The Complexity of Recognizing Geometric Hypergraphs

Abstract

As set systems, hypergraphs are omnipresent and have various representations. In a geometric representation of a hypergraph H=(V,E), each vertex v∈V is a associated with a point pv∈Rd and each hyperedge e∈E is associated with a connected set se⊂Rd such that {pv∣v∈V}∩se={pv∣v∈e} for all e∈E. We say that a given hypergraph H is representable by some (infinite) family F of sets in Rd, if there exist P⊂Rd and S⊆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 ER-hard for halfspaces in Rd. We study the families of balls and ellipsoids in Rd, as well as other convex sets, and show that their RECOGNITION problems are also ER-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.