Tight Lower Bounds for Approximate & Exact $k$-Center in $\mathbb{R}^d$

TL;DR

This paper proves that the known algorithms for the discrete $k$-center problem in Euclidean space are essentially optimal under ETH, establishing tight lower bounds for both approximate and exact solutions in higher dimensions.

Contribution

The paper introduces tight lower bounds for approximate and exact algorithms for the discrete $k$-center problem in Euclidean space, matching the upper bounds under ETH.

Findings

Lower bounds match existing algorithms, assuming ETH.

Exact algorithm is asymptotically optimal in $d$-dimensional space.

Approximation algorithms cannot be significantly improved without ETH failure.

Abstract

In the discrete $k$-center problem, we are given a metric space $(P,\texttt{dist})$ where $|P|=n$ and the goal is to select a set $C\subseteq P$ of $k$ centers which minimizes the maximum distance of a point in $P$ from its nearest center. For any $\epsilon>0$, Agarwal and Procopiuc [SODA '98, Algorithmica '02] designed an $(1+\epsilon)$-approximation algorithm for this problem in $d$-dimensional Euclidean space which runs in $O(dn\log k) + \left(\dfrac{k}{\epsilon}\right)^{O\left(k^{1-1/d}\right)}\cdot n^{O(1)}$ time. In this paper we show that their algorithm is essentially optimal: if for some $d\geq 2$ and some computable function $f$, there is an $f(k)\cdot \left(\dfrac{1}{\epsilon}\right)^{o\left(k^{1-1/d}\right)} \cdot n^{o\left(k^{1-1/d}\right)}$ time algorithm for $(1+\epsilon)$-approximating the discrete $k$-center on $n$ points in $d$-dimensional Euclidean space then the Exponential Time Hypothesis (ETH) fails. We obtain our lower bound by designing a gap reduction from a $d$-dimensional constraint satisfaction problem (CSP) defined by Marx and Sidiropoulos [SoCG '14] to discrete $d$-dimensional $k$-center. As a byproduct of our reduction, we also obtain that the exact algorithm of Agarwal and Procopiuc [SODA '98, Algorithmica '02] which runs in $n^{O\left(d\cdot k^{1-1/d}\right)}$ time for discrete $k$-center on $n$ points in $d$-dimensional Euclidean space is asymptotically optimal. Formally, we show that if for some $d\geq 2$ and some computable function $f$, there is an $f(k)\cdot n^{o\left(k^{1-1/d}\right)}$ time exact algorithm for the discrete $k$-center problem on $n$ points in $d$-dimensional Euclidean space then the Exponential Time Hypothesis (ETH) fails. Previously, such a lower bound was only known for $d=2$ and was implicit in the work of Marx [IWPEC '06]. [see paper for full abstract]