Abstract
In this work we study the problem of Bipartite Correlation Clustering (BCC), a natural bipartite counterpart of the well studied Correlation Clustering (CC) problem. Given a bipartite graph, the objective of BCC is to generate a set of vertex-disjoint bi-cliques (clusters) which minimizes the symmetric difference to it. The best known approximation algorithm for BCC due to Amit (2004) guarantees an 11-approximation ratio.
In this paper we present two algorithms. The first is an improved 4-approximation algorithm. However, like the previous approximation algorithm, it requires solving a large convex problem which becomes prohibitive even for modestly sized tasks.
The second algorithm, and our main contribution, is a simple randomized combinatorial algorithm. It also achieves an expected 4-approximation factor, it is trivial to implement and highly scalable. The analysis extends a method developed by Ailon, Charikar and Newman in 2008, where a randomized pivoting algorithm was analyzed for obtaining a 3-approximation algorithm for CC. For analyzing our algorithm for BCC, considerably more sophisticated arguments are required in order to take advantage of the bipartite structure.
Whether it is possible to achieve (or beat) the 4-approximation factor using a scalable and deterministic algorithm remains an open problem.
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Ailon, N., Avigdor-Elgrabli, N., Liberty, E., van Zuylen, A. (2011). Improved Approximation Algorithms for Bipartite Correlation Clustering. In: Demetrescu, C., Halldórsson, M.M. (eds) Algorithms – ESA 2011. ESA 2011. Lecture Notes in Computer Science, vol 6942. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-23719-5_3
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DOI: https://doi.org/10.1007/978-3-642-23719-5_3
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