The k-means++ algorithm by Arthur and Vassilvitskii [SODA 2007] is a classical and time-tested algorithm for the k-means problem. While being very practical, the algorithm also has good theoretical guarantees: its solution is O(log k)-approximate, in expectation. In a recent work, Bhattacharya, Eube, Roglin, and Schmidt [ESA 2020] considered the following question: does the algorithm retain its guarantees if we allow for a slight adversarial noise in the sampling probability distributions used by the algorithm? This is motivated e.g. by the fact that computations with real numbers in k-means++ implementations are inexact. Surprisingly, the analysis under this scenario gets substantially more difficult and the authors were able to prove only a weaker approximation guarantee of O(log² k). In this paper, we close the gap by providing a tight, O(log k)-approximate guarantee for the k-means++ algorithm with noise.

The famous k-means++ algorithm of Arthur and Vassilvitskii [SODA 2007] is the most popular way of solving the k-means problem in practice. The algorithm is very simple: it samples the first center uniformly at random and each of the following k — 1 centers is then always sampled proportional to its squared distance to the closest center so far. Afterward, Lloyd's iterative algorithm is run. The k-means++ algorithm is known to return Θ(log k) approximate solution in expectation. In their seminal work, Arthur and Vassilvitskii [SODA 2007] asked about the guarantees for its following greedy variant: in every step, we sample ℓ candidate centers instead of one and then pick the one that minimizes the new cost. This is also how k-means++ is implemented in e.g. the popular Scikit-learn library [Pedregosa et al.; JMLR 2011]. We present nearly matching lower and upper bounds for the greedy k-means++: We prove that it is an O(ℓ3 log3 k)-approximation algorithm. On the other hand, we prove a lower bound of Ω(ℓ3 log3 k/ log2 (ℓ log k)). Previously, only an Ω(ℓ log k) lower bound was known [Bhattacharya, Eube, Röglin, Schmidt; ESA 2020] and there was no known upper bound.