# The Complexity of Recognizing Geometric Hypergraphs

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## Abstract

As set systems, hypergraphs are omnipresent and have various representations ranging from Euler and Venn diagrams to contact representations. In a geometric representation of a hypergraph $$H=(V,E)$$, each vertex $$v\in V$$ is associated with a point $$p_v\in \mathbb {R}^d$$ and each hyperedge $$e\in E$$ is associated with a connected set $$s_e\subset \mathbb {R}^d$$ such that $$\{p_v\mid v\in V\}\,\cap \, s_e=\{p_v\mid v\in e\}$$ for all $$e\in E$$. We say that a given hypergraph H is representable by some (infinite) family $$\mathcal {F}$$ of sets in $${\mathbb {R}^d}$$, if there exist $$P\subset \mathbb {R}^d$$ and $$S \subseteq \mathcal {F}$$ such that (PS) is a geometric representation of H. For a family $$\mathcal {F}$$, we define Recognition ($$\mathcal {F}$$) as the problem to determine if a given hypergraph is representable by $$\mathcal {F}$$. It is known that the Recognition problem is $$\exists \mathbb {R}$$-hard for halfspaces in $$\mathbb {R}^d$$. We study the families of translates of balls and ellipsoids in $$\mathbb {R}^d$$, as well as of other convex sets, and show that their Recognition problems are also $$\exists \mathbb {R}$$-complete. This means that these recognition problems are equivalent to deciding whether a multivariate system of polynomial equations with integer coefficients has a real solution.

The full version of this paper can be found on arXiv: 2302.13597v2.

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Bertschinger, D., El Maalouly, N., Kleist, L., Miltzow, T., Weber, S. (2023). The Complexity of Recognizing Geometric Hypergraphs. In: Bekos, M.A., Chimani, M. (eds) Graph Drawing and Network Visualization. GD 2023. Lecture Notes in Computer Science, vol 14465. Springer, Cham. https://doi.org/10.1007/978-3-031-49272-3_12

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