Abstract
This paper is related to the problem of finding the maximal quasi-bicliques in a bipartite graph (bigraph). A quasi-biclique in the bigraph is its “almost” complete subgraph. The relaxation of completeness can be understood variously; here, we assume that the subgraph is a \(\gamma \)-quasi-biclique if it lacks a certain number of edges to form a biclique such that its density is at least \(\gamma \in (0,1]\). For a bigraph and fixed \(\gamma \), the problem of searching for the maximal quasi-biclique consists of finding a subset of vertices of the bigraph such that the induced subgraph is a quasi-biclique and its size is maximal for a given graph. Several models based on Mixed Integer Programming (MIP) to search for a quasi-biclique are proposed and tested for working efficiency. An alternative model inspired by biclustering is formulated and tested; this model simultaneously maximises both the size of the quasi-biclique and its density, using the least-square criterion similar to the one exploited by triclustering TriBox.
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Notes
- 1.
- 2.
The size column in Table 3 shown as the result of summation \(|U'|\) and \(|V'|\).
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Acknowledgements
The work of Dmitry I. Ignatov shown in all the sections, except 5 and 6, has been supported by the Russian Science Foundation grant no. 17-11-01276 and performed at St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences, Russia. The authors would like to thank Boris Mirkin, Vladimir Kalyagin, Panos Pardalos, and Oleg Prokopyev for their piece of advice and inspirational discussions. Last but not least, we are thankful to anonymous reviewers for their useful feedback.
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Ignatov, D.I., Ivanova, P., Zamaletdinova, A. (2020). Mixed Integer Programming for Searching Maximum Quasi-Bicliques. In: Bychkov, I., Kalyagin, V., Pardalos, P., Prokopyev, O. (eds) Network Algorithms, Data Mining, and Applications. NET 2018. Springer Proceedings in Mathematics & Statistics, vol 315. Springer, Cham. https://doi.org/10.1007/978-3-030-37157-9_2
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