Computational Aspects of the Colorful Carathéodory Theorem



Let \(C_1,\dots ,C_{d+1}\subset \mathbb {R}^d\) be \(d+1\) point sets, each containing the origin in its convex hull. We call these sets color classes, and we call a sequence \(p_1, \dots , p_{d+1}\) with \(p_i \in C_i\), for \(i = 1, \dots , d+1\), 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 \(\varepsilon > 0\), outputs an \(\lceil \varepsilon d\rceil \)-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 \(C_1,\dots ,C_n\subset \mathbb {R}^d\) 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.


Colorful Carathéodory Theorem PLS Approximation 

Mathematics Subject Classification

68Q25 68W25 68W40 



We would like to thank Frédéric Meunier and Pauline Sarrabezolles for interesting discussions on the colorful Carathéodory problem and for hosting us during multiple research stays at the École Nationale des Ponts et Chaussées. Furthermore, we would like to thank the anonymous reviewers for their detailed reading of our paper and for their helpful and encouraging comments on previous versions.


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Authors and Affiliations

  1. 1.Institut für InformatikFreie Universität BerlinBerlinGermany

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