Linearly ordered colourings of hypergraphs
A linearly ordered (LO) k-colouring of an r-uniform hypergraph assigns an integer from {1, …, k} to every vertex so that, in every edge, the (multi)set of colours has a unique maximum. Equivalently, for r = 3, if two vertices in an edge are assigned the same colour, then the third vertex is assigned...
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| Format: | Conference item |
| Language: | English |
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Leibniz International Proceedings in Informatics, LIPIcs
2022
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| _version_ | 1797107289268682752 |
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| author | Nakajima, T-V Zivny, S |
| author_facet | Nakajima, T-V Zivny, S |
| author_sort | Nakajima, T-V |
| collection | OXFORD |
| description | A linearly ordered (LO) k-colouring of an r-uniform hypergraph assigns an integer from {1, …, k} to every vertex so that, in every edge, the (multi)set of colours has a unique maximum. Equivalently, for r = 3, if two vertices in an edge are assigned the same colour, then the third vertex is assigned a larger colour (as opposed to a different colour, as in classic non-monochromatic colouring). Barto, Battistelli, and Berg [STACS'21] studied LO colourings on 3-uniform hypergraphs in the context of promise constraint satisfaction problems (PCSPs). We show two results.
First, given a 3-uniform hypergraph that admits an LO 2-colouring, one can find in polynomial time an LO k-colouring with k = O(√{nlog log n}/log n), where n is the number of vertices of the input hypergraph. This is established by building on ideas from algorithms designed for approximate graph colourings.
Second, given an r-uniform hypergraph that admits an LO 2-colouring, we establish NP-hardness of finding an LO 3-colouring for every constant uniformity r ≥ 5. In fact, we determine the precise relationship of polymorphism minions for all uniformities r ≥ 3, which reveals a key difference between r = 3,4 and r ≥ 5 and which may be of independent interest. Using the algebraic approach to PCSPs, we actually show a more general result establishing NP-hardness of finding an LO (k+1)-colouring for LO k-colourable r-uniform hypergraphs for k ≥ 2 and r ≥ 5. |
| first_indexed | 2024-03-07T07:13:59Z |
| format | Conference item |
| id | oxford-uuid:86b24d89-ef66-45f8-8a80-8f7507eb601d |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T07:13:59Z |
| publishDate | 2022 |
| publisher | Leibniz International Proceedings in Informatics, LIPIcs |
| record_format | dspace |
| spelling | oxford-uuid:86b24d89-ef66-45f8-8a80-8f7507eb601d2022-07-14T10:00:43ZLinearly ordered colourings of hypergraphsConference itemhttp://purl.org/coar/resource_type/c_5794uuid:86b24d89-ef66-45f8-8a80-8f7507eb601dEnglishSymplectic ElementsLeibniz International Proceedings in Informatics, LIPIcs2022Nakajima, T-VZivny, SA linearly ordered (LO) k-colouring of an r-uniform hypergraph assigns an integer from {1, …, k} to every vertex so that, in every edge, the (multi)set of colours has a unique maximum. Equivalently, for r = 3, if two vertices in an edge are assigned the same colour, then the third vertex is assigned a larger colour (as opposed to a different colour, as in classic non-monochromatic colouring). Barto, Battistelli, and Berg [STACS'21] studied LO colourings on 3-uniform hypergraphs in the context of promise constraint satisfaction problems (PCSPs). We show two results. First, given a 3-uniform hypergraph that admits an LO 2-colouring, one can find in polynomial time an LO k-colouring with k = O(√{nlog log n}/log n), where n is the number of vertices of the input hypergraph. This is established by building on ideas from algorithms designed for approximate graph colourings. Second, given an r-uniform hypergraph that admits an LO 2-colouring, we establish NP-hardness of finding an LO 3-colouring for every constant uniformity r ≥ 5. In fact, we determine the precise relationship of polymorphism minions for all uniformities r ≥ 3, which reveals a key difference between r = 3,4 and r ≥ 5 and which may be of independent interest. Using the algebraic approach to PCSPs, we actually show a more general result establishing NP-hardness of finding an LO (k+1)-colouring for LO k-colourable r-uniform hypergraphs for k ≥ 2 and r ≥ 5. |
| spellingShingle | Nakajima, T-V Zivny, S Linearly ordered colourings of hypergraphs |
| title | Linearly ordered colourings of hypergraphs |
| title_full | Linearly ordered colourings of hypergraphs |
| title_fullStr | Linearly ordered colourings of hypergraphs |
| title_full_unstemmed | Linearly ordered colourings of hypergraphs |
| title_short | Linearly ordered colourings of hypergraphs |
| title_sort | linearly ordered colourings of hypergraphs |
| work_keys_str_mv | AT nakajimatv linearlyorderedcolouringsofhypergraphs AT zivnys linearlyorderedcolouringsofhypergraphs |