Embeddability of simplicial complexes is undecidable

M. Filakovský, U. Wagner, S.Y. Zhechev, in:, Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms, SIAM, 2020, pp. 767–785.


Conference Paper | Published | English
Department
Abstract
We consider the following decision problem EMBEDk→d in computational topology (where k ≤ d are fixed positive integers): Given a finite simplicial complex K of dimension k, does there exist a (piecewise-linear) embedding of K into ℝd? The special case EMBED1→2 is graph planarity, which is decidable in linear time, as shown by Hopcroft and Tarjan. In higher dimensions, EMBED2→3 and EMBED3→3 are known to be decidable (as well as NP-hard), and recent results of Čadek et al. in computational homotopy theory, in combination with the classical Haefliger–Weber theorem in geometric topology, imply that EMBEDk→d can be solved in polynomial time for any fixed pair (k, d) of dimensions in the so-called metastable range . Here, by contrast, we prove that EMBEDk→d is algorithmically undecidable for almost all pairs of dimensions outside the metastable range, namely for . This almost completely resolves the decidability vs. undecidability of EMBEDk→d in higher dimensions and establishes a sharp dichotomy between polynomial-time solvability and undecidability. Our result complements (and in a wide range of dimensions strengthens) earlier results of Matoušek, Tancer, and the second author, who showed that EMBEDk→d is undecidable for 4 ≤ k ϵ {d – 1, d}, and NP-hard for all remaining pairs (k, d) outside the metastable range and satisfying d ≥ 4.
Publishing Year
Date Published
2020-01-01
Proceedings Title
Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms
Volume
2020-January
Page
767-785
Conference
SODA: Symposium on Discrete Algorithms
Conference Location
Salt Lake City, UT, United States
Conference Date
2020-01-05 – 2020-01-08
IST-REx-ID

Cite this

Filakovský M, Wagner U, Zhechev SY. Embeddability of simplicial complexes is undecidable. In: Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms. Vol 2020-January. SIAM; 2020:767-785. doi:10.1137/1.9781611975994.47
Filakovský, M., Wagner, U., & Zhechev, S. Y. (2020). Embeddability of simplicial complexes is undecidable. In Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms (Vol. 2020–January, pp. 767–785). Salt Lake City, UT, United States: SIAM. https://doi.org/10.1137/1.9781611975994.47
Filakovský, Marek, Uli Wagner, and Stephan Y Zhechev. “Embeddability of Simplicial Complexes Is Undecidable.” In Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms, 2020–January:767–85. SIAM, 2020. https://doi.org/10.1137/1.9781611975994.47.
M. Filakovský, U. Wagner, and S. Y. Zhechev, “Embeddability of simplicial complexes is undecidable,” in Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms, Salt Lake City, UT, United States, 2020, vol. 2020–January, pp. 767–785.
Filakovský M, Wagner U, Zhechev SY. 2020. Embeddability of simplicial complexes is undecidable. Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms. SODA: Symposium on Discrete Algorithms vol. 2020–January. 767–785.
Filakovský, Marek, et al. “Embeddability of Simplicial Complexes Is Undecidable.” Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms, vol. 2020–January, SIAM, 2020, pp. 767–85, doi:10.1137/1.9781611975994.47.

Link(s) to Main File(s)
Access Level
OA Open Access

Export

Marked Publications

Open Data IST Research Explorer

Search this title in

Google Scholar
ISBN Search