{"title":"Hardness of embedding simplicial complexes in ℝd","date_updated":"2021-01-12T06:57:27Z","author":[{"full_name":"Matoušek, Jiří","last_name":"Matoušek","first_name":"Jiří"},{"first_name":"Martin","id":"38AC689C-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-1191-6714","full_name":"Martin Tancer","last_name":"Tancer"},{"first_name":"Uli","full_name":"Uli Wagner","last_name":"Wagner","id":"36690CA2-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-1494-0568"}],"month":"01","_id":"2433","date_published":"2009-01-01T00:00:00Z","publication_status":"published","date_created":"2018-12-11T11:57:38Z","day":"01","citation":{"chicago":"Matoušek, Jiří, Martin Tancer, and Uli Wagner. “Hardness of Embedding Simplicial Complexes in ℝd,” 855–64. SIAM, 2009.","apa":"Matoušek, J., Tancer, M., &#38; Wagner, U. (2009). Hardness of embedding simplicial complexes in ℝd (pp. 855–864). Presented at the SODA: Symposium on Discrete Algorithms, SIAM.","ieee":"J. Matoušek, M. Tancer, and U. Wagner, “Hardness of embedding simplicial complexes in ℝd,” presented at the SODA: Symposium on Discrete Algorithms, 2009, pp. 855–864.","ama":"Matoušek J, Tancer M, Wagner U. Hardness of embedding simplicial complexes in ℝd. In: SIAM; 2009:855-864.","ista":"Matoušek J, Tancer M, Wagner U. 2009. Hardness of embedding simplicial complexes in ℝd. SODA: Symposium on Discrete Algorithms, 855–864.","short":"J. Matoušek, M. Tancer, U. Wagner, in:, SIAM, 2009, pp. 855–864.","mla":"Matoušek, Jiří, et al. <i>Hardness of Embedding Simplicial Complexes in ℝd</i>. SIAM, 2009, pp. 855–64."},"type":"conference","conference":{"name":"SODA: Symposium on Discrete Algorithms"},"page":"855 - 864","extern":1,"status":"public","quality_controlled":0,"publisher":"SIAM","publist_id":"4476","year":"2009","oa":1,"abstract":[{"text":"Let EMBEDk→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 ℝd? Known results easily imply polynomiality of EMBEDk→2 (k = 1, 2; the case k = 1, d = 2 is graph planarity) and of EMBEDk→2k for all k ≥ 3 (even if k is not considered fixed). 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 NP-hardness of EMBED2→4 and, more generally, of EMBEDk→d for all k, d with d ≥ 4 and d ≥ k ≥ (2d - 2)/3.","lang":"eng"}],"main_file_link":[{"url":"http://arxiv.org/abs/0807.0336","open_access":"1"}]}