Multi-colored Spanning Graphs

Abstract

We study a problem proposed by Hurtado et al. [10] motivated by sparse set visualization. Given n points in the plane, each labeled with one or more primary colors, a colored spanning graph (CSG) is a graph such that for each primary color, the vertices of that color induce a connected subgraph. The 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.