Given a metric space $(V, d)$ along with an integer $k$, the $k$-Median problem asks to open $k$ centers $C \subseteq V$ to minimize $\sum_{v \in V} d(v, C)$, where $d(v, C) := \min_{c \in C} d(v, c)$. While the best-known approximation ratio of $2.613$ holds for the more general supplier version where an additional set $F \subseteq V$ is given with the restriction $C \subseteq F$, the best known hardness for these two versions are $1+1/e \approx 1.36$ and $1+2/e \approx 1.73$ respectively, using the same reduction from Max $k$-Coverage. We prove the following two results separating them. First, we show a $1.546$-parameterized approximation algorithm that runs in time $f(k) n^{O(1)}$. Since $1+2/e$ is proved to be the optimal approximation ratio for the supplier version in the parameterized setting, this result separates the original $k$-Median from the supplier version. Next, we prove a $1.416$-hardness for polynomial-time algorithms assuming the Unique Games Conjecture. This is achieved via a new fine-grained hardness of Max-$k$-Coverage for small set sizes. Our upper bound and lower bound are derived from almost the same expression, with the only difference coming from the well-known separation between the powers of LP and SDP on (hypergraph) vertex cover.

翻译：给定度量空间$(V, d)$和整数$k$，$k$-中位数问题要求选择$k$个中心$C \subseteq V$，以最小化$\sum_{v \in V} d(v, C)$，其中$d(v, C) := \min_{c \in C} d(v, c)$。对于更一般的供应商版本（额外给定集合$F \subseteq V$并限制$C \subseteq F$），目前已知的最佳近似比为$2.613$，而这两个版本的最佳下界分别为$1+1/e \approx 1.36$和$1+2/e \approx 1.73$，且均基于最大$k$-覆盖问题的相同归约。我们证明了以下两个分离结果。首先，我们提出一个运行时间为$f(k) n^{O(1)}$的$1.546$-参数化近似算法。由于$1+2/e$已被证明是供应商版本在参数化设置下的最优近似比，该结果将原始$k$-中位数与供应商版本区分开。其次，我们假设唯一游戏猜想，证明了一个多项式时间算法的$1.416$-下界。这一结果通过针对小集合大小的最大$k$-覆盖问题的新细粒度下界实现。我们的上界和下界源于几乎相同的表达式，唯一差异来自LP与SDP在（超图）顶点覆盖问题上众所周知的分离能力。