Schloss Dagstuhl - Leibniz-Zentrum für Informatik GmbH Schloss Dagstuhl - Leibniz-Zentrum für Informatik GmbH scholarly article en Nakajima, Tamio-Vesa; Živný, Stanislav https://www.dagstuhl.de/lipics License: Creative Commons Attribution 4.0 license (CC BY 4.0)
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URN: urn:nbn:de:0030-drops-164692
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Linearly Ordered Colourings of Hypergraphs

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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(√{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.

BibTeX - Entry

@InProceedings{nakajima_et_al:LIPIcs.ICALP.2022.128,
  author =	{Nakajima, Tamio-Vesa and \v{Z}ivn\'{y}, Stanislav},
  title =	{{Linearly Ordered Colourings of Hypergraphs}},
  booktitle =	{49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)},
  pages =	{128:1--128:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-235-8},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{229},
  editor =	{Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2022/16469},
  URN =		{urn:nbn:de:0030-drops-164692},
  doi =		{10.4230/LIPIcs.ICALP.2022.128},
  annote =	{Keywords: hypegraph colourings, promise constraint satisfaction, PCSP, polymorphisms, minions, algebraic approach}
}

Keywords: hypegraph colourings, promise constraint satisfaction, PCSP, polymorphisms, minions, algebraic approach
Seminar: 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)
Issue date: 2022
Date of publication: 28.06.2022


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