Part of Advances in Neural Information Processing Systems 36 (NeurIPS 2023) Main Conference Track
Konstantin Makarychev, Liren Shan
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).