Fine-Grained Complexity of Continuous Euclidean k-Center

2026-03-30

AI Summary

This work investigates the fine-grained complexity of the continuous Euclidean k-center problem in constant dimensions, aiming to establish tight lower bounds on the running time of algorithms for this fundamental clustering task. By leveraging conditional complexity assumptions such as the Exponential Time Hypothesis (ETH) and the 3-SUM conjecture, and combining them with geometric embedding techniques that reduce the feasibility of linear systems to clustering instances, the study employs a parameterized algorithmic framework to derive the first rigorous time lower bounds for both exact and approximation algorithms. These results resolve a long-standing open question regarding the problem’s computational complexity and demonstrate that existing algorithms are nearly optimal with respect to the parameter k, particularly in the practically relevant regime of small k.

Abstract

In the (continuous) Euclidean $k$-center problem, given $n$ points in $\mathbb{R}^d$ and an integer $k$, the goal is to find $k$ center points in $\mathbb{R}^d$ that minimize the maximum Euclidean distance from any input point to its closest center. In this paper, we establish conditional lower bounds for this problem in constant dimensions in two settings. $\bullet$ Parameterized by $k$: Assuming the Exponential Time Hypothesis (ETH), we show that there is no $f(k)n^{o(k^{1-1/d})}$-time algorithm for the Euclidean $k$-center problem. This result shows that the algorithm of Agarwal and Procopiuc [SODA 1998; Algorithmica 2002] is essentially optimal. Furthermore, our lower bound rules out any $(1+\varepsilon)$-approximation algorithm running in time $(k/\varepsilon)^{o(k^{1-1/d})}n^{O(1)}$, thereby establishing near-optimality of the corresponding approximation scheme by the same authors. $\bullet$ Small $k$: Assuming the 3-SUM hypothesis, we prove that for any $\varepsilon>0$ there is no $O(n^{2-\varepsilon})$-time algorithm for the Euclidean $2$-center problem in $\mathbb{R}^3$. This settles an open question posed by Agarwal, Ben Avraham, and Sharir [SoCG 2010; Computational Geometry 2013]. In addition, under the same hypothesis, we prove that for any $\varepsilon > 0$, the Euclidean $6$-center problem in $\mathbb{R}^2$ also admits no $O(n^{2-\varepsilon})$-time algorithm. The technical core of all our proofs is a novel geometric embedding of a system of linear equations. We construct a point set where each variable corresponds to a specific collection of points, and the geometric structure ensures that a small-radius clustering is possible if and only if the system has a valid solution.

Problem

• Euclidean k-center
• fine-grained complexity
• Exponential Time Hypothesis
• 3-SUM hypothesis
• computational geometry

Innovation

• fine-grained complexity
• Euclidean k-center
• conditional lower bounds
• geometric embedding
• 3-SUM hypothesis