在这篇文章中，我们研究了欧氏最小生成树问题的不精确设置。该问题在文献中称为\emph{Minimum Spanning Tree Problem with Neighborhoods}。我们研究的问题表示为非交叉线段的邻域。给定$I\!\!R^2$中$n$不相交线段的集合${\cal S}$，目标是找到一个最小生成树（MST），该最小生成树包含$\cal S$中每个线段的一个端点，并且该最小生成树的代价在$2^n$可能的最小生成树中最小.我们发现，找到这样的MST是NP-困难的一般情况下，并提出了一个$2α$因子近似算法相同，其中$α$是最有名的近似算法的近似因子，以计算最小成本Steiner树在无向图与非负边权重。作为我们约简的一个含义，我们可以证明问题的无限制版本（即，必须从每个段中选择一个点，使得MST的成本尽可能最小）也是NP难的。我们还提出了一个参数化算法的问题的基础上的“可分性”参数定义的部分。
英文摘要：
In this article, we study the Euclidean minimum spanning tree problem in an imprecise setup. The problem is known as the \emph{Minimum Spanning Tree Problem with Neighborhoods} in the literature. We study the problem where the neighborhoods are represented as non-crossing line segments. Given a set ${\cal S}$ of $n$ disjoint line segments in $I\!\!R^2$, the objective is to find a minimum spanning tree (MST) that contains exactly one end-point from each segment in $\cal S$ and the cost of the MST is minimum among $2^n$ possible MSTs. We show that finding such an MST is NP-hard in general, and propose a $2α$-factor approximation algorithm for the same, where $α$ is the approximation factor of the best-known approximation algorithm to compute a minimum cost Steiner tree in an undirected graph with non-negative edge weights. As an implication of our reduction, we can show that the unrestricted version of the problem (i.e., one point must be chosen from each segment such that the cost of MST is as minimum as possible) is also NP-hard. We also propose a parameterized algorithm for the problem based on the "separability" parameter defined for segments.
A Surprisingly Simple Method for Distributed Euclidean-Minimum Spanning Tree / Single Linkage Dendrogram Construction from High Dimensional Embeddings via Distance Decomposition
