Random Cuts are Optimal for Explainable k-Medians

Published: 21 Sept 2023, Last Modified: 02 Nov 2023NeurIPS 2023 oralEveryoneRevisionsBibTeX
Keywords: Clustering, k-medians, Decision Tree, Explainability
TL;DR: We provide the optimal competitive ratio for explainable k-medians.
Abstract: We show that the RandomCoordinateCut algorithm gives the optimal competitive ratio for explainable $k$-medians in $\ell_1$. The problem of explainable $k$-medians was introduced by Dasgupta, Frost, Moshkovitz, and Rashtchian in 2020. Several groups of authors independently proposed a simple polynomial-time randomized algorithm for the problem and showed that this algorithm is $O(\log k \log\log k)$ competitive. We provide a tight analysis of the algorithm and prove that its competitive ratio is upper bounded by $2\ln k+2$. This bound matches the $\Omega(\log k)$ lower bound by Dasgupta et al (2020).
Supplementary Material: pdf
Submission Number: 3589