Solving the Chromatic Cone Clustering Problem via Minimum Spanning Sphere

In this paper, we study the following Chromatic Cone Clustering (CCC) problem: Given n point-sets with each containing k points in the first quadrant of the d-dimensional space R, find k cones apexed at the origin such that each cone contains at least one distinct point (i.e., different from other cones) from every point-set and the total size of the k cones is minimized, where the size of a cone is the angle from any boundary ray to its center line. CCC is motivated by an important biological problem and finds applications in several other areas. For the CCC problem, we consider in this paper the cases that k is either 2 or some other constant number. Our approaches for solving the CCC problem relies on the solutions to the problem of computing a Minimum Spanning Sphere (MinSS) for point-sets. For the MinSS problem, we present two (1 + )-approximation algorithms based on core-sets and -net respectively. With these algorithms, we then show that the CCC problem admits (1 + )-approximation solutions for both cases. Our results use several interesting geometric techniques in high dimensional space, and are the first solutions to these problems.