A Logarithmic Approximation of Linearly-Ordered Colourings

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



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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

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.

Cite AsGet BibTex

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

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.

Subject Classification

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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References

  1. Libor Barto, Diego Battistelli, and Kevin M. Berg. Symmetric Promise Constraint Satisfaction Problems: Beyond the Boolean Case. In Proc. 38th International Symposium on Theoretical Aspects of Computer Science (STACS'21), volume 187 of LIPIcs, pages 10:1-10:16, 2021. URL: https://doi.org/10.4230/LIPIcs.STACS.2021.10.
  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.
  3. Manuel Bodirsky and Jan Kára. The complexity of temporal constraint satisfaction problems. J. ACM, 57(2):9:1-9:41, 2010. URL: https://doi.org/10.1145/1667053.1667058.
  4. Joshua Brakensiek and Venkatesan Guruswami. An algorithmic blend of LPs and ring equations for promise CSPs. In Proc. 30th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA'19), pages 436-455, 2019. URL: https://doi.org/10.1137/1.9781611975482.28.
  5. Joshua Brakensiek and Venkatesan Guruswami. Promise Constraint Satisfaction: Algebraic Structure and a Symmetric Boolean Dichotomy. SIAM J. Comput., 50(6):1663-1700, 2021. URL: https://doi.org/10.1137/19M128212X.
  6. Eden Chlamtac and Gyanit Singh. Improved approximation guarantees through higher levels of SDP hierarchies. In Proc. 11th International Workshiop on Approximation, Randomization and Combinatorial Optimization (APPROX'08), volume 5171 of Lecture Notes in Computer Science, pages 49-62. Springer, 2008. URL: https://doi.org/10.1007/978-3-540-85363-3_5.
  7. Irit Dinur, Oded Regev, and Clifford Smyth. The hardness of 3-uniform hypergraph coloring. Comb., 25(5):519-535, September 2005. URL: https://doi.org/10.1007/s00493-005-0032-4.
  8. Marek Filakovský, Tamio-Vesa Nakajima, Jakub Opršal, Gianluca Tasinato, and Uli Wagner. Hardness of Linearly Ordered 4-Colouring of 3-Colourable 3-Uniform Hypergraphs. In Proc. 41st International Symposium on Theoretical Aspects of Computer Science (STACS'24), volume 289 of Leibniz International Proceedings in Informatics (LIPIcs), pages 34:1-34:19, 2024. URL: https://doi.org/10.4230/LIPIcs.STACS.2024.34.
  9. Ken-ichi Kawarabayashi, Mikkel Thorup, and Hirotaka Yoneda. Better coloring of 3-Colorable graphs. In Proceedings of the 56th Annual ACM Symposium on Theory of Computing, STOC 2024, pages 331-339, New York, NY, USA, 2024. Association for Computing Machinery. URL: https://doi.org/10.1145/3618260.3649768.
  10. Michael Krivelevich, Ram Nathaniel, and Benny Sudakov. Approximating coloring and maximum independent sets in 3-uniform hypergraphs. J. Algorithms, 41(1):99-113, 2001. URL: https://doi.org/10.1006/jagm.2001.1173.
  11. Michael Krivelevich and Benny Sudakov. Approximate coloring of uniform hypergraphs. J. Algorithms, 49(1):2-12, 2003. URL: https://doi.org/10.1016/S0196-6774(03)00077-4.
  12. Anand Louis, Alantha Newman, and Arka Ray. Improved linearly ordered colorings of hypergraphs via SDP rounding, 2024. URL: https://doi.org/10.48550/arXiv.2405.00427.
  13. Tamio-Vesa Nakajima and Stanislav Živný. Linearly Ordered Colourings of Hypergraphs. In Proc. 49th International Colloquium on Automata, Languages, and Programming (ICALP'22), volume 229 of LIPIcs, pages 128:1-128:18, 2022. URL: https://doi.org/10.4230/LIPIcs.ICALP.2022.128.
  14. Tamio-Vesa Nakajima and Stanislav Živný. Linearly Ordered Colourings of Hypergraphs. ACM Trans. Comput. Theory, 13(3-4), 2023. URL: https://doi.org/10.1145/3570909.
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