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The Complexity of Recognizing Geometric Hypergraphs

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Graph Drawing and Network Visualization (GD 2023)


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.

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