We consider the classic 1-center problem: Given a set $P$ of $n$ points in a metric space find the point in $P$ that minimizes the maximum distance to the other points of $P$. We study the complexity of this problem in $d$-dimensional $\ell_p$-metrics and in edit and Ulam metrics over strings of length $d$. Our results for the 1-center problem may be classified based on $d$ as follows. $\bullet$ Small $d$: Assuming the hitting set conjecture (HSC), we show that when $d=\omega(\log n)$, no subquadratic algorithm can solve 1-center problem in any of the $\ell_p$-metrics, or in edit or Ulam metrics. $\bullet$ Large $d$: When $d=\Omega(n)$, we extend our conditional lower bound to rule out subquartic algorithms for 1-center problem in edit metric (assuming Quantified SETH). On the other hand, we give a $(1+\epsilon)$-approximation for 1-center in Ulam metric with running time $\tilde{O_{\varepsilon}}(nd+n^2\sqrt{d})$. We also strengthen some of the above lower bounds by allowing approximations or by reducing the dimension $d$, but only against a weaker class of algorithms which list all requisite solutions. Moreover, we extend one of our hardness results to rule out subquartic algorithms for the well-studied 1-median problem in the edit metric, where given a set of $n$ strings each of length $n$, the goal is to find a string in the set that minimizes the sum of the edit distances to the rest of the strings in the set.

翻译：我们考虑经典的1-中心问题：给定一个度量空间中$n$个点的集合$P$，寻找$P中$使得到$P$中其他点的最大距离最小的点。我们研究了该问题在$d$维$\ell_p$度量以及长度为$d$的字符串编辑度量和Ulam度量下的复杂度。针对1-中心问题的结果可根据$d$分类如下：$\bullet$ 小$d$情形：假设击中集猜想（HSC），我们证明当$d=\omega(\log n)$时，在任意$\ell_p$度量、编辑度量或Ulam度量下，不存在次二次算法能解决1-中心问题。$\bullet$ 大$d$情形：当$d=\Omega(n)$时，我们扩展了条件下界，排除编辑度量下1-中心问题存在次四次算法（基于量化SETH）。另一方面，我们给出了Ulam度量下1-中心问题的$(1+\epsilon)$-近似算法，运行时间为$\tilde{O_{\varepsilon}}(nd+n^2\sqrt{d})$。我们还通过允许近似或降低维度$d$加强了部分下界，但仅针对需列举所有必要解的较弱算法类别。此外，我们将一个硬度结果扩展至编辑度量中广泛研究的1-中位数问题（给定$n$个长度为$n$的字符串集，目标是找到集合中与其余字符串编辑距离之和最小的字符串），排除了该问题的次四次算法。