The Traveling Salesman Problem (TSP) in the two-dimensional Euclidean plane is among the oldest and most famous NP-hard optimization problems. In breakthrough works, Arora [J. ACM 1998] and Mitchell [SICOMP 1999] gave the first polynomial time approximation schemes. The running time of their approximation schemes was improved by Rao and Smith [STOC 1998] to $(1/\varepsilon)^{O(1/\varepsilon)} n \log n$. Bartal and Gottlieb [FOCS 2013] gave an approximation scheme of running time $2^{(1/\varepsilon)^{O(1)}} n$, which is optimal in $n$. Recently, Kisfaludi-Bak, Nederlof, and W\k{e}grzycki [FOCS 2021] gave a $2^{O(1/\varepsilon)} n \log n$ time approximation scheme, achieving the optimal running time in $\varepsilon$ under the Gap-ETH conjecture. In our work, we give a $2^{O(1/\varepsilon)} n$ time approximation scheme, achieving the optimal running time both in $n$ and in $\varepsilon$ under the Gap-ETH conjecture.

翻译：二维欧几里得平面上的旅行商问题是最古老且最著名的NP难优化问题之一。在突破性工作中，Arora [J. ACM 1998] 和 Mitchell [SICOMP 1999] 提出了首个多项式时间近似方案。Rao和Smith [STOC 1998] 将其近似方案的运行时间改进为$(1/\varepsilon)^{O(1/\varepsilon)} n \log n$。Bartal和Gottlieb [FOCS 2013] 提出了运行时间为$2^{(1/\varepsilon)^{O(1)}} n$的近似方案，该方案在$n$上达到最优。最近，Kisfaludi-Bak、Nederlof和Węgrzycki [FOCS 2021] 提出了$2^{O(1/\varepsilon)} n \log n$时间的近似方案，在Gap-ETH猜想下实现了$\varepsilon$方面的最优运行时间。在本工作中，我们提出了$2^{O(1/\varepsilon)} n$时间的近似方案，在Gap-ETH猜想下同时实现了$n$和$\varepsilon$方面的最优运行时间。