In the Euclidean $k$-TSP (resp. Euclidean $k$-MST), we are given $n$ points in the $d$-dimensional Euclidean space (for any fixed constant $d\geq 2$) and a positive integer $k$, the goal is to find a shortest tour visiting at least $k$ points (resp. a minimum tree spanning at least $k$ points). We give approximation schemes for both Euclidean $k$-TSP and Euclidean $k$-MST in time $2^{O(1/\varepsilon^{d-1})}\cdot n \cdot(\log n)^{d\cdot 4^{d}}$. This improves the running time of the previous approximation schemes due to Arora [J. ACM 1998] and Mitchell [SICOMP 1999]. Our algorithms can be derandomized by increasing the running time by a factor $O(n^d)$. In addition, our algorithm for Euclidean $k$-TSP is Gap-ETH tight, given the matching Gap-ETH lower bound due to Kisfaludi-Bak, Nederlof, and W\k{e}grzycki [FOCS 2021].

翻译：在欧几里得$k$-TSP（分别地，欧几里得$k$-MST）问题中，给定$d$维欧几里得空间中的$n$个点（对于任意固定常数$d\geq 2$）和一个正整数$k$，目标是找到一条访问至少$k$个点的最短游览路径（分别地，一棵覆盖至少$k$个点的最小树）。我们针对欧几里得$k$-TSP和欧几里得$k$-MST两者，给出了时间复杂度为$2^{O(1/\varepsilon^{d-1})}\cdot n \cdot(\log n)^{d\cdot 4^{d}}$的近似方案。这改进了Arora [J. ACM 1998] 和Mitchell [SICOMP 1999] 先前近似方案的运行时间。我们的算法可以通过将运行时间增加$O(n^d)$因子来去随机化。此外，针对欧几里得$k$-TSP的算法是Gap-ETH紧的，这与Kisfaludi-Bak、Nederlof和W\k{e}grzycki [FOCS 2021] 给出的匹配性Gap-ETH下界一致。