Abstract
Given a simple, undirected graph G, a biclique is a subset of vertices inducing a complete bipartite subgraph in G. In this paper, we consider two associated optimization problems, the maximum biclique problem, which asks for a biclique of the maximum cardinality in the graph, and the maximum edge biclique problem, aiming to find a biclique with the maximum number of edges in the graph. These NP-hard problems find applications in biclustering-type tasks arising in complex network analysis. Real-life instances of these problems often involve massive, but sparse networks. We develop exact approaches for detecting optimal bicliques in large-scale graphs that combine effective scale reduction techniques with integer programming methodology. Results of computational experiments with numerous real-life network instances demonstrate the performance of the proposed approach.
Highlights
Network-based analysis offers a powerful approach for modeling elements and their interactions in complex systems
We proposed a scale reduction technique for the maximum biclique (MB) and maximum edge biclique (MEB) problems, which can be used to expand the applicability of exact algorithms for this problem to larger instances
Through experiments with 33 real-life instances with the number of vertices ranging from 198 to 1,048,576, we have shown that the proposed method is very effective in reducing the considered instances to scales that can be handled by modern mixed integer programming (MIP) solvers
Summary
Network-based analysis offers a powerful approach for modeling elements and their interactions in complex systems. From a practical perspective, one is not interested in independent set solutions when searching for large bicliques in a graph This observation is reflected in the definitions of the corresponding optimization problems given next. Lin [37] has proved that the problem of deciding whether a given graph G contains a complete bipartite subgraph Kk,k is W[1]-hard, meaning that an fpt-algorithm is unlikely to exist In another recent paper, Feng et al [38] studied the parameterized edge biclique problem, which asks if a given bipartite graph G contains a biclique subgraph with at least k edges, where k is a given integer parameter.
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