We study exact algorithms for Euclidean TSP in $\mathbb{R}^d$. In the early 1990s algorithms with $n^{O(\sqrt{n})}$ running time were presented for the planar case, and some years later an algorithm with $n^{O(n^{1-1/d})}$ running time was presented for any $d\geq 2$. Despite significant interest in subexponential exact algorithms over the past decade, there has been no progress on Euclidean TSP, except for a lower bound stating that the problem admits no $2^{O(n^{1-1/d-\epsilon})}$ algorithm unless ETH fails. Up to constant factors in the exponent, we settle the complexity of Euclidean TSP by giving a $2^{O(n^{1-1/d})}$ algorithm and by showing that a $2^{o(n^{1-1/d})}$ algorithm does not exist unless ETH fails.

翻译：我们研究$\mathbb{R}^d$中欧几里得旅行商问题（Euclidean TSP）的精确算法。在20世纪90年代初期，针对平面情形提出了运行时间为$n^{O(\sqrt{n})}$的算法，随后几年，针对任意$d\geq 2$给出了运行时间为$n^{O(n^{1-1/d})}$的算法。尽管在过去十年间，亚指数精确算法引起了广泛关注，但欧几里得TSP问题却未取得任何进展，仅有一个下界表明：除非ETH失败，否则该问题不存在$2^{O(n^{1-1/d-\epsilon})}$算法。在指数项的常数因子范围内，我们通过给出一个$2^{O(n^{1-1/d})}$算法，并证明除非ETH失败，否则不存在$2^{o(n^{1-1/d})}$算法，从而确定了欧几里得TSP的复杂度。