The Complexity of Recognizing Geometric Hypergraphs

TL;DR

This paper investigates the computational complexity of recognizing geometric hypergraphs, proving that recognizing hypergraphs represented by translates of convex sets like balls and ellipsoids is -complete, indicating high computational difficulty.

Contribution

It establishes -completeness of recognition problems for hypergraphs represented by translates of convex sets, extending known results to new geometric families.

Findings

Recognition is -complete for translates of balls and ellipsoids.

Recognition problems are equivalent to solving systems of polynomial equations.

These problems are computationally as hard as deciding real solutions to polynomial systems.

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 $F$ 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 $\exists\mathbb{R}$-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.