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Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ICALP.2022.128
URN: urn:nbn:de:0030-drops-164692
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2022/16469/
Nakajima, Tamio-Vesa ;
Živný, Stanislav
Linearly Ordered Colourings of Hypergraphs
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}
}
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Keywords: |
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hypegraph colourings, promise constraint satisfaction, PCSP, polymorphisms, minions, algebraic approach |
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Collection: |
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49th International Colloquium on Automata, Languages, and Programming (ICALP 2022) |
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Issue Date: |
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2022 |
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Date of publication: |
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28.06.2022 |
