The Complexity of Recognizing Geometric Hypergraphs

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 $F$ of sets in ${\mathbb{R}}^d$, if there exist $P\subset {\mathbb{R}}^d$ and $S\subseteq F$ such that $(P,S)$ is a geometric representation of $H$. For a family F, we define RECOGNITION(F) as the problem to determine if a given hypergraph is representable by F. It is known that the RECOGNITION problem is $\mathrm{\exists }\mathbb{R}$-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 $\mathrm{\exists }\mathbb{R}$-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.