We consider the geometric version of the well-known Generalized Traveling Salesman Problem introduced in 2015 by Bhattacharya et al. that is called the Euclidean Generalized Traveling Salesman Problem in Grid Clusters (EGTSP-GC). They proved the intractability of the problem and proposed first polynomial time algorithms with fixed approximation factors. The extension of these results in the field of constructing the polynomial time approximation schemes (PTAS) and the description of non-trivial polynomial time solvable subclasses for the EGTSP-GC appear to be relevant in the light of the classic C. Papadimitriou result on the intractability of the Euclidean TSP and recent inapproximability results for the Traveling Salesman Problem with Neighborhoods (TSPN) in the case of discrete neighborhoods. In this paper, we propose Efficient Polynomial Time Approximation Schemes (EPTAS) for two special cases of the EGTSP-GC, when the number of clusters k = O ( log n ) $k=O(\log n)$ and k = n − O ( log n ) $k=n-O(\log n)$ . Also, we show that any time, when one of the grid dimensions (height or width) is fixed, the EGTSP-GC can be solved to optimality in polynomial time. As a consequence, we specify a novel non-trivial polynomially solvable subclass of the Euclidean TSP in the plane.

中文翻译：

---

网格群中欧几里得广义旅行商问题的复杂性和逼近性

我们考虑 Bhattacharya 等人于 2015 年引入的著名广义旅行商问题的几何版本。这被称为网格集群中的欧几里得广义旅行商问题（EGTSP-GC）。他们证明了问题的棘手性，并提出了第一个具有固定近似因子的多项式时间算法。这些结果在构建多项式时间近似方案 (PTAS) 领域的扩展以及 EGTSP-GC 的非平凡多项式时间可解子类的描述似乎与经典的 C. Papadimitriou 结果相关在离散邻域的情况下，欧几里得 TSP 的难处理性和最近邻域旅行商问题 (TSPN) 的不可逼近性结果。在本文中，我们为 EGTSP-GC 的两种特殊情况提出了有效的多项式时间近似方案 (EPTAS)，当簇数 k = O ( log n ) $k=O(\log n)$ 和 k = n − O ( log n ) $k=nO(\log n)$ 。此外，我们表明，任何时候，当网格维度（高度或宽度）之一固定时，EGTSP-GC 可以在多项式时间内求解为最优。因此，我们在平面中指定了欧几里得 TSP 的一个新的非平凡多项式可解子类。