In this work, we study the hardness of approximation of the fair $k$-center problem. In this problem, we are given a set of data points in a metric space that is partitioned into groups and the task is to choose a subset of $k$-data points, called centers, such that a prescribed number of data points from each group are chosen while minimizing the maximum distance from any point to its closest center. Although a polynomial-time $3$-approximation is known for fair $k$-center in general metrics, it has remained open whether this approximation guarantee is tight or could be further improved, especially since the classical unconstrained $k$-center problem admits a polynomial-time factor-$2$ approximation. We resolve this open question by proving that, assuming $\mathsf{P} \neq \mathsf{NP}$, for any $ε>0$, no polynomial-time algorithm can approximate fair $k$-center to $(3-ε)$-factor. Our inapproximability results hold even when only two disjoint groups are present and at least one center must be chosen from each group. Further, it extends to the canonical one-per-group setting with $k$-groups (for arbitrary $k$), where exactly one center must be selected from each group. Consequently, the factor-$3$ barrier for fair $k$-center in general metric spaces is inherent, and existing $3$-approximation algorithms are optimal up to lower-order terms even in these restricted regimes. This result stands in sharp contrast to the $k$-supplier formulation, where both the unconstrained and fair variants admit factor-$3$ approximation in polynomial time.

翻译：本文研究公平k中心问题的近似难度。在该问题中，我们给定一个度量空间中按组划分的数据点集合，任务是从每组数据点中选取规定数量的点作为中心（共选取k个中心），同时最小化任意数据点到其最近中心的最大距离。尽管在一般度量空间中，公平k中心问题存在多项式时间的3-近似算法，但该近似保证是否紧界或能否进一步改进始终是开放问题——特别是考虑到经典无约束k中心问题存在多项式时间的2-近似算法。通过证明在假设$\mathsf{P} \neq \mathsf{NP}$的前提下，对于任意$ε>0$，不存在多项式时间算法能以$(3-ε)$因子近似公平k中心问题，我们解决了这一开放问题。我们的不可近似性结果甚至适用于仅存在两个不相交组且每组至少需选取一个中心的情形。该结论进一步可推广到具有k个组（任意k值）的经典"每组一中心"设定，即必须从每个组中恰好选取一个中心。因此，一般度量空间中公平k中心问题的3因子障碍具有内在必然性，现有3-近似算法即使在这些受限情形下也达到最优（仅存在低阶项差异）。这一结果与k供应点问题形成鲜明对比：无论是无约束形式还是公平变体，k供应点问题都存在多项式时间的3-近似算法。