Using the algebraic approach to promise constraint satisfaction problems, we establish complexity classifications of three natural variants of hypergraph colourings: standard nonmonochromatic colourings, conflict-free colourings, and linearly-ordered colourings. Firstly, we show that finding an 𝓁-colouring of a k-colourable r-uniform hypergraph is NP-hard for all constant 2 ≤ k ≤ 𝓁 and r ≥ 3. This provides a shorter proof of a celebrated result by Dinur et al. [FOCS'02/Combinatorica'05]. Secondly, we show that finding an 𝓁-conflict-free colouring of an r-uniform hypergraph that admits a k-conflict-free colouring is NP-hard for all constant 2 ≤ k ≤ 𝓁 and r ≥ 4, except for r = 4 and k = 2 (and any 𝓁); this case is solvable in polynomial time. The case of r = 3 is the standard nonmonochromatic colouring, and the case of r = 2 is the notoriously difficult open problem of approximate graph colouring. Thirdly, we show that finding an 𝓁-linearly-ordered colouring of an r-uniform hypergraph that admits a k-linearly-ordered colouring is NP-hard for all constant 3 ≤ k ≤ 𝓁 and r ≥ 4, thus improving on the results of Nakajima and Živný [ICALP'22/ACM TocT'23].
@InProceedings{nakajima_et_al:LIPIcs.ICALP.2025.169, author = {Nakajima, Tamio-Vesa and Verwimp, Zephyr and Wrochna, Marcin and \v{Z}ivn\'{y}, Stanislav}, title = {{Complexity of Approximate Conflict-Free, Linearly-Ordered, and Nonmonochromatic Hypergraph Colourings}}, booktitle = {52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)}, pages = {169:1--169:10}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-372-0}, ISSN = {1868-8969}, year = {2025}, volume = {334}, editor = {Censor-Hillel, Keren and Grandoni, Fabrizio and Ouaknine, Jo\"{e}l and Puppis, Gabriele}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2025.169}, URN = {urn:nbn:de:0030-drops-235460}, doi = {10.4230/LIPIcs.ICALP.2025.169}, annote = {Keywords: hypergraph colourings, conflict-free colourings, unique-maximum colourings, linearly-ordered colourings} }
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