On Approximability of Clustering Problems Without Candidate Centers

TL;DR

This paper proves stronger NP-hardness of approximation for continuous clustering problems like k-median and k-means in metric spaces, showing they are harder to approximate than previously known, and highlights differences between continuous and discrete settings.

Contribution

It significantly improves the known hardness of approximation factors for continuous clustering problems in metric spaces, revealing new computational complexity insights.

Findings

Continuous k-median is NP-hard to approximate within 2 - o(1)

Continuous k-means is NP-hard to approximate within 4 - o(1)

k-minsum has a hardness of 1.415 approximation factor

Abstract

The k-means objective is arguably the most widely-used cost function for modeling clustering tasks in a metric space. In practice and historically, k-means is thought of in a continuous setting, namely where the centers can be located anywhere in the metric space. For example, the popular Lloyd's heuristic locates a center at the mean of each cluster. Despite persistent efforts on understanding the approximability of k-means, and other classic clustering problems such as k-median and k-minsum, our knowledge of the hardness of approximation factors of these problems remains quite poor. In this paper, we significantly improve upon the hardness of approximation factors known in the literature for these objectives. We show that if the input lies in a general metric space, it is NP-hard to approximate: $\bullet$ Continuous k-median to a factor of $2-o(1)$; this improves upon the previous inapproximability factor of 1.36 shown by Guha and Khuller (J. Algorithms '99). $\bullet$ Continuous k-means to a factor of $4- o(1)$; this improves upon the previous inapproximability factor of 2.10 shown by Guha and Khuller (J. Algorithms '99). $\bullet$ k-minsum to a factor of $1.415$; this improves upon the APX-hardness shown by Guruswami and Indyk (SODA '03). Our results shed new and perhaps counter-intuitive light on the differences between clustering problems in the continuous setting versus the discrete setting (where the candidate centers are given as part of the input).