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 ( log k ) , where k denotes the number of clusters. Examples are known on which the expected approximation ratio of k-means++ is Ω ( log k ) , showing that the upper bound is asymptotically tight. However, it remained open whether k-means++ yields a constant 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 no better than ( 2 / 3 − ε ) ⋅ log k with probability exponentially close to 1 .