In the Traveling Salesperson Problem with Neighborhoods (TSPN), we are given a collection of geometric regions in some space. The goal is to output a tour of minimum length that visits at least one point in each region. Even in the Euclidean plane, TSPN is known to be APX-hard [27{, which gives rise to studying more tractable special cases of the problem. In this article, we focus on the fundamental special case of regions that are hyperplanes in the d -dimensional Euclidean space. This case contrasts the much-better understood case of so-called fat regions [20, 40{. While for d = 2, an exact algorithm with a running time of O(n 5 ) is known [34{, settling the exact approximability of the problem for d = 3 has been repeatedly posed as an open question [29, 30, 40, 47{. To date, only an approximation algorithm with guarantee exponential in d is known [30{, and NP-hardness remains open. For arbitrary fixed d , we develop a Polynomial Time Approximation Scheme (PTAS) that works for both the tour and path version of the problem. Our algorithm is based on approximating the convex hull of an optimal tour by a convex polytope of bounded complexity. After enumerating a number of structural properties of these polytopes, a linear program finds one of them that minimizes the length of the tour. As the approximation guarantee approaches 1, our scheme adjusts the complexity of the considered polytopes accordingly. In the analysis of our approximation scheme, we show that our search space includes a sufficiently good approximation of the optimum. To do so, we develop a novel and general sparsification technique that transforms an arbitrary convex polytope into one with a constant number of vertices, and, subsequently, into one of bounded complexity in the above sense. We show that this transformation does not increase the tour length by too much, while the transformed tour visits any hyperplane that it visited before the transformation.

中文翻译：

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具有超平面邻域的欧几里得 TSP 的 PTAS

在邻里旅行销售员问题 (TSPN) 中，我们得到了某个空间中的几何区域集合。目标是输出访问每个区域中至少一个点的最小长度的游览。即使在欧几里得平面上，TSPN 也被认为是 APX-hard [27{，这导致研究更容易处理的问题特殊情况。在这篇文章中，我们关注的是区域中超平面的基本特例。d维欧几里得空间。这种情况与更好理解的所谓脂肪区域的情况形成对比 [20, 40{。而对于d= 2，运行时间为 O(n) 的精确算法5) 是已知的 [34{，解决问题的精确近似性d= 3 已作为一个开放问题反复提出 [29, 30, 40, 47{。迄今为止，只有一种保证指数的近似算法d已知 [30{，并且 NP 硬度保持开放。对于任意固定d，我们开发了一个多项式时间近似方案（PTAS），它适用于问题的游览和路径版本。我们的算法基于通过有限复杂度的凸多面体来近似最优游览的凸包。在列举了这些多面体的许多结构特性之后，线性程序会找到其中一个使旅行的长度最小化。随着近似保证接近 1，我们的方案相应地调整了所考虑的多面体的复杂性。在对我们的近似方案的分析中，我们表明我们的搜索空间包括一个足够好的最优近似。为此，我们开发了一种新颖且通用的稀疏化技术，该技术将任意凸多面体转换为具有恒定顶点数的多面体，并且随后，进入上述意义上的有限复杂性之一。我们表明，这种变换不会过多地增加旅行长度，而变换后的旅行会访问它在变换之前访问过的任何超平面。