Hierarchical Agglomerative Clustering via Relaxed Representatives

Due to their simplicity and interpretability, linkage methods are a central class of hierarchical agglomerative clustering (HAC). In particular, representative-based methods, such as centroid and medoid approaches, are appealing because their representatives provide intuitive summaries of the clusters. However, many representative-based linkage methods suffer from important drawbacks, such as high computational cost and the generation of inversions, which limit their broader applicability. We introduce *relaxed-representative clustering*, a principled generalization of representative-based linkages that preserves their interpretability while addressing these limitations. This framework yields (i) RMM*, a quadratic-time variant of the cubic-time Minimax method, and (ii) RC*, an inversion-free variant of the Centroid method whose representatives belong to the input set. On the theoretical side, we prove that for every $k$ our methods are $O(\min\{k,\log n\})$-approximation algorithms for the $k$-center objective. Empirically, experiments on tabular, biological, and textual datasets show that RMM* consistently outperforms several baselines and achieves clustering quality comparable to Minimax at a fraction of the computational cost. Meanwhile, RC* remains competitive with the Centroid method while producing inversion-free dendrograms and interpretable representatives.