--- _id: '15168' abstract: - lang: eng text: 'A linearly ordered (LO) k-colouring of a hypergraph is a colouring of its vertices with colours 1, … , k such that each edge contains a unique maximal colour. Deciding whether an input hypergraph admits LO k-colouring with a fixed number of colours is NP-complete (and in the special case of graphs, LO colouring coincides with the usual graph colouring). Here, we investigate the complexity of approximating the "linearly ordered chromatic number" of a hypergraph. We prove that the following promise problem is NP-complete: Given a 3-uniform hypergraph, distinguish between the case that it is LO 3-colourable, and the case that it is not even LO 4-colourable. We prove this result by a combination of algebraic, topological, and combinatorial methods, building on and extending a topological approach for studying approximate graph colouring introduced by Krokhin, Opršal, Wrochna, and Živný (2023).' acknowledgement: "Marek Filakovský: This research was supported by Charles University (project PRIMUS/\r\n21/SCI/014), the Austrian Science Fund (FWF project P31312-N35), and MSCAfellow5_MUNI\r\n(CZ.02.01.01/00/22_010/0003229). Tamio-Vesa Nakajima: This research was funded by UKRI EP/X024431/1 and by a Clarendon Fund Scholarship. All data is provided in full in the results section of this paper. Jakub Opršal: This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement No 101034413. Uli Wagner: This research was supported by the Austrian Science Fund (FWF project P31312-N35)." alternative_title: - LIPIcs article_number: '34' article_processing_charge: No author: - first_name: Marek full_name: Filakovský, Marek id: 3E8AF77E-F248-11E8-B48F-1D18A9856A87 last_name: Filakovský - first_name: Tamio Vesa full_name: Nakajima, Tamio Vesa last_name: Nakajima - first_name: Jakub full_name: Opršal, Jakub id: ec596741-c539-11ec-b829-c79322a91242 last_name: Opršal orcid: 0000-0003-1245-3456 - first_name: Gianluca full_name: Tasinato, Gianluca id: 0433290C-AF8F-11E9-A4C7-F729E6697425 last_name: Tasinato - first_name: Uli full_name: Wagner, Uli id: 36690CA2-F248-11E8-B48F-1D18A9856A87 last_name: Wagner orcid: 0000-0002-1494-0568 citation: ama: 'Filakovský M, Nakajima TV, Opršal J, Tasinato G, Wagner U. Hardness of linearly ordered 4-colouring of 3-colourable 3-uniform hypergraphs. In: 41st International Symposium on Theoretical Aspects of Computer Science. Vol 289. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2024. doi:10.4230/LIPIcs.STACS.2024.34' apa: 'Filakovský, M., Nakajima, T. V., Opršal, J., Tasinato, G., & Wagner, U. (2024). Hardness of linearly ordered 4-colouring of 3-colourable 3-uniform hypergraphs. In 41st International Symposium on Theoretical Aspects of Computer Science (Vol. 289). Clermont-Ferrand, France: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. https://doi.org/10.4230/LIPIcs.STACS.2024.34' chicago: Filakovský, Marek, Tamio Vesa Nakajima, Jakub Opršal, Gianluca Tasinato, and Uli Wagner. “Hardness of Linearly Ordered 4-Colouring of 3-Colourable 3-Uniform Hypergraphs.” In 41st International Symposium on Theoretical Aspects of Computer Science, Vol. 289. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. https://doi.org/10.4230/LIPIcs.STACS.2024.34. ieee: M. Filakovský, T. V. Nakajima, J. Opršal, G. Tasinato, and U. Wagner, “Hardness of linearly ordered 4-colouring of 3-colourable 3-uniform hypergraphs,” in 41st International Symposium on Theoretical Aspects of Computer Science, Clermont-Ferrand, France, 2024, vol. 289. ista: 'Filakovský M, Nakajima TV, Opršal J, Tasinato G, Wagner U. 2024. Hardness of linearly ordered 4-colouring of 3-colourable 3-uniform hypergraphs. 41st International Symposium on Theoretical Aspects of Computer Science. STACS: Symposium on Theoretical Aspects of Computer Science, LIPIcs, vol. 289, 34.' mla: Filakovský, Marek, et al. “Hardness of Linearly Ordered 4-Colouring of 3-Colourable 3-Uniform Hypergraphs.” 41st International Symposium on Theoretical Aspects of Computer Science, vol. 289, 34, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024, doi:10.4230/LIPIcs.STACS.2024.34. short: M. Filakovský, T.V. Nakajima, J. Opršal, G. Tasinato, U. Wagner, in:, 41st International Symposium on Theoretical Aspects of Computer Science, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. conference: end_date: 2024-03-14 location: Clermont-Ferrand, France name: 'STACS: Symposium on Theoretical Aspects of Computer Science' start_date: 2024-03-12 date_created: 2024-03-24T23:00:59Z date_published: 2024-03-01T00:00:00Z date_updated: 2024-03-25T07:45:54Z day: '01' ddc: - '510' department: - _id: UlWa doi: 10.4230/LIPIcs.STACS.2024.34 ec_funded: 1 external_id: arxiv: - '2312.12981' file: - access_level: open_access checksum: 0524d4189fd1ed08989546511343edf3 content_type: application/pdf creator: dernst date_created: 2024-03-25T07:44:30Z date_updated: 2024-03-25T07:44:30Z file_id: '15175' file_name: 2024_LIPICs_Filakovsky.pdf file_size: 927290 relation: main_file success: 1 file_date_updated: 2024-03-25T07:44:30Z has_accepted_license: '1' intvolume: ' 289' language: - iso: eng license: https://creativecommons.org/licenses/by/4.0/ month: '03' oa: 1 oa_version: Published Version project: - _id: 26611F5C-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: P31312 name: Algorithms for Embeddings and Homotopy Theory - _id: fc2ed2f7-9c52-11eb-aca3-c01059dda49c call_identifier: H2020 grant_number: '101034413' name: 'IST-BRIDGE: International postdoctoral program' publication: 41st International Symposium on Theoretical Aspects of Computer Science publication_identifier: eissn: - 1868-8969 isbn: - '9783959773119' publication_status: published publisher: Schloss Dagstuhl - Leibniz-Zentrum für Informatik quality_controlled: '1' scopus_import: '1' status: public title: Hardness of linearly ordered 4-colouring of 3-colourable 3-uniform hypergraphs tmp: image: /images/cc_by.png legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0) short: CC BY (4.0) type: conference user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 289 year: '2024' ... --- _id: '7806' 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." article_processing_charge: No author: - first_name: Marek full_name: Filakovský, Marek id: 3E8AF77E-F248-11E8-B48F-1D18A9856A87 last_name: Filakovský - first_name: Uli full_name: Wagner, Uli id: 36690CA2-F248-11E8-B48F-1D18A9856A87 last_name: Wagner orcid: 0000-0002-1494-0568 - first_name: Stephan Y full_name: Zhechev, Stephan Y id: 3AA52972-F248-11E8-B48F-1D18A9856A87 last_name: Zhechev citation: ama: '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' apa: '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' chicago: 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. ieee: 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. 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.' mla: 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. short: M. Filakovský, U. Wagner, S.Y. Zhechev, in:, Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms, SIAM, 2020, pp. 767–785. conference: end_date: 2020-01-08 location: Salt Lake City, UT, United States name: 'SODA: Symposium on Discrete Algorithms' start_date: 2020-01-05 date_created: 2020-05-10T22:00:48Z date_published: 2020-01-01T00:00:00Z date_updated: 2021-01-12T08:15:38Z day: '01' department: - _id: UlWa doi: 10.1137/1.9781611975994.47 language: - iso: eng main_file_link: - open_access: '1' url: https://doi.org/10.1137/1.9781611975994.47 month: '01' oa: 1 oa_version: Published Version page: 767-785 project: - _id: 26611F5C-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: P31312 name: Algorithms for Embeddings and Homotopy Theory publication: Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms publication_identifier: isbn: - '9781611975994' publication_status: published publisher: SIAM quality_controlled: '1' scopus_import: 1 status: public title: Embeddability of simplicial complexes is undecidable type: conference user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 2020-January year: '2020' ... --- _id: '6563' abstract: - lang: eng text: "This paper presents two algorithms. The first decides the existence of a pointed homotopy between given simplicial maps \U0001D453,\U0001D454:\U0001D44B→\U0001D44C, and the second computes the group [\U0001D6F4\U0001D44B,\U0001D44C]∗ of pointed homotopy classes of maps from a suspension; in both cases, the target Y is assumed simply connected. More generally, these algorithms work relative to \U0001D434⊆\U0001D44B." article_processing_charge: No article_type: original author: - first_name: Marek full_name: Filakovský, Marek id: 3E8AF77E-F248-11E8-B48F-1D18A9856A87 last_name: Filakovský - first_name: Lukas full_name: Vokřínek, Lukas last_name: Vokřínek citation: ama: Filakovský M, Vokřínek L. Are two given maps homotopic? An algorithmic viewpoint. Foundations of Computational Mathematics. 2020;20:311-330. doi:10.1007/s10208-019-09419-x apa: Filakovský, M., & Vokřínek, L. (2020). Are two given maps homotopic? An algorithmic viewpoint. Foundations of Computational Mathematics. Springer Nature. https://doi.org/10.1007/s10208-019-09419-x chicago: Filakovský, Marek, and Lukas Vokřínek. “Are Two given Maps Homotopic? An Algorithmic Viewpoint.” Foundations of Computational Mathematics. Springer Nature, 2020. https://doi.org/10.1007/s10208-019-09419-x. ieee: M. Filakovský and L. Vokřínek, “Are two given maps homotopic? An algorithmic viewpoint,” Foundations of Computational Mathematics, vol. 20. Springer Nature, pp. 311–330, 2020. ista: Filakovský M, Vokřínek L. 2020. Are two given maps homotopic? An algorithmic viewpoint. Foundations of Computational Mathematics. 20, 311–330. mla: Filakovský, Marek, and Lukas Vokřínek. “Are Two given Maps Homotopic? An Algorithmic Viewpoint.” Foundations of Computational Mathematics, vol. 20, Springer Nature, 2020, pp. 311–30, doi:10.1007/s10208-019-09419-x. short: M. Filakovský, L. Vokřínek, Foundations of Computational Mathematics 20 (2020) 311–330. date_created: 2019-06-16T21:59:14Z date_published: 2020-04-01T00:00:00Z date_updated: 2023-08-17T13:50:44Z day: '01' department: - _id: UlWa doi: 10.1007/s10208-019-09419-x external_id: arxiv: - '1312.2337' isi: - '000522437400004' intvolume: ' 20' isi: 1 language: - iso: eng main_file_link: - open_access: '1' url: https://arxiv.org/abs/1312.2337 month: '04' oa: 1 oa_version: Preprint page: 311-330 project: - _id: 26611F5C-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: P31312 name: Algorithms for Embeddings and Homotopy Theory publication: Foundations of Computational Mathematics publication_identifier: eissn: - '16153383' issn: - '16153375' publication_status: published publisher: Springer Nature quality_controlled: '1' scopus_import: '1' status: public title: Are two given maps homotopic? An algorithmic viewpoint type: journal_article user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8 volume: 20 year: '2020' ... --- _id: '6774' abstract: - lang: eng text: "A central problem of algebraic topology is to understand the homotopy groups \ \U0001D70B\U0001D451(\U0001D44B) of a topological space X. For the computational version of the problem, it is well known that there is no algorithm to decide whether the fundamental group \U0001D70B1(\U0001D44B) of a given finite simplicial complex X is trivial. On the other hand, there are several algorithms that, given a finite simplicial complex X that is simply connected (i.e., with \U0001D70B1(\U0001D44B) \ trivial), compute the higher homotopy group \U0001D70B\U0001D451(\U0001D44B) \ for any given \U0001D451≥2 . However, these algorithms come with a caveat: They compute the isomorphism type of \U0001D70B\U0001D451(\U0001D44B) , \U0001D451≥2 \ as an abstract finitely generated abelian group given by generators and relations, but they work with very implicit representations of the elements of \U0001D70B\U0001D451(\U0001D44B) . Converting elements of this abstract group into explicit geometric maps from the d-dimensional sphere \U0001D446\U0001D451 to X has been one of the main unsolved problems in the emerging field of computational homotopy theory. Here we present an algorithm that, given a simply connected space X, computes \U0001D70B\U0001D451(\U0001D44B) \ and represents its elements as simplicial maps from a suitable triangulation of the d-sphere \U0001D446\U0001D451 to X. For fixed d, the algorithm runs in time exponential in size(\U0001D44B) , the number of simplices of X. Moreover, we prove that this is optimal: For every fixed \U0001D451≥2 , we construct a family of simply connected spaces X such that for any simplicial map representing a generator of \U0001D70B\U0001D451(\U0001D44B) , the size of the triangulation of \U0001D446\U0001D451 on which the map is defined, is exponential in size(\U0001D44B) ." article_type: original author: - first_name: Marek full_name: Filakovský, Marek id: 3E8AF77E-F248-11E8-B48F-1D18A9856A87 last_name: Filakovský - first_name: Peter full_name: Franek, Peter id: 473294AE-F248-11E8-B48F-1D18A9856A87 last_name: Franek orcid: 0000-0001-8878-8397 - first_name: Uli full_name: Wagner, Uli id: 36690CA2-F248-11E8-B48F-1D18A9856A87 last_name: Wagner orcid: 0000-0002-1494-0568 - first_name: Stephan Y full_name: Zhechev, Stephan Y id: 3AA52972-F248-11E8-B48F-1D18A9856A87 last_name: Zhechev citation: ama: Filakovský M, Franek P, Wagner U, Zhechev SY. Computing simplicial representatives of homotopy group elements. Journal of Applied and Computational Topology. 2018;2(3-4):177-231. doi:10.1007/s41468-018-0021-5 apa: Filakovský, M., Franek, P., Wagner, U., & Zhechev, S. Y. (2018). Computing simplicial representatives of homotopy group elements. Journal of Applied and Computational Topology. Springer. https://doi.org/10.1007/s41468-018-0021-5 chicago: Filakovský, Marek, Peter Franek, Uli Wagner, and Stephan Y Zhechev. “Computing Simplicial Representatives of Homotopy Group Elements.” Journal of Applied and Computational Topology. Springer, 2018. https://doi.org/10.1007/s41468-018-0021-5. ieee: M. Filakovský, P. Franek, U. Wagner, and S. Y. Zhechev, “Computing simplicial representatives of homotopy group elements,” Journal of Applied and Computational Topology, vol. 2, no. 3–4. Springer, pp. 177–231, 2018. ista: Filakovský M, Franek P, Wagner U, Zhechev SY. 2018. Computing simplicial representatives of homotopy group elements. Journal of Applied and Computational Topology. 2(3–4), 177–231. mla: Filakovský, Marek, et al. “Computing Simplicial Representatives of Homotopy Group Elements.” Journal of Applied and Computational Topology, vol. 2, no. 3–4, Springer, 2018, pp. 177–231, doi:10.1007/s41468-018-0021-5. short: M. Filakovský, P. Franek, U. Wagner, S.Y. Zhechev, Journal of Applied and Computational Topology 2 (2018) 177–231. date_created: 2019-08-08T06:47:40Z date_published: 2018-12-01T00:00:00Z date_updated: 2023-09-07T13:10:36Z day: '01' ddc: - '514' department: - _id: UlWa doi: 10.1007/s41468-018-0021-5 file: - access_level: open_access checksum: cf9e7fcd2a113dd4828774fc75cdb7e8 content_type: application/pdf creator: dernst date_created: 2019-08-08T06:55:21Z date_updated: 2020-07-14T12:47:40Z file_id: '6775' file_name: 2018_JourAppliedComputTopology_Filakovsky.pdf file_size: 1056278 relation: main_file file_date_updated: 2020-07-14T12:47:40Z has_accepted_license: '1' intvolume: ' 2' issue: 3-4 language: - iso: eng month: '12' oa: 1 oa_version: Published Version page: 177-231 project: - _id: 25F8B9BC-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: M01980 name: Robust invariants of Nonlinear Systems - _id: 3AC91DDA-15DF-11EA-824D-93A3E7B544D1 call_identifier: FWF name: FWF Open Access Fund publication: Journal of Applied and Computational Topology publication_identifier: eissn: - 2367-1734 issn: - 2367-1726 publication_status: published publisher: Springer quality_controlled: '1' related_material: record: - id: '6681' relation: dissertation_contains status: public status: public title: Computing simplicial representatives of homotopy group elements tmp: image: /images/cc_by.png legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0) short: CC BY (4.0) type: journal_article user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 2 year: '2018' ...