In this work, we study the hardness of approximation of the fair $k$-center problem. Here the data points are partitioned into groups and the task is to choose a prescribed number of data points from each group, called centers, while minimizing the maximum distance from any point to its closest center. Although a polynomial-time $3$-approximation is known for this problem in general metrics, it has remained open whether this approximation guarantee is tight or could be further improved, especially since the unconstrained $k$-center problem admits a polynomial-time factor-$2$ approximation. We resolve this open question by proving that, for every $ε>0$, achieving a $(3-ε)$-approximation is NP-hard, assuming $\text{P} \neq \text{NP}$. 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$k$-中心问题的近似计算难度。该问题中数据点被划分为若干组，任务是从每组中选取规定数量的数据点作为中心点，同时最小化任意点到其最近中心点的最大距离。尽管在一般度量空间中该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可进一步改进的问题始终悬而未决——特别是考虑到无约束$k$-中心问题存在多项式时间的$2$-近似算法。我们通过证明以下结论解决了这一开放性问题：在$\text{P} \neq \text{NP}$的假设下，对于任意$ε>0$，实现$(3-ε)$-近似都是NP难的。我们的不可近似性结果甚至适用于仅存在两个不相交分组且每组至少需选取一个中心点的情形。该结论进一步可推广至具有$k$个分组（任意$k$值）的经典"每组恰选一中心"场景。因此，一般度量空间中公平$k$-中心问题的$3$倍近似障碍具有内在必然性，现有$3$-近似算法即使在这些受限场景下也是最优的（仅存在低阶项差异）。该结论与$k$-供应商问题形成鲜明对比：在多项式时间内，无约束与公平变体的$k$-供应商问题均可实现$3$-近似。