{"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","article_processing_charge":"No","language":[{"iso":"eng"}],"abstract":[{"lang":"eng","text":"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?\r\nThe 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 .\r\nHere, 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.\r\nOur 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."}],"publisher":"SIAM","conference":{"start_date":"2020-01-05","name":"SODA: Symposium on Discrete Algorithms","end_date":"2020-01-08","location":"Salt Lake City, UT, United States"},"citation":{"ieee":"M. Filakovský, U. Wagner, and S. Y. Zhechev, “Embeddability of simplicial complexes is undecidable,” in <i>Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms</i>, Salt Lake City, UT, United States, 2020, vol. 2020–January, pp. 767–785.","ama":"Filakovský M, Wagner U, Zhechev SY. Embeddability of simplicial complexes is undecidable. In: <i>Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms</i>. Vol 2020-January. SIAM; 2020:767-785. doi:<a href=\"https://doi.org/10.1137/1.9781611975994.47\">10.1137/1.9781611975994.47</a>","chicago":"Filakovský, Marek, Uli Wagner, and Stephan Y Zhechev. “Embeddability of Simplicial Complexes Is Undecidable.” In <i>Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms</i>, 2020–January:767–85. SIAM, 2020. <a href=\"https://doi.org/10.1137/1.9781611975994.47\">https://doi.org/10.1137/1.9781611975994.47</a>.","short":"M. Filakovský, U. Wagner, S.Y. Zhechev, in:, Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms, SIAM, 2020, pp. 767–785.","apa":"Filakovský, M., Wagner, U., &#38; Zhechev, S. Y. (2020). Embeddability of simplicial complexes is undecidable. In <i>Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms</i> (Vol. 2020–January, pp. 767–785). Salt Lake City, UT, United States: SIAM. <a href=\"https://doi.org/10.1137/1.9781611975994.47\">https://doi.org/10.1137/1.9781611975994.47</a>","mla":"Filakovský, Marek, et al. “Embeddability of Simplicial Complexes Is Undecidable.” <i>Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms</i>, vol. 2020–January, SIAM, 2020, pp. 767–85, doi:<a href=\"https://doi.org/10.1137/1.9781611975994.47\">10.1137/1.9781611975994.47</a>.","ista":"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."},"doi":"10.1137/1.9781611975994.47","scopus_import":1,"main_file_link":[{"open_access":"1","url":"https://doi.org/10.1137/1.9781611975994.47"}],"project":[{"name":"Algorithms for Embeddings and Homotopy Theory","call_identifier":"FWF","_id":"26611F5C-B435-11E9-9278-68D0E5697425","grant_number":"P31312"}],"date_published":"2020-01-01T00:00:00Z","publication_identifier":{"isbn":["9781611975994"]},"day":"01","status":"public","publication":"Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms","quality_controlled":"1","publication_status":"published","oa_version":"Published Version","oa":1,"author":[{"full_name":"Filakovský, Marek","last_name":"Filakovský","first_name":"Marek","id":"3E8AF77E-F248-11E8-B48F-1D18A9856A87"},{"orcid":"0000-0002-1494-0568","last_name":"Wagner","full_name":"Wagner, Uli","first_name":"Uli","id":"36690CA2-F248-11E8-B48F-1D18A9856A87"},{"last_name":"Zhechev","full_name":"Zhechev, Stephan Y","first_name":"Stephan Y","id":"3AA52972-F248-11E8-B48F-1D18A9856A87"}],"type":"conference","year":"2020","page":"767-785","date_updated":"2021-01-12T08:15:38Z","month":"01","volume":"2020-January","date_created":"2020-05-10T22:00:48Z","department":[{"_id":"UlWa"}],"_id":"7806","title":"Embeddability of simplicial complexes is undecidable"}