal cliques are widely used to solve problems in areas such as 3-D protein structure alignment, genome mapping, gene expression analysis, and detection of social hierarchies. Even the most efficient serial MCE algorithms require large amounts of time to enumerate the maximal cliques in networks arising from these problems that contain hundreds, thousands, or larger numbers of vertices. The previous attempts to provide practical solutions to the MCE problem through parallel implementation have had limited success, largely due to a number of challenges inherent to the nature of the MCE combinatorial search space. On the one hand, MCE algorithms often create a backtracking search tree that has a highly irregular and hard-or-impossible to predict structure; therefore, almost any static decomposition of the search tree by parallel processors results in highly unbalanced processor execution times. On the other hand, the data-intensive nature of the MCE problem often makes naive dynamic load distribution strategies that require extensive data movement prohibitively expensive. As a result, good scaling of the overall execution time of parallel MCE algorithms has been reported for only up to a couple hundred processors. In this paper, we propose a parallel, scalable, and memory-efficient MCE algorithm for distributed and/or shared memory high performance computing architectures, whose runtime scales linearly for thousands of processors on real-world application graphs with hundreds and thousands of nodes. Its scalability and efficiency are attributed to the proposed: (a) representation of the search tree decomposition to enable parallelization; (b) parallel depth-first backtracking search to both constrain the search space and minimize memory requirement; (c) least stringent synchronization to minimize data movement; and (d) on-demand work stealing intelligently coupled with work stack splitting to minimize computing elements’ idle time. To the best of our knowledge, the proposed parallel MCE algorithm is the first to achieve a linear scaling runtime using up to 2048 processors on Cray XT machines for a number of real-world biological networks.

Coupling graph perturbation theory with scalable parallel algorithms for large-scale enumeration of maximal cliques in biological graphs

Data-driven construction of predictive models for biological systems faces challenges from data intensity, uncertainty, and computational complexity. Data-driven model inference is often considered a combinatorial graph problem where an enumeration of all feasible models is sought. The data-intensive and the NP-hard nature of such problems, however, challenges existing methods to meet the required scale of data size and uncertainty, even on modern supercomputers. Maximal clique enumeration (MCE) in a graph derived from such biological data is often a rate-limiting step in detecting protein complexes in protein interaction data, finding clusters of co-expressed genes in microarray data, or identifying clusters of orthologous genes in protein sequence data. We report two key advances that address this challenge. We designed and implemented the first (to the best of our knowledge) parallel MCE algorithm that scales linearly on thousands of processors running MCE on real-world biological networks with thousands and hundreds of thousands of vertices. In addition, we proposed and developed the Graph Perturbation Theory (GPT) that establishes a foundation for efficiently solving the MCE problem in perturbed graphs, which model the uncertainty in the data. GPT formulates necessary and sufficient conditions for detecting the differences between the sets of maximal cliques in the original and perturbed graphs and reduces the enumeration time by more than 80% compared to complete recomputation.