# Tight Lower Bounds for Approximate Exact k-Center in ℝ^d

In the discrete k-center problem, we are given a metric space (P,) where |P|=n and the goal is to select a set C⊆ P of k centers which minimizes the maximum distance of a point in P from its nearest center. For any ϵ>0, Agarwal and Procopiuc [SODA '98, Algorithmica '02] designed an (1+ϵ)-approximation algorithm for this problem in d-dimensional Euclidean space which runs in O(dnlog k) + (kϵ)^O(k^1-1/d)· n^O(1) time. In this paper we show that their algorithm is essentially optimal: if for some d≥ 2 and some computable function f, there is an f(k)·(1ϵ)^o(k^1-1/d)· n^o(k^1-1/d) time algorithm for (1+ϵ)-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(d· k^1-1/d) time for discrete k-center on n points in d-dimensional Euclidean space is asymptotically optimal. Formally, we show that if for some d≥ 2 and some computable function f, there is an f(k)· n^o(k^1-1/d) 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]

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