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Complexity of Approximate Conflict-Free, Linearly-Ordered, and Nonmonochromatic Hypergraph Colourings

Authors Tamio-Vesa Nakajima , Zephyr Verwimp , Marcin Wrochna , Stanislav Živný

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Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
  • Theory of computation → Problems, reductions and completeness
  • Theory of computation → Constraint and logic programming
Keywords
  • hypergraph colourings
  • conflict-free colourings
  • unique-maximum colourings
  • linearly-ordered colourings

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Abstract

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].

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Tamio-Vesa Nakajima, Zephyr Verwimp, Marcin Wrochna, and Stanislav Živný. Complexity of Approximate Conflict-Free, Linearly-Ordered, and Nonmonochromatic Hypergraph Colourings. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 169:1-169:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ICALP.2025.169

Author Details

Tamio-Vesa Nakajima
  • Department of Computer Science, University of Oxford, UK
Zephyr Verwimp
  • Department of Computer Science, University of Oxford, UK
Marcin Wrochna
  • Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Poland
Stanislav Živný
  • Department of Computer Science, University of Oxford, UK

Funding

  • Nakajima, Tamio-Vesa: UKRI EP/X024431/1, Clarendon Fund Scholarship.
  • Verwimp, Zephyr: UKRI EP/X024431/1.
  • Živný, Stanislav: UKRI EP/X024431/1.

Acknowledgements

We thank the anonymous reviewers for their feedback.

References

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