Abstract
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.
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Notes
Recall that the Real-Ram is the standard model of computational geometry where memory cells store arbitrary real numbers and operations on them can be performed at unit cost. We emphasize that there is no known algorithm for solving linear programs that needs a polynomial number of steps on the Real-Ram. Thus, our algorithms avoid the use of LPs.
On the Real-Ram, we need not worry about the bit-complexity of Gaussian elimination.
Recall that A and B are relations between problem instances and candidate solutions.
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Acknowledgements
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.
Funding
WM was supported in part by DFG Grants MU 3501/1 and MU 3501/2 and ERC StG 757609. YS was supported by the Deutsche Forschungsgemeinschaft within the research training group ‘Methods for Discrete Structures’ (GRK 1408) and by GIF Grant 1161.
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Editor in Charge: János Pach
A preliminary version of this article appeared as W. Mulzer and Y. Stein. Computational Aspects of the Colorful Carathéodory Theorem. Proc. 31st SoCG, 2015.
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Mulzer, W., Stein, Y. Computational Aspects of the Colorful Carathéodory Theorem. Discrete Comput Geom 60, 720–755 (2018). https://doi.org/10.1007/s00454-018-9979-y
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DOI: https://doi.org/10.1007/s00454-018-9979-y