Bipartite Correlation Clustering -- Maximizing Agreements

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. 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 \({(1-\delta)}\)-factor from the optimal, for any desired accuracy \(\delta\). It relies on solving a combinatorially constrained bilinear maximization on the bi-adjacency matrix of \(G\). It runs in time exponential in \(k\) and \(\delta^{-1}\), but linear in the size of the input. Further, we show that, in the (unconstrained) BCC setting, an \({(1-\delta)}\)-approximation can be achieved by \(O(\delta^{-1})\) 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.