Hardness of embedding simplicial complexes in $\mathbb{R}^d$

Abstract

Abstract. Let EMBED k→d be the following algorithmic problem: Given a finite simplicial complex K of dimension at most k, does there exist a (piecewise linear) embedding of K into R d ?Known results easily imply the polynomiality of EMBED k→2 (k = 1, 2; the case k = 1, d = 2 is graph planarity) and of EMBED k→2k for all k ≥ 3.We show that the celebrated result of Novikov on the algorithmic unsolvability of recognizing the 5-sphere implies that EMBED d→d and EMBED (d−1)→d are undecidable for each d ≥ 5. Our main result is the NP-hardness of EMBED 2→4 and, more generally, of EMBED k→d for all k, d with d ≥ 4 and d ≥ k ≥ (2d − 2)/3. These dimensions fall outside the metastable range of a theorem of Haefliger and Weber, which characterizes embeddability using the deleted product obstruction. Our reductions are based on examples, due to Segal, Spież, Freedman, Krushkal, Teichner, and Skopenkov, showing that outside the metastable range the deleted product obstruction is not sufficient to characterize embeddability.

Invariants of Graph Drawings in the Plane

“…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.…”

“…For history, more motivation, references to classical results, more proofs, related problems and generalizations see surveys [BBZ,Zi11,Sk16], [BZ16, §1- §3] (to §3.1) and [Sk08,Sk14], [MTW11,§1], [Sk, §5 'Realizability of hypergraphs'] (to §3.2). Discussion of those related problems and generalizations is outside purposes of this survey.…”

Minors in random and expanding hypergraphs

“…Which definition is "the right one" depends on the context and on the aspects of the theory of graph minors that one wishes to generalize. We stress right away that we are not aiming for a characterization of embeddable complexes in terms of finitely many forbidden minors (indeed, recent NPhardness results [33] regarding embeddability of simplicial complexes are an indication that such a goal might be too ambitious, especially for embeddings of 2-dimensional complexes into R 4 ). Instead, our principal motivation are topological extremal problems for sparse simplicial complexes and the corresponding threshold questions for random complexes (we think of a k-dimensional complex as sparse if…”

“…For further background on embeddings of simplicial complexes, see the survey [43]. For an introduction especially geared towards combinatorialists and computer scientists and and with an emphasis on algorithmic aspects of embeddability, see also [33].…”

“…It is important to remark, however, that ultimately, the van Kampen obstruction boils down to a very concrete system of linear equations given by the simplicial complex, see, e.g. [33] for an elementary exposition.…”