Geometric Embeddability of Complexes is $\exists \mathbb R$-complete

We show that the decision problem of determining whether a given (abstract simplicial) k-complex has a geometric embedding in R is complete for the Existential Theory of the Reals for all d ≥ 3 and k ∈ {d− 1, d}. This implies that the problem is polynomial time equivalent to determining whether a polynomial equation system has a real solution. Moreover, this implies NP-hardness and constitutes the first hardness result for the algorithmic problem of geometric embedding (abstract simplicial) complexes. Figure 1: Illustration of different embeddings of a complex; figure taken from Bing [8, Annals of Mathematics 1959]. ar X iv :2 10 8. 02 58 5v 2 [ cs .C C ] 5 N ov 2 02 1

[1]  Erik D. Demaine,et al.  Who Needs Crossings? Hardness of Plane Graph Rigidity , 2016, Symposium on Computational Geometry.

[2]  Debajyoti Mondal,et al.  The Complexity of Drawing a Graph in a Polygonal Region , 2018, GD.

[3]  J. Dieudonne,et al.  A History of Algebraic and Differential Topology, 1900 - 1960 , 1989 .

[4]  Tillmann Miltzow,et al.  A Universality Theorem for Nested Polytopes , 2019, ArXiv.

[5]  Uli Wagner,et al.  Embeddability of Simplicial Complexes is Undecidable , 2020, SODA.

[6]  Jirí Matousek,et al.  Embeddability in the 3-Sphere Is Decidable , 2014, J. ACM.

[7]  Tillmann Miltzow,et al.  The art gallery problem is ∃ ℝ-complete , 2018, STOC.

[8]  Tillmann Miltzow,et al.  Framework for ER-Completeness of Two-Dimensional Packing Problems , 2020, 2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS).

[9]  Jirí Matousek,et al.  Computing All Maps into a Sphere , 2011, J. ACM.

[10]  Yaroslav Shitov,et al.  A universality theorem for nonnegative matrix factorizations , 2016, 1606.09068.

[11]  Mikhail Skopenkov,et al.  Embedding products of graphs into Euclidean spaces , 2008, 0808.1199.

[12]  E. R. Kampen.  Komplexe in euklidischen Räumen , 1933 .

[13]  Jorge L. Ramírez Alfonsín,et al.  Knots and links in spatial graphs: A survey , 2005, Discret. Math..

[14]  B. Grünbaum.  Imbeddings of simplicial complexes , 1969 .

[15]  Arnold Shapiro,et al.  Obstructions to the Imbedding of a Complex in a Euclidean Space.: I. The First Obstruction , 1957 .

[16]  Tillmann Miltzow,et al.  Smoothing the gap between NP and ER , 2019, 2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS).

[17]  Jean Cardinal,et al.  Recognition and Complexity of Point Visibility Graphs , 2015, Discret. Comput. Geom..

[18]  Mikkel Abrahamsen,et al.  Covering Polygons is Even Harder , 2021, 2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS).

[19]  Tillmann Miltzow,et al.  On Classifying Continuous Constraint Satisfaction problems , 2021, ArXiv.

[20]  Salman Parsa,et al.  On the Links of Vertices in Simplicial d-Complexes Embeddable in the Euclidean 2d-Space , 2017, Discrete & Computational Geometry.

[21]  Arkadiy Skopenkov,et al.  Realizability of hypergraphs and Ramsey link theory , 2014 .

[22]  A. Skopenkov.  Extendability of simplicial maps is undecidable , 2020, ArXiv.

[23]  Jirí Matousek,et al.  Polynomial-Time Homology for Simplicial Eilenberg–MacLane Spaces , 2012, Found. Comput. Math..

[24]  Mikkel Abrahamsen.  Training Neural Networks is ∃R-complete , 2021 .

[25]  Jirí Matousek,et al.  Extendability of Continuous Maps Is Undecidable , 2013, Discret. Comput. Geom..

[26]  Brian R. Ummel.  The product of nonplanar complexes does not imbed in 4-space , 1978 .

[27]  Jirí Matousek,et al.  Hardness of embedding simplicial complexes in Rd , 2009, SODA.

[28]  Lukas Fins hiy,et al.  Generation of Oriented Matroids , 2002 .

[29]  Gunter M. Ziegler,et al.  Realization spaces of 4-polytopes are universal , 1995 .

[30]  Ulrich Brehm,et al.  A nonpolyhedral triangulated Möbius strip , 1983 .

[31]  Michael H. Freedman,et al.  Van Kampen’s embedding Obstruction is incomplete for $2$-Complexes in $\rz^{4}$ , 1994 .

[32]  Jürgen Richter-Gebert.  Realization Spaces of Polytopes , 1996 .

[33]  Robert E. Tarjan,et al.  Efficient Planarity Testing , 1974, JACM.

[34]  Ruta Mehta,et al.  ∃R-Completeness for Decision Versions of Multi-Player (Symmetric) Nash Equilibria , 2018, ACM Trans. Economics and Comput..

[35]  Jirí Matousek,et al.  Polynomial-Time Computation of Homotopy Groups and Postnikov Systems in Fixed Dimension , 2012, SIAM J. Comput..

[36]  Arkadiy Skopenkov,et al.  Invariants of Graph Drawings in the Plane , 2018, ArXiv.

[37]  Arkadiy Skopenkov,et al.  A Short Exposition of Salman Parsa’s Theorems on Intrinsic Linking and Non-realizability , 2018, Discret. Comput. Geom..

[38]  R. Bing.  An Alternative Proof that 3-Manifolds Can be Triangulated , 1959 .

[39]  J. L. Bryant.  Approximating embeddings of polyhedra in codimension three , 1972 .

[40]  D. Timmreck.  Realization Problems for Point Configurations and Polyhedral Surfaces , 2015 .

[41]  Marcus Schaefer,et al.  Complexity of Some Geometric and Topological Problems , 2009, GD.

[42]  Jeremy Gray,et al.  A history of algebraic and differential topology, 1900–1960: By Jean Dieudonné. Boston and Basel (Birkhäuser). 1989. xxi + 648 pp , 1991 .

[43]  Patrice Ossona de Mendez,et al.  Realization of Posets , 2002, J. Graph Algorithms Appl..

[44]  Colin McDiarmid,et al.  Integer realizations of disk and segment graphs , 2011, J. Comb. Theory, Ser. B.

[45]  J. Carmesin.  Embedding simply connected 2-complexes in 3-space , 2017, 2309.15504.

[46]  N. Mnev.  The universality theorems on the classification problem of configuration varieties and convex polytopes varieties , 1988 .

[47]  Martin Tancer,et al.  Hardness of almost embedding simplicial complexes in Rd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^d$$ , 2018, Discrete & Computational Geometry.

[48]  B. Grünbaum.  Polytopes, graphs, and complexes , 1970 .

[49]  Peter W. Shor,et al.  Stretchability of Pseudolines is NP-Hard , 1990, Applied Geometry And Discrete Mathematics.

[50]  Vittorio Bilò,et al.  A Catalog of EXISTS-R-Complete Decision Problems About Nash Equilibria in Multi-Player Games , 2016, STACS.

[51]  Egon Schulte,et al.  Polyhedral Maps , 2004, Handbook of Discrete and Computational Geometry, 2nd Ed..

[52]  Jean Cardinal,et al.  Computational Geometry Column 62 , 2015, SIGACT News.

[53]  Isabella Novik,et al.  A Note on Geometric Embeddings of Simplicial Complexes in a Euclidean Space , 2000, Discret. Comput. Geom..

[54]  A. Gundert,et al.  On the Complexity of Embeddable Simplicial Complexes , 2018, 1812.08447.

[55]  D. Timmreck.  Necessary Conditions for Geometric Realizability of Simplicial Complexes , 2007, 0705.1912.

[56]  John F. Canny,et al.  Some algebraic and geometric computations in PSPACE , 1988, STOC '88.

[57]  Marek Krcál,et al.  Algorithmic Solvability of the Lifting-Extension Problem , 2013, Discrete & Computational Geometry.