Multi-Colored Spanning Graphs

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 $n$ 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.