{"title":"超图的线性有序着色","authors":"Tamio-Vesa Nakajima, Stanislav Živný","doi":"10.1145/3570909","DOIUrl":null,"url":null,"abstract":"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(\\sqrt [3]{n \\log \\log n / \\log n} \\) . Second, given an r-uniform hypergraph that admits an LO 2-colouring, we establish NP-hardness of finding an LO k-colouring for every constant uniformity r≥k+2. In fact, we determine relationships between polymorphism minions for all uniformities r≥ 3, which reveals a key difference between r< k+2 and r≥ k+2 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-colouring for LO ℓ-colourable r-uniform hypergraphs for 2 ≤ ℓ ≤ k and r ≥ k - ℓ + 4.","PeriodicalId":44045,"journal":{"name":"ACM Transactions on Computation Theory","volume":null,"pages":null},"PeriodicalIF":0.8000,"publicationDate":"2022-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"8","resultStr":"{\"title\":\"Linearly Ordered Colourings of Hypergraphs\",\"authors\":\"Tamio-Vesa Nakajima, Stanislav Živný\",\"doi\":\"10.1145/3570909\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"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(\\\\sqrt [3]{n \\\\log \\\\log n / \\\\log n} \\\\) . Second, given an r-uniform hypergraph that admits an LO 2-colouring, we establish NP-hardness of finding an LO k-colouring for every constant uniformity r≥k+2. In fact, we determine relationships between polymorphism minions for all uniformities r≥ 3, which reveals a key difference between r< k+2 and r≥ k+2 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-colouring for LO ℓ-colourable r-uniform hypergraphs for 2 ≤ ℓ ≤ k and r ≥ k - ℓ + 4.\",\"PeriodicalId\":44045,\"journal\":{\"name\":\"ACM Transactions on Computation Theory\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.8000,\"publicationDate\":\"2022-04-12\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"8\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"ACM Transactions on Computation Theory\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1145/3570909\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"COMPUTER SCIENCE, THEORY & METHODS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"ACM Transactions on Computation Theory","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1145/3570909","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"COMPUTER SCIENCE, THEORY & METHODS","Score":null,"Total":0}
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 \( k=O(\sqrt [3]{n \log \log n / \log n} \) . Second, given an r-uniform hypergraph that admits an LO 2-colouring, we establish NP-hardness of finding an LO k-colouring for every constant uniformity r≥k+2. In fact, we determine relationships between polymorphism minions for all uniformities r≥ 3, which reveals a key difference between r< k+2 and r≥ k+2 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-colouring for LO ℓ-colourable r-uniform hypergraphs for 2 ≤ ℓ ≤ k and r ≥ k - ℓ + 4.