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Enumerating maximal biclique subgraphs from a graph is a computationally challenging problem. In this paper, we efficiently enumerate them through the use of closed patterns of the adjacency matrix of the graph. For an undirected graph $G$ without self-loops, we prove that: (i) the number of closed patterns in the adjacency matrix of $G$ is even; and (ii) for every maximal biclique subgraph, there always exists a unique pair of closed patterns that matches the two vertex sets of the subgraph. Therefore, the problem of enumerating maximal bicliques can be solved by using efficient algorithms for mining closed patterns, which are algorithms extensively studied in the data mining field. However, this direct use of existing algorithms causes a duplicated enumeration. To achieve high efficiency, we propose an $O(mn)$ time delay algorithm for a non-duplicated enumeration, in particular for enumerating those maximal bicliques with a large size, where $m$ and $n$ are the number of edges and vertices of the graph respectively. We evaluate the high efficiency of our algorithm by comparing it to state-of-the-art algorithms on many graphs.
Mining methods and algorithms, Graph algorithms
Guimei Liu, Haiquan Li, Limsoon Wong, Jinyan Li, "Maximal Biclique Subgraphs and Closed Pattern Pairs of the Adjacency Matrix: A One-to-One Correspondence and Mining Algorithms", IEEE Transactions on Knowledge & Data Engineering, vol. 19, no. , pp. 1625-1637, December 2007, doi:10.1109/TKDE.2007.190660
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