Random Cuts are Optimal for Explainable k-Medians

We show that the RandomCoordinateCut algorithm gives the optimal competitive ratio for explainable k-medians in ℓ 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 Ω(log k) lower bound by Dasgupta et al (2020) .