Complexity of Finding Nearest Colorful Polytopes

Let P1, . . . , Pd+1 ⊂ R be point sets whose convex hulls each contain the origin. Each set represents a color class. The Colorful Carathéodory theorem guarantees the existence of a colorful choice, i.e., a set that contains exactly one point from each color class, whose convex hull also contains the origin. The computational complexity of finding such a colorful choice is still unknown. We study a natural generalization of the problem: in the Nearest Colorful Polytope problem (NCP), we are given sets P1, . . . , Pn ⊂ R, and we would like to find a colorful choice whose convex hull minimizes the distance to the origin. We show that computing local optima of the NCP problem is PLS-complete, while computing a global optimum is NP-hard.