Multi-colored spanning graphs

Abstract

We study a problem motivated by sparse set visualization. Given n points in the plane, each labeled with one or more primary colors, a colored spanning graph (for short, CSG) is a graph in which the vertices of each primary 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≥3 and provide a (2−13+2ϱ)-approximation algorithm for k=3 that runs in polynomial time, where ϱ is the Steiner ratio. Further, we give an O(n) time algorithm in the special case that the given points are collinear and k is constant.