Multi-Colored Spanning Graphs
Hugo A. Akitaya, Maarten L\"offler, Csaba D. T\'oth

TL;DR
This paper investigates the computational complexity of the Min-CSG problem, proving NP-hardness for three or more colors, and provides approximation and specialized algorithms for certain cases.
Contribution
It establishes NP-hardness for Min-CSG with three or more colors and introduces a new approximation algorithm for the case of three colors.
Findings
NP-hardness for k≥3 primary colors
A polynomial-time approximation algorithm for k=3
An O(n) algorithm for collinear points with constant k
Abstract
We study a problem proposed by Hurtado et al. (2016) motivated by sparse set visualization. Given 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 primary colors when and provide a -approximation algorithm for that runs in polynomial time, where is the Steiner ratio. Further, we give a time algorithm in the special case that the input points are collinear and is constant.
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