The Price of Explainability for Clustering

Given a set of points in d-dimensional space, an explainable clustering is one where the clusters are specified by a tree of axis-aligned threshold cuts. Dasgupta et al. (ICML 2020) posed the question of the price of explainability: the worst-case ratio between the cost of the best explainable clusterings to that of the best clusterings. We show that the price of explainability for k-medians is at most 1 + Hk-1; in fact, we show that the popular Random Thresholds algorithm has exactly this price of explainability, matching the known lower bound constructions. We complement our tight analysis of this particular algorithm by constructing instances where the price of explainability (using any algorithm) is at least (1 - o(1)) ln k, showing that our result is best possible, up to lower-order terms. We also improve the price of explainability for the k-means problem to O(k ln ln k) from the previous O(k ln k), considerably closing the gap to the lower bounds of Omega(k). Finally, we study the algorithmic question of finding the best explainable clustering: We show that explainable k-medians and k-means cannot be approximated better than O(ln k), under standard complexity-theoretic conjectures. This essentially settles the approximability of explainable k-medians and leaves open the intriguing possibility to get significantly better approximation algorithms for k-means than its price of explainability.