Work

Geometric Embeddability of Complexes is ∃ℝ-complete

Year: 2024

Type: article

Abstract: We show that the decision problem of determining whether a given (abstract simplicial) k -complex has a geometric embedding in ℝ d is complete for the Existential Theory of the Reals for all d ≥ 3 and k ∈ { d -1, d } by reducing from pseudoline stretchability. Consequently, 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 geometrically embedding (abstract simplicial) complexes. This complements recent breakthroughs for the computational complexity of piece-wise linear embeddability [Matoušek, Sedgwick, Tancer, and Wagner, J. ACM 2018, and de Mesmay, Rieck, Sedgwick and Tancer, J. ACM 2020] and establishes connections to computational topology.

Source: Journal of the ACM

DOI: 10.1145/3707201

Authors: Mikkel Abrahamsen, Linda Kleist, Tillmann Miltzow

Institutions: University of Copenhagen, University of Potsdam, Utrecht University

Language: English

Topic: Computational Geometry and Mesh Generation

Subfield: Computer Graphics and Computer-Aided Design

Field: Computer Science

Domain: Physical Sciences

Open Access status: hybrid

Funders: Villum Fonden, Deutscher Akademischer Austauschdienst, Nederlandse Organisatie voor Wetenschappelijk Onderzoek, Bhabha Atomic Research Centre

Awards: Metallic clay: shape-matching orthopaedic implants, EAGER – Algorithmic R[EA]l Al[GE]braic Lowe[R] Bounds, Solving Continuous Problems with Guarantees, G4422840520, G7010378412