Internal measures that are used to assess the quality of a clustering usually take into account intra-group and/or inter-group criteria. There are many papers in the literature that propose algorithms with provable approximation guarantees for optimizing the former. However, the optimization of inter-group criteria is much less understood. Here, we contribute to the state-of-the-art of this literature by devising algorithms with provable guarantees for the maximization of two natural inter-group criteria, namely the minimum spacing and the minimum spanning tree spacing. The former is the minimum distance between points in different groups while the latter captures separability through the cost of the minimum spanning tree that connects all groups. We obtain results for both the unrestricted case, in which no constraint on the clusters is imposed, and for the constrained case where each group is required to have a minimum number of points. Our constraint is motivated by the fact that the popular Single-Linkage, which optimizes both criteria in the unrestricted case, produces clustering with many tiny groups. To complement our work, we present an empirical study with 10 real datasets that provides evidence that our methods work very well in practical settings.

The price of explainability for a clustering task can be defined as the unavoidable loss, in terms of the objective function, if we force the final partition to be explainable. Here, we study this price for the following clustering problems: $k$-means, $k$-medians, $k$-centers and maximum-spacing. We provide upper and lower bounds for a natural model where explainability is achieved via decision trees. For the $k$-means and $k$-medians problems our upper bounds improve those obtained by [Dasgupta et. al, ICML 20] for low dimensions. Another contribution is a simple and efficient algorithm for building explainable clusterings for the $k$-means problem. We provide empirical evidence that its performance is better than the current state of the art for decision-tree based explainable clustering.

We study the problem of optimizing the clustering of a set of vectors when the quality of the clustering is measured by the Entropy or the Gini impurity measure. Our results contribute to the state of the art both in terms of best known approximation guarantees and inapproximability bounds: (i) we give the first polynomial time algorithm for Entropy impurity based clustering with approximation guarantee independent of the number of vectors and (ii) we show that the problem of clustering based on entropy impurity does not admit a PTAS. This also implies an inapproximability result in information theoretic clustering for probability distributions closing a problem left open in [Chaudhury and McGregor, COLT08] and [Ackermann et al., ECCC11]. We also report experiments with a new clustering method that was designed on top of the theoretical tools leading to the above results. These experiments suggest a practical applicability for our method, in particular, when the number of clusters is large.