# Bipartite Correlation Clustering: Maximizing Agreements

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Megasthenis Asteris,
Anastasios Kyrillidis,
Dimitris Papailiopoulos,
Alexandros Dimakis
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Proceedings of the 19th International Conference on Artificial Intelligence and Statistics, PMLR 51:121-129, 2016.

#### Abstract

In Bipartite Correlation Clustering (BCC) we are given a complete bipartite graph G with ’+’ and ’-’ edges, and we seek a vertex clustering that maximizes the number of agreements: the number of all ’+’ edges within clusters plus all ’-’ edges cut across clusters. BCC is known to be NP-hard [5]. We present a novel approximation algorithm for k-BCC, a variant of BCC with an upper bound k on the number of clusters. Our algorithm outputs a k-clustering that provably achieves a number of agreements within a multiplicative (1-δ)-factor from the optimal, for any desired accuracy δ. It relies on solving a combinatorially constrained bilinear maximization on the bi-adjacency matrix of G. It runs in time exponential in k and 1/δ, but linear in the size of the input. Further, we show that, in the (unconstrained) BCC setting, an (1-δ)-approximation can be achieved by O(1/δ) clusters regardless of the size of the graph. In turn, our k-BCC algorithm implies an Efficient PTAS for the BCC objective of maximizing agreements.

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