The maximum traveling salesman problem (Max TSP) is one of the intensively researched combinatorial optimization problems. It consists of finding a maximum-weight Hamiltonian cycle in a given complete weighted graph. This problem is APX-hard in the general and metric cases but admits approximation schemes in the geometric setting, when the vertices of the input graph are some points in fixed-dimensional real space and the edge weights are induced by some vector norm. In this paper, we suggest the first polynomial-time approximation scheme for Max TSP in arbitrary metric space of fixed doubling dimension. In fact, the suggested algorithm implements a scheme EPTAS: for any fixed $\varepsilon\in(0,1)$, it computes a $(1-\varepsilon)$-approximate solution of the considered problem in cubic time. Also, we propose a polynomial-time algorithm which finds asymptotically optimal solutions of the metric Max TSP in fixed and sublogarithmic doubling dimensions.

翻译：最大旅行销售员问题(Max TSP)是经过深入研究的组合优化问题之一。 它包括在一个特定的完整加权图中找到一个最大重量的汉密尔顿周期。 这个问题在一般和计量案例中是APX硬的, 但在几何设置中承认近似方案, 当输入图的顶点是固定维实际空间的某些点, 边缘重量是由某种矢量规范引起的。 在本文中, 我们建议为 Max TSP 在固定双倍的任意计量空间中的第一个多元时近似方案。 事实上, 建议的算法会实施一种 ECTAS 方案: 对于任何固定的 $\ varepsilon\ in ( 0. 1 $), 它计算出一个( 1\ varepsilon) $- 近似几何方法在立方位时所考虑的问题的解决方案。 另外, 我们提议一个多位时算法, 它将找到在固定和子对称的 Max TSP 在固定和子对称双倍的双倍的维维维维维维维维度上 。