The maximum traveling salesman problem (Max TSP) consists of finding a Hamiltonian cycle with the maximum total weight of the edges in a given complete weighted graph. This problem is APX-hard in the general metric case but admits polynomial-time approximation schemes in the geometric setting, when the edge weights are induced by a vector norm in fixed-dimensional real space. We propose the first approximation scheme for Max TSP in an arbitrary metric space of fixed doubling dimension. The proposed algorithm implements an efficient PTAS which, for any fixed $\varepsilon\in(0,1)$, computes a $(1-\varepsilon)$-approximate solution of the problem in cubic time. Additionally, we suggest a cubic-time algorithm which finds asymptotically optimal solutions of the metric Max TSP in fixed and sublogarithmic doubling dimensions.

翻译：旅行销售员的最大问题(Max TSP) 在于找到一个具有给定的完整加权图中边缘最大总重量的汉密尔顿周期。 这个问题在一般公制案例中是APX硬的, 但在几何环境中却承认了多元时间近似方案, 当边缘重量是由固定维实际空间的矢量规范引起时。 我们提议了马克斯TSP的第一个近似方案, 其任意的尺度空间为固定双倍。 提议的算法实施高效的 PTAS, 对任何固定的 $varepsilon\in( 0. 1美元) 来说, 计算出立方时问题的一个( 1- varepsilon) 近似值的解决方案。 此外, 我们建议了一种三次算法, 它在固定和子对数双倍维中找到矩阵 Max TSP 的不同时最佳解决方案 。