Bipartite Correlation Clustering -- Maximizing Agreements

TL;DR

This paper introduces a novel approximation algorithm for bipartite correlation clustering with a fixed number of clusters, achieving near-optimal agreements efficiently, and extends results to the unconstrained case with an efficient PTAS.

Contribution

The paper presents a new approximation algorithm for k-BCC with provable guarantees and extends the approach to the unconstrained BCC setting, providing an efficient PTAS.

Findings

Achieves a (1-δ)-approximation for k-BCC with exponential dependence on k and δ^{-1}.

In the unconstrained BCC, a (1-δ)-approximation can be achieved with O(δ^{-1}) clusters.

Provides an efficient PTAS for maximizing agreements in BCC.

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. 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.