Faster Approximation Schemes for $k$-TSP and $k$-MST in the Euclidean Space

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].