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

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}. This implies that the problem is polynomial time equivalent to determining whether a polynomial equation system has a real root. Moreover, this implies NP-hardness and constitutes the first hardness results for the algorithmic problem of geometric embedding (abstract simplicial) complexes.

Topological Art in Simple Galleries

“…The proof can be found in Sect. 4. There are two observations that may also hold for other algorithmic problems.…”

“…, X n ]. As we will not use ∃R-completeness, except for pointing out its link to homotopyuniversality, we merely refer to some surveys and recent developments [4,5,15,22,33,35,38,39].…”

“…x 2 Fig. 4 An example of a union of hypercube faces in dimension 3 consisting of three maximal faces. The blue edge is a non-maximal face, as it is a subset of the green face automatically guards the whole polygon, thus the solution space is the same when considering the two versions of the problem.…”

“…Observation 4. 4 For every non-empty union of hypercube faces U there exists a facial disjunctive formula ϕ such that U = {x ∈ [0, 1] n : ϕ(x)}. Similarly, every facial disjunctive formula ϕ defines a union of hypercube faces {x ∈ [0, 1] n : ϕ(x)}.…”

The Complexity of the Hausdorff Distance

“…Famous examples from discrete geometry are the recognition of geometric structures, such as unit disk graphs [34], segment intersection graphs [33], visibility graphs [20], stretchability of pseudoline arrangements [37,49], and order type realizability [33]. Other ∃R-complete problems are related to graph drawing [32], Nash-Equilibria [15,28], geometric packing [6], the art gallery problem [3], convex covers [2], non-negative matrix factorization [48], polytopes [25,42], geometric embeddings of simplicial complexes [4], geometric linkage constructions [1], training neural networks [5], and continuous constraint satisfaction problems [35]. We refer the reader to the lecture notes by Matoušek [33] and surveys by Schaefer [45] and Cardinal [19] for more information on the complexity class ∃R.…”