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 − 1 3 + 2 ϱ ) -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.

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