一种关于欧几里得TSP的Gap-ETH紧逼近方案
A Gap-ETH-Tight Approximation Scheme for Euclidean TSP
AI总结：
本研究针对d维欧几里得TSP问题，基于四叉树方法提出稀疏度敏感修补技术，得到了运行时间关于ε依赖更优的(1+ε)近似算法，并证明该结果在Gap-ETH假设下是紧的，且技术可推广至Steiner树相关问题。
摘要：
我们重新研究了在d维欧几里得空间中寻找n个点的最短环游这一经典任务，其中d≥2为任意固定常数。在一个合理假设下，我们确定了计算(1+ε)近似环游的算法运行时间中关于ε的最优依赖关系。具体而言，我们提出了一种运行时间为2^(O(1/ε^(d-1)))n log n的算法。这一结果改进了Rao和Smith（STOC 1998）算法此前最小的ε依赖运行时间(1/ε)^(O(1/ε^(d-1)))n log n。我们还证明，若存在2^(o(1/ε^(d-1)))poly(n)的算法，则会违反Gap-指数时间假设（Gap-Exponential Time Hypothesis, Gap-ETH）。\n我们的新算法建立在Arora（J. ACM 1998）最初提出的著名四叉树方法基础之上，但引入了一个我们称之为“稀疏度敏感修补（sparsity-sensitive patching）”的新思路。大体而言，这使得我们简化环游的粒度取决于其局部的稀疏程度。我们通过证明该技术对Steiner树（Steiner Tree）和直角Steiner树（Rectilinear Steiner Tree）也能得到相同的运行时间，展示了其可推广到其他问题。我们还给出了直角Steiner树匹配的Gap-ETH下界，完善了研究结果。
英文摘要：
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 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. We demonstrate that our technique extends to other problems, by showing that for Steiner Tree and Rectilinear Steiner Tree it yields the same running time. We complement our results with a matching Gap-ETH lower bound for Rectilinear Steiner Tree.
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