A Bad Instance for k-Means++

Tobias Brunsch, Heiko Röglin

Abstract: k-means++ is a seeding technique for the k-means method with an expected approximation ratio of O(logk), where k denotes the number of clusters. Examples are known on which the expected approximation ratio of k-means++ is Ω(logk), showing that the upper bound is asymptotically tight. However, it remained open whether k-means++ yields an O(1)-approximation with probability 1/poly(k) or even with constant probability. We settle this question and present instances on which k-means++ achieves an approximation ratio of (2/3 − ε)·logk only with exponentially small probability.