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The Complexity of Recognizing Geometric Hypergraphs
Authors
Daniel Bertschinger
Linda Kleist
+3Β more
Nicolas El Maalouly
Tillmann Miltzow
Simon Weber
Publication date
17 August 2023
Publisher
View
on
arXiv
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
)
H=(V,E)
H
=
(
V
,
E
)
, each vertex
v
β
V
v\in V
v
β
V
is associated with a point
p
v
β
R
d
p_v\in \mathbb{R}^d
p
v
β
β
R
d
and each hyperedge
e
β
E
e\in E
e
β
E
is associated with a connected set
s
e
β
R
d
s_e\subset \mathbb{R}^d
s
e
β
β
R
d
such that
{
p
v
β£
v
β
V
}
β©
s
e
=
{
p
v
β£
v
β
e
}
\{p_v\mid v\in V\}\cap s_e=\{p_v\mid v\in e\}
{
p
v
β
β£
v
β
V
}
β©
s
e
β
=
{
p
v
β
β£
v
β
e
}
for all
e
β
E
e\in E
e
β
E
. We say that a given hypergraph
H
H
H
is representable by some (infinite) family
F
F
F
of sets in
R
d
\mathbb{R}^d
R
d
, if there exist
P
β
R
d
P\subset \mathbb{R}^d
P
β
R
d
and
S
β
F
S \subseteq F
S
β
F
such that
(
P
,
S
)
(P,S)
(
P
,
S
)
is a geometric representation of
H
H
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
β
R
\exists\mathbb{R}
β
R
-hard for halfspaces in
R
d
\mathbb{R}^d
R
d
. We study the families of translates of balls and ellipsoids in
R
d
\mathbb{R}^d
R
d
, as well as of other convex sets, and show that their RECOGNITION problems are also
β
R
\exists\mathbb{R}
β
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.Comment: Appears in the Proceedings of the 31st International Symposium on Graph Drawing and Network Visualization (GD 2023) 17 pages, 11 figure
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Last time updated on 18/03/2023