The Colored Points Traveling Salesman Problem (Colored Points TSP) is introduced in this work as a novel variation of the traditional Traveling Salesman Problem (TSP) in which the set of points is partitioned into multiple classes, each of which is represented by a distinct color (or label). The goal is to find a minimum cost cycle $C$ that visits all the colors and only makes each one appears once. This issue has various applications in the fields of transportation, goods distribution network, postal network, inspection, insurance, banking, etc. By reducing the traditional TSP to it, we can demonstrate that Colored Points TSP is NP-hard. Here, we offer a $\frac{2\pi r}{3}$-approximation algorithm to solve this issue, where $r$ denotes the radius of the points' smallest color-spanning circle. The algorithm has been implemented, executed on random datasets, and compared against the brute force method.

翻译：本文引入有色点旅行商问题（Colored Points TSP）作为传统旅行商问题（TSP）的一种新颖变体，其中点集被划分为多个类别，每个类别由不同颜色（或标签）表示。目标是找到一个最小代价的循环路径$C$，该路径访问所有颜色且每种颜色仅出现一次。该问题在交通、货物配送网络、邮政网络、巡检、保险、银行等领域具有广泛的应用。通过将传统TSP归约到该问题，我们可以证明有色点TSP是NP难的。本文提供了一种$\frac{2\pi r}{3}$近似算法来解决该问题，其中$r$表示点集最小颜色覆盖圆的半径。该算法已实现并在随机数据集上执行，与暴力方法进行了比较。