Realizability of combinatorial types of convex polyhedra over fields

Mnëv, Nikolai

doi:10.1007/bf02104991

Cited by 7 publications

(9 citation statements)

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“…This theorem is also a straightforward corollary of a deep topological result by Mnëv [14]. Shor's proof is more direct.…”

Section: Theorem 9 (Shor [21]) Simple Stretchability Is Np-hardmentioning

confidence: 62%

“…Mnëv's work [14,15] (see also [3,21]) actually proves that SIMPLE STRETCH-ABILITY is polynomially equivalent to the decision problem for the "existential theory of the reals" (ERT). ERT is the problem of deciding, given as input a set of polynomial equalities, inequalities and strict inequalities with integer coefficients, whether there exists a simultaneous real solution.…”

Section: Discussion and Further Workmentioning

confidence: 99%

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Sphere and Dot Product Representations of Graphs

Kang

Müller

2012

Discrete Comput Geom

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A graph G is a k-sphere graph if there are k-dimensional real vectors v 1 , . . . , v n such that ij ∈ E(G) if and only if the distance between v i and v j is at most 1. A graph G is a k-dot product graph if there are k-dimensional real vectors v 1 , . . . , v n such that ij ∈ E(G) if and only if the dot product of v i and v j is at least 1.By relating these two geometric graph constructions to oriented k-hyperplane arrangements, we prove that the problems of deciding, given a graph G, whether G is a k-sphere or a k-dot product graph are NP-hard for all k > 1. In the former case, this proves a conjecture of Breu and Kirkpatrick (Comput. Geom. 9:3-24, 1998). In the latter, this answers a question of Fiduccia et al. (Discrete Math. 181:113-138, 1998).Furthermore, motivated by the question of whether these two recognition problems are in NP, as well as by the implicit graph conjecture, we demonstrate that, for all k > 1, there exist k-sphere graphs and k-dot product graphs such that each representation in k-dimensional real vectors needs at least an exponential number of bits to be stored in the memory of a computer. On the other hand, we show that exponentially many bits are always enough. This resolves a question of Spinrad (Efficient Graph Representations, 2003).

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“…This theorem is also a straightforward corollary of a deep topological result by Mnëv [14]. Shor's proof is more direct.…”

Section: Theorem 9 (Shor [21]) Simple Stretchability Is Np-hardmentioning

confidence: 62%

Section: Discussion and Further Workmentioning

confidence: 99%

Sphere and Dot Product Representations of Graphs

Kang

Müller

2012

Discrete Comput Geom

View full text Add to dashboard Cite

show abstract

“…Mnëv's work [14,15] (see also [1,21]) moreover proves that SIMPLE STRETCHABILITY is polynomially equivalent to the decision problem for the "existential theory of the reals" (ERT). ERT is the problem of deciding, given as input a set of polynomial equalities, inequalities and strict inequalities with integer coefficients, whether there exists a simultaneous real solution.…”

Section: Discussion and Further Workmentioning

confidence: 98%

Sphere and dot product representations of graphs

Kang

Müller

2011

Proceedings of the Twenty-Seventh Annual Symposium on Computational Geometry

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A graph G is a k-sphere graph if there are k-dimensional real vectors v1, . . . , vn such that ij ∈ E(G) if and only if the distance between vi and vj is at most 1. A graph G is a k-dot product graph if there are k-dimensional real vectors v1, . . . , vn such that ij ∈ E(G) if and only if the dot product of vi and vj is at least 1.By relating these two geometric graph constructions to oriented k-hyperplane arrangements, we prove that the problems of deciding, given a graph G, whether G is a k-sphere or a k-dot product graph are NP-hard for all k > 1. In the former case, this proves a conjecture of Breu and Kirkpatrick [2]. In the latter, this answers a question of Fiduccia, Scheinerman, Trenk and Zito [5].Furthermore, motivated by the question of whether these two recognition problems are in NP, as well as by the implicit graph conjecture, we demonstrate that, for all k > 1, there exist k-sphere graphs and k-dot product graphs such that each representation in k-dimensional real vectors needs at least an exponential number of bits to be stored in the memory of a computer. On the other hand, we show that exponentially many bits are always enough. This resolves a question of Spinrad [22].

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“…If all sets o i j are singletons, the same problem is called Simple-Stretchability. Both variants of the problem are complete for ∃R [5,6].…”

Section: Stretchability and Combinatorial Descriptionsmentioning

confidence: 99%

“…One prototypical ∃R-complete problem that serves as the starting point of many reductions is Stretchability, which was among the first known ∃R-hard problems. The original hardness-proof is due to Mnëv [6], and it was restated in terms of ∃R by Matoušek [5].…”

Section: Introductionmentioning

confidence: 99%

Recognizing Generalized Transmission Graphs of Line Segments and Circular Sectors

Klost

Mulzer

2018

LATIN 2018: Theoretical Informatics

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Suppose we have an arrangement A of n geometric objects x1, . . . , xn ⊆ R 2 in the plane, with a distinguished point pi in each object xi. The generalized transmission graph of A has vertex set {x1, . . . , xn} and a directed edge xixj if and only if pj ∈ xi. Generalized transmission graphs provide a generalized model of the connectivity in networks of directional antennas.The complexity class ∃R contains all problems that can be reduced in polynomial time to an existential sentence of the form ∃x1, . . . , xn : φ(x1, . . . , xn), where x1, . . . , xn range over R and φ is a propositional formula with signature (+, −, ·, 0, 1). The class ∃R aims to capture the complexity of the existential theory of the reals. It lies between NP and PSPACE.Many geometric decision problems, such as recognition of disk graphs and of intersection graphs of lines, are complete for ∃R. Continuing this line of research, we show that the recognition problem of generalized transmission graphs of line segments and of circular sectors is hard for ∃R. As far as we know, this constitutes the first such result for a class of directed graphs. * Supported in part by ERC StG 757609.

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Realizability of combinatorial types of convex polyhedra over fields

Cited by 7 publications

References 0 publications

Sphere and Dot Product Representations of Graphs

Sphere and Dot Product Representations of Graphs

Sphere and dot product representations of graphs

Recognizing Generalized Transmission Graphs of Line Segments and Circular Sectors

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