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A Logarithmic Approximation of Linearly-Ordered Colourings

Authors Johan Håstad , Björn Martinsson , Tamio-Vesa Nakajima , Stanislav Živný

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ACM Subject Classification
  • Theory of computation → Approximation algorithms analysis
  • Theory of computation → Constraint and logic programming
Keywords
  • Linear ordered colouring
  • Hypergraph
  • Approximation
  • Promise Constraint Satisfaction Problems

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Abstract

A linearly ordered (LO) k-colouring of a hypergraph assigns to each vertex a colour from the set {0,1,…,k-1} in such a way that each hyperedge has a unique maximum element. Barto, Batistelli, and Berg conjectured that it is NP-hard to find an LO k-colouring of an LO 2-colourable 3-uniform hypergraph for any constant k ≥ 2 [STACS'21] but even the case k = 3 is still open. Nakajima and Živný gave polynomial-time algorithms for finding, given an LO 2-colourable 3-uniform hypergraph, an LO colouring with O^*(√n) colours [ICALP'22] and an LO colouring with O^*(n^(1/3)) colours [ACM ToCT'23]. Very recently, Louis, Newman, and Ray gave an SDP-based algorithm with O^*(n^(1/5)) colours. We present two simple polynomial-time algorithms that find an LO colouring with O(log₂(n)) colours, which is an exponential improvement.

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Johan Håstad, Björn Martinsson, Tamio-Vesa Nakajima, and Stanislav Živný. A Logarithmic Approximation of Linearly-Ordered Colourings. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 7:1-7:6, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2024.7

Author Details

Johan Håstad
  • KTH Royal Institute of Technology, Stockholm, Sweden
Björn Martinsson
  • KTH Royal Institute of Technology, Stockholm, Sweden
Tamio-Vesa Nakajima
  • Department of Computer Science, University of Oxford, UK
Stanislav Živný
  • Department of Computer Science, University of Oxford, UK

Funding

This work was supported by UKRI EP/X024431/1, by a Clarendon Fund Scholarship and by the Knut and Alice Wallenberg Foundation. For the purpose of Open Access, the authors have applied a CC BY public copyright licence to any Author Accepted Manuscript version arising from this submission. All data is provided in full in the results section of this paper.

Acknowledgements

This paper is a merger of independent work by Håstad and Martinsson, and by Nakajima and Živný respectively. We are grateful to Venkat Guruswami for noting and informing the authors of the fact that we independently had found the same algorithm.

References

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  2. Libor Barto, Jakub Bulín, Andrei A. Krokhin, and Jakub Opršal. Algebraic approach to promise constraint satisfaction. J. ACM, 68(4):28:1-28:66, 2021. URL: https://doi.org/10.1145/3457606.
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