We revisit the classic task of finding the shortest tour of $n$ points in $d$-dimensional Euclidean space, for any fixed constant $d \geq 2$. We determine the optimal dependence on $\varepsilon$ in the running time of an algorithm that computes a $(1+\varepsilon)$-approximate tour, under a plausible assumption. Specifically, we give an algorithm that runs in $2^{\mathcal{O}(1/\varepsilon^{d-1})} n\log n$ time. This improves the previously smallest dependence on $\varepsilon$ in the running time $(1/\varepsilon)^{\mathcal{O}(1/\varepsilon^{d-1})}n \log n$ of the algorithm by Rao and Smith (STOC 1998). We also show that a $2^{o(1/\varepsilon^{d-1})}\text{poly}(n)$ algorithm would violate the Gap-Exponential Time Hypothesis (Gap-ETH). Our new algorithm builds upon the celebrated quadtree-based methods initially proposed by Arora (J. ACM 1998), but it adds a simple new idea that we call \emph{sparsity-sensitive patching}. On a high level this lets the granularity with which we simplify the tour depend on how sparse it is locally. Our approach is (arguably) simpler than the one by Rao and Smith since it can work without geometric spanners. We demonstrate the technique extends easily to other problems, by showing as an example that it also yields a Gap-ETH-tight approximation scheme for Rectilinear Steiner Tree.

中文翻译：

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欧几里得 TSP 的 Gap-ETH-Tight 近似方案

我们重新审视在 $d$ 维欧几里得空间中寻找 $n$ 点的最短路径的经典任务，对于任何固定常数 $d\geq 2$。我们在一个合理的假设下，确定在计算 $(1+\varepsilon)$-approximate tour 的算法的运行时间中对 $\varepsilon$ 的最佳依赖。具体来说，我们给出了一个在 $2^{\mathcal{O}(1/\varepsilon^{d-1})} n\log n$ 时间内运行的算法。这改进了先前在算法的运行时间 $(1/\varepsilon)^{\mathcal{O}(1/\varepsilon^{d-1})}n \log n$ 中对 $\varepsilon$ 的最小依赖拉奥和史密斯 (STOC 1998)。我们还表明 $2^{o(1/\varepsilon^{d-1})}\text{poly}(n)$ 算法会违反间隙指数时间假设（Gap-ETH）。我们的新算法建立在 Arora (J. ACM 1998) 最初提出的著名的基于四叉树的方法之上，但它增加了一个简单的新想法，我们称之为 \emph{sparsity-sensitive patching}。在高层次上，这让我们简化游览的粒度取决于它在本地的稀疏程度。我们的方法（可以说）比 Rao 和 Smith 的方法简单，因为它可以在没有几何扳手的情况下工作。我们展示了该技术可以轻松扩展到其他问题，作为一个例子，它也为 Rectilinear Steiner Tree 生成了一个 Gap-ETH-tight 近似方案。