  We study a problem proposed by Hurtado et al. (2016) motivated by sparse set
visualization. Given $n$ points in the plane, each labeled with one or more
primary colors, a \emph{colored spanning graph} (CSG) is a graph such that for
each primary color, the vertices of that color induce a connected subgraph. The
\textsc{Min-CSG} problem asks for the minimum sum of edge lengths in a colored
spanning graph. We show that the problem is NP-hard for $k$ primary colors when
$k\ge 3$ and provide a $(2-\frac{1}{3+2\varrho})$-approximation algorithm for
$k=3$ that runs in polynomial time, where $\varrho$ is the Steiner ratio.
Further, we give a $O(n)$ time algorithm in the special case that the input
points are collinear and $k$ is constant.
