Bipartite Correlation Clustering -- Maximizing Agreements
This provides an algorithmic solution for clustering bipartite graphs with correlation constraints, which is incremental as it builds on known hardness and approximation frameworks.
The paper tackles the NP-hard Bipartite Correlation Clustering problem by developing a novel approximation algorithm for its variant with a cluster number bound, achieving a (1-δ)-factor approximation for any desired accuracy δ, and shows this leads to an Efficient PTAS for the unconstrained version.
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 ${(1-δ)}$-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 $δ^{-1}$, 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 $O(δ^{-1})$ 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.