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 [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 - 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 1/delta, 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.