AI 中文总结
研究 k 色非交叉欧几里得旅行商问题，此问题是 NP 难的。核心方法是提出多项式时间(k + ε)近似算法，主要贡献为解决 k - ETSP 难题，给出了相应近似算法。
AI 中文摘要
给定一个 k 色点集\(P\subseteq \mathbb{R}^2\)，k 色非交叉欧几里得旅行商问题（简称 k - ETSP）要求 k 条非交叉封闭曲线，每条曲线跨越一个对应颜色类，使曲线两两不交叉且欧几里得长度之和最小。由于 1 - ETSP 是标准欧几里得旅行商问题，该问题是 NP 难的。我们提出了 k - ETSP 的多项式时间(k + ε)近似算法。
英文摘要
Given a $k$-coloured point set $P\subseteq \mathbb{R}^2$, the $k$-coloured Non-crossing Euclidean Travelling Salesperson Problem (short $k$-ETSP) asks for $k$ non-crossing closed curves, where one curve spans one corresponding colour class, such that the curves are pairwise non-crossing and the sum of their Euclidean lengths is minimised. This problem is NP-hard as $1$-ETSP is the standard Euclidean Travelling Salesperson Problem. We present a polynomial-time $(k+ε)$-approximation for $k$-ETSP.
