Embeddability in ℝ<sup>3</sup> is NP-hard

Mesmay, Arnaud de; Rieck, Yo'av; Sedgwick, Eric; Tancer, Martin

doi:10.1137/1.9781611975031.86

Cited by 9 publications

(8 citation statements)

References 32 publications

(44 reference statements)

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“…For every fixed d, k such that 3 ≤ d ≤ 3k 2 + 1 the algorithmic problem of recognizing linear embeddability of k-complexes into R d is NP hard. 24 Theorem 3.2.4 (NP -hardness; [MTW11,MRS+]). For every fixed d, k such that 3 ≤ d ≤ 3k 2 + 1 the algorithmic problem EMBED(k,d) of recognizing PL embeddability of k-complexes into R d is NP -hard.…”

Section: Algorithmic Recognition Of Realizablity Of Hypergraphsmentioning

confidence: 99%

Invariants of Graph Drawings in the Plane

Skopenkov

2020

Arnold Math J.

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We present a simplified exposition of some classical and modern results on graph drawings in the plane. These results are chosen so that they illustrate some spectacular recent higher-dimensional results on the border of topology and combinatorics. We define a Z 2 -valued self-intersection invariant (i.e. the van Kampen number) and its generalizations. We present elementary formulations and arguments, so we do not require any knowledge of algebraic topology. This survey is accessible to mathematicians not specialized in any of the areas discussed. It may serve as an introduction into algebraic topology motivated by algorithmic, combinatorial and geometric problems.

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Section: Algorithmic Recognition Of Realizablity Of Hypergraphsmentioning

confidence: 99%

Invariants of Graph Drawings in the Plane

Skopenkov

2020

Arnold Math J.

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“…Furthermore, due to the same argument, a polynomial-time algorithm for deciding whether a 2-dimensional simplicial complex embeds in R 3 would already imply a polynomial-time algorithm for our problem if we restrict ourselves to orientable surfaces. However, this problem is NP-hard [9], so the existence of such an algorithm is highly unlikely.…”

Section: Constructing An Embeddingmentioning

confidence: 99%

Recognizing Weak Embeddings of Graphs

Akitaya

Fulek

Tóth

2019

ACM Trans. Algorithms

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We present an efficient algorithm for a problem in the interface between clustering and graph embeddings. An embedding φ : G → M of a graph G into a 2-manifold M maps the vertices in V (G) to distinct points and the edges in E (G) to interior-disjoint Jordan arcs between the corresponding vertices. In applications in clustering, cartography, and visualization, nearby vertices and edges are often bundled to the same point or overlapping arcs due to data compression or low resolution. This raises the computational problem of deciding whether a given map φ : G → M comes from an embedding. A map φ : G → M is a weak embedding if it can be perturbed into an embedding ψ ε : G → M with φ − ψ ε < ε for every ε > 0, where. is the unform norm. A polynomial-time algorithm for recognizing weak embeddings has recently been found by Fulek and Kynčl. It reduces the problem to solving a system of linear equations over Z 2. It runs in O (n 2ω) ≤ O (n 4.75) time, where ω ∈ [2, 2.373) is the matrix multiplication exponent and n is the number of vertices and edges of G. We improve the running time to O (n log n). Our algorithm is also conceptually simpler: We perform a sequence of local operations that gradually "untangles" the image φ(G) into an embedding ψ (G) or reports that φ is not a weak embedding. It combines local constraints on the orientation of subgraphs directly, thereby eliminating the need for solving large systems of linear equations. CCS Concepts: • Mathematics of computing → Graphs and surfaces; • Theory of computation → Computational geometry;

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“…Furthermore, due to the same argument, a polynomial-time algorithm for deciding whether a 2-dimensional simplicial complex embeds in R 3 would already imply a polynomial-time algorithm for our problem if we restrict ourselves to orientable surfaces. However, this problem is NP-hard [7], so the existence of such an algorithm is highly unlikely.…”

Section: Atomicmentioning

confidence: 99%

Recognizing Weak Embeddings of Graphs

Akitaya

Fulek

Tóth

2018

Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms

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We present an efficient algorithm for a problem in the interface between clustering and graph embeddings. An embedding ϕ : G → M of a graph G into a 2-manifold M maps the vertices in V (G) to distinct points and the edges in E(G) to interior-disjoint Jordan arcs between the corresponding vertices. In applications in clustering, cartography, and visualization, nearby vertices and edges are often bundled to a common node or arc, due to data compression or low resolution. This raises the computational problem of deciding whether a given map ϕ : G → M comes from an embedding. A map ϕ : G → M is a weak embedding if it can be perturbed into an embedding ψ ε : G → M with ϕ − ψ ε < ε for every ε > 0.A polynomial-time algorithm for recognizing weak embeddings was recently found by Fulek and Kynčl [14], which reduces to solving a system of linear equations over Z 2 . It runs in O(n 2ω ) ≤ O(n 4.75 ) time, where ω ≈ 2.373 is the matrix multiplication exponent and n is the number of vertices and edges of G. We improve the running time to O(n log n). Our algorithm is also conceptually simpler than [14]: We perform a sequence of local operations that gradually "untangles" the image ϕ(G) into an embedding ψ(G), or reports that ϕ is not a weak embedding. It generalizes a recent technique developed for the case that G is a cycle and the embedding is a simple polygon [1], and combines local constraints on the orientation of subgraphs directly, thereby eliminating the need for solving large systems of linear equations.

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Embeddability in ℝ³ is NP-hard

Cited by 9 publications

References 32 publications

Invariants of Graph Drawings in the Plane

Invariants of Graph Drawings in the Plane

Recognizing Weak Embeddings of Graphs

Recognizing Weak Embeddings of Graphs

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Embeddability in ℝ3 is NP-hard

Cited by 9 publications

References 32 publications

Invariants of Graph Drawings in the Plane

Invariants of Graph Drawings in the Plane

Recognizing Weak Embeddings of Graphs

Recognizing Weak Embeddings of Graphs

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Product

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Embeddability in ℝ³ is NP-hard