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Quasi-cliques are dense incomplete subgraphs of a graph that generalize the notion of cliques. Enumerating quasi-cliques from a graph is a robust way to detect densely connected structures with applications in bioinformatics and social network analysis. However, enumerating quasi-cliques in a graph is a challenging problem, even harder than the problem of enumerating cliques. We consider the enumeration of top- k degree-based quasi-cliques and make the following contributions: (1) we show that even the problem of detecting whether a given quasi-clique is maximal (i.e., not contained within another quasi-clique) is NP-hard. (2) We present a novel heuristic algorithm K ernel QC to enumerate the k largest quasi-cliques in a graph. Our method is based on identifying kernels of extremely dense subgraphs within a graph, followed by growing subgraphs around these kernels, to arrive at quasi-cliques with the required densities. (3) Experimental results show that our algorithm accurately enumerates quasi-cliques from a graph, is much faster than current state-of-the-art methods for quasi-clique enumeration (often more than three orders of magnitude faster), and can scale to larger graphs than current methods. 

We present shared memory parallel algorithms for maximal biclique enumeration (MBE), the task of enumerating all complete dense subgraphs (maximal bicliques) from a bipartite graph, which is widely used in the analysis of social, biological, and transactional networks. Since MBE is computationally expensive, it is necessary to use parallel computing to scale to large graphs. Our parallel algorithm ParMBE efficiently uses the power of multiple cores that share memory. From a theoretical view, ParMBE is work-efficient with respect to a state-of-the-art sequential algorithm. Our experimental evaluation shows that ParMBE scales well up to 64 cores, and is significantly faster than current parallel algorithms. Since ParMBE was yielding a super-linear speedup compared to the sequential algorithm on which it was based (MineLMBC), we develop an improved sequential algorithm FMBE, through "sequentializing" ParMBE. 

We present shared-memory parallel methods for Maximal Clique Enumeration (MCE) from a graph. MCE is a fundamental and well-studied graph analytics task, and is a widely used primitive for identifying dense structures in a graph. Due to its computationally intensive nature, parallel methods are imperative for dealing with large graphs. However, surprisingly, there do not yet exist scalable and parallel methods for MCE on a shared-memory parallel machine. In this work, we present efficient shared-memory parallel algorithms for MCE, with the following properties: (1) the parallel algorithms are provably work-efficient relative to a state-of-the-art sequential algorithm (2) the algorithms have a provably small parallel depth, showing that they can scale to a large number of processors, and (3) our implementations on a multicore machine shows a good speedup and scaling behavior with increasing number of cores, and are substantially faster than prior shared-memory parallel algorithms for MCE.