We address point sets in the $d$-dimensional space, in which every point is colored with at least one color out of the set $C$. A Color-Spanning Set or Rainbow Set is a set of points that covers all colors of $C$. The diameter of a set is the maximum distance between two points in the set. In this paper we answer some open questions about Minimum Diameter Color-Spanning Sets in $d$-dimensional space, which were studied by Fleischer and Xu (2010), and extend this concept to general graphs. We show that the problem is W[1] hard for parameter $\left|C\right|$ and not in PTAS if the number of dimensions is part of the input. Furthermore we demonstrate the membership in W[2]. Most importantly we present two exact solution methods, which both can also be used for the Largest Closest Color-Spanning Set Problem and compare them in an experimental evaluation. For general graphs we show that Minimum Diameter Color-Spanning Set Problem is NP-hard and W[1]-hard for parameter $\left|C\right|$ but give a polynomial time algorithm for trees. In addition we develop a polynomial time 2-approximation algorithm for general graphs and prove that this problem admits no better approximation factor, unless $P=NP$.

我们在 $d$维空间，其中每个点都用至少一种颜色着色 $C$。跨色集或彩虹集是涵盖所有颜色的点集$C$。集合的直径是集合中两点之间的最大距离。在本文中，我们回答一些有关最小直径跨度集的公开问题。$d$维空间（由Fleischer和Xu（2010）研究），并将此概念扩展到一般图。我们证明问题在于参数W [1]很难$\left|C\right|$如果尺寸数是输入的一部分，则不在PTAS中。此外，我们证明了W [2]中的成员资格。最重要的是，我们提出了两种精确的求解方法，它们也都可以用于最大最近色跨集问题，并在实验评估中进行比较。对于一般图形，我们显示最小直径跨色集问题对于参数来说是NP-hard和W [1] -hard$\left|C\right|$但给出了树的多项式时间算法。此外，我们为一般图开发了多项式时间2逼近算法，并证明了这个问题没有更好的逼近因子，除非$P=ñP$。