Computational Aspects of the Colorful Carathéodory Theorem

Abstract

Let C1, ⋯ , Cd+1⊂ Rd be d+ 1 point sets, each containing the origin in its convex hull. We call these sets color classes, and we call a sequence p1, ⋯ , pd+1 with pi∈ Ci, for i= 1 , ⋯ , d+ 1 , a colorful choice. The colorful Carathéodory theorem guarantees the existence of a colorful choice that also contains the origin in its convex hull. The computational complexity of finding such a colorful choice (ColorfulCarathéodory) is unknown. This is particularly interesting in the light of polynomial-time reductions from several related problems, such as computing centerpoints, to ColorfulCarathéodory. We define a novel notion of approximation that is compatible with the polynomial-time reductions to ColorfulCarathéodory: a sequence that contains at most k points from each color class is called a k-colorful choice. We present an algorithm that for any fixed ε> 0 , outputs an ⌈ εd⌉ -colorful choice containing the origin in its convex hull in polynomial time. Furthermore, we consider a related problem of ColorfulCarathéodory: in the nearest colorful polytope problem (Ncp), we are given sets C1, ⋯ , Cn⊂ Rd that do not necessarily contain the origin in their convex hulls. The goal is to find a colorful choice whose convex hull minimizes the distance to the origin. We show that computing a local optimum for Ncp is PLS-complete, while computing a global optimum is NP-hard.