In this paper, we present a new approach of creating PTAS to the TSP problems by defining a bounded-curvature surface embedded spaces. Using this definition we prove: - A bounded-curvature surface embedded spaces TSP admits to a PTAS. - Every bounded doubling dimension space can be embedded into a bounded-curvature surface. - Every uniform metric space can be embedded into a bounded-curvature surface. Thus, the algorithm generalizes arXiv:1112.0699 (and therefore [7] and [8] as well, w.r.t PTAS of TSP). But, the algorithm is much broader as uniform metric spaces aren't bounded doubling dimension spaces. It should be mentioned that our definition of a surface is derived from Riemannian geometry, but doesn't match it exactly. therefore, our definitions and basic geometry algorithm is given here in full. [7] Sanjeev Arora. 1998. Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems. J. ACM 45, 5 (September 1998), 753-782. DOI=http://dx.doi.org/10.1145/290179.290180 [8] Joseph S. B. Mitchell. Guillotine subdivisions approximate polygonal subdivisions: A simple polynomial time approximation scheme for geometric TSP, k- MST, and related problems. SIAM J. Comput., 28(4):1298-1309, 1999.

翻译：本文提出了一种通过定义有界曲率曲面嵌入空间来为旅行商问题（TSP）创建多项式时间近似方案（PTAS）的新方法。基于这一定义，我们证明了：- 有界曲率曲面嵌入空间中的TSP承认一个PTAS。- 每个有界加倍维空间均可嵌入到有界曲率曲面中。- 每个一致度量空间均可嵌入到有界曲率曲面中。因此，该算法推广了arXiv:1112.0699（进而也推广了[7]和[8]中关于TSP的PTAS）。然而，该算法的适用范围更为广泛，因为一致度量空间并不属于有界加倍维空间。需要指出的是，我们对曲面的定义源于黎曼几何，但与其并不完全吻合。因此，本文完整给出了我们的定义及基本几何算法。[7] Sanjeev Arora. 1998. 欧几里得旅行商问题及其他几何问题的多项式时间近似方案. J. ACM 45, 5 (1998年9月), 753-782. DOI=http://dx.doi.org/10.1145/290179.290180 [8] Joseph S. B. Mitchell. 吉洛丁细分逼近多边形细分：面向几何TSP、k-MST及相关问题的简单多项式时间近似方案. SIAM J. Comput., 28(4):1298-1309, 1999.