Almost-Optimal Upper and Lower Bounds for Clustering in Low Dimensional Euclidean Spaces

TL;DR

This paper improves the computational efficiency of near-optimal clustering algorithms in low-dimensional Euclidean spaces and establishes tight lower bounds, advancing theoretical understanding of clustering complexity.

Contribution

It presents a faster algorithm for $(1+ ext{epsilon})$-approximate $k$-median and $k$-means clustering in low dimensions and proves a matching lower bound under the Gap Exponential Time Hypothesis.

Findings

Improved running time to $2^{ ilde{O}(1/ ext{epsilon})^{d-1}} imes n imes ext{polylog}(n)$.

Established a near-matching lower bound assuming Gap ETH.

Enhanced understanding of the computational complexity of clustering in low-dimensional spaces.

Abstract

The $k$-median and $k$-means clustering objectives are classic objectives for modeling clustering in a metric space. Given a set of points in a metric space, the goal of the $k$-median (resp. $k$-means) problem is to find $k$ representative points so as to minimize the sum of the distances (resp. sum of squared distances) from each point to its closest representative. Cohen-Addad, Feldmann, and Saulpic [JACM'21] showed how to obtain a $(1+\varepsilon)$-factor approximation in low-dimensional Euclidean metric for both the $k$-median and $k$-means problems in near-linear time $2^{(1/\varepsilon)^{O(d^2)}} n \cdot \text{polylog}(n)$ (where $d$ is the dimension and $n$ is the number of input points). We improve this running time to $2^{\tilde{O}(1/\varepsilon)^{d-1}} \cdot n \cdot \text{polylog}(n)$, and show an almost matching lower bound: under the Gap Exponential Time Hypothesis for 3-SAT, there is no $2^{{o}(1/\varepsilon^{d-1})} n^{O(1)}$ algorithm achieving a $(1+\varepsilon)$-approximation for $k$-means.