skip to main content
research-article

Linearly Ordered Colourings of Hypergraphs

Published:01 February 2023Publication History
Skip Abstract Section

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 \( k=O(\sqrt [3]{n \log \log n / \log n} \).

Second, given an r-uniform hypergraph that admits an LO 2-colouring, we establish NP-hardness of finding an LO k-colouring for every constant uniformity rk+2. In fact, we determine relationships between polymorphism minions for all uniformities r≥ 3, which reveals a key difference between r< k+2 and rk+2 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-colouring for LO ℓ-colourable r-uniform hypergraphs for 2 ≤ ℓ ≤ k and rk - ℓ + 4.

REFERENCES

  1. [1] Austrin Per, Bhangale Amey, and Potukuchi Aditya. 2019. Simplified inpproximability of hypergraph coloring via \(t\)-agreeing families. arXiv:1904.01163. Retrieved from https://arxiv.org/abs/1904.01163.Google ScholarGoogle Scholar
  2. [2] Austrin Per, Bhangale Amey, and Potukuchi Aditya. 2020. Improved inapproximability of rainbow coloring. In Proceedings of the 31st Annual ACM-SIAM Symposium on Discrete Algorithms (SODA’20). 14791495. DOI:Google ScholarGoogle ScholarCross RefCross Ref
  3. [3] Austrin Per, Guruswami Venkatesan, and Håstad Johan. 2017. (2+\(\epsilon\))-sat is NP-hard. SIAM J. Comput. 46, 5 (2017), 15541573. DOI:Google ScholarGoogle ScholarDigital LibraryDigital Library
  4. [4] Barto Libor, Battistelli Diego, and Berg Kevin M.. 2021. Symmetric promise constraint satisfaction problems: Beyond the boolean case. In Proceedings of the 38th International Symposium on Theoretical Aspects of Computer Science (STACS’21), LIPIcs Vol. 187. 10:1–10:16. DOI:Google ScholarGoogle ScholarCross RefCross Ref
  5. [5] Barto Libor, Bulín Jakub, Krokhin Andrei A., and Opršal Jakub. 2021. Algebraic approach to promise constraint satisfaction. J. ACM 68, 4 (2021), 28:1–28:66. DOI:Google ScholarGoogle ScholarDigital LibraryDigital Library
  6. [6] Barto Libor and Kozik Marcin. 2022. Combinatorial gap theorem and reductions between promise CSPs. In Proceedings of the ACM-SIAM Symposium on Discrete Algorithms (SODA’22). 12041220. DOI:Google ScholarGoogle ScholarCross RefCross Ref
  7. [7] Barto Libor, Krokhin Andrei, and Willard Ross. 2017. Polymorphisms, and how to use them. In The Constraint Satisfaction Problem: Complexity and Approximability, Krokhin Andrei and Živný Stanislav (Eds.). Dagstuhl Follow-Ups, Vol. 7. Schloss Dagstuhl–Leibniz-Zentrum für Informatik, Dagstuhl, Germany, 144. DOI:Google ScholarGoogle ScholarCross RefCross Ref
  8. [8] Barto Libor, Opršal Jakub, and Pinsker Michael. 2018. The wonderland of reflections. Isr. J. Math. 223, 1 (February2018), 363398. DOI:Google ScholarGoogle ScholarCross RefCross Ref
  9. [9] Berger Bonnie and Rompel John. 1990. A better performance guarantee for approximate graph coloring. Algorithmica 5, 3 (1990), 459466. DOI:Google ScholarGoogle ScholarDigital LibraryDigital Library
  10. [10] Bhangale Amey. 2018. NP-hardness of coloring 2-colorable hypergraph with poly-logarithmically many colors. In Proceedings of the 45th International Colloquium on Automata, Languages, and Programming (ICALP’18), LIPIcs Vol. 107. 15:1–15:11. DOI:Google ScholarGoogle ScholarCross RefCross Ref
  11. [11] Blum Avrim. 1994. New approximation algorithms for graph coloring. J. ACM 41, 3 (1994), 470516. DOI:Google ScholarGoogle ScholarDigital LibraryDigital Library
  12. [12] Brakensiek Joshua and Guruswami Venkatesan. 2016. New hardness results for graph and hypergraph colorings. In Proceedings of the 31st Conference on Computational Complexity (CCC’16), LIPIcs Vol. 50. 14:1–14:27. DOI:Google ScholarGoogle ScholarCross RefCross Ref
  13. [13] Brakensiek Joshua and Guruswami Venkatesan. 2021. Promise constraint satisfaction: Algebraic structure and a symmetric boolean dichotomy. SIAM J. Comput. 50, 6 (2021), 16631700. DOI:Google ScholarGoogle ScholarCross RefCross Ref
  14. [14] Brakensiek Joshua and Guruswami Venkatesan. 2021. The quest for strong inapproximability results with perfect completeness. ACM Trans. Algor. 17, 3 (2021), 27:1–27:35. DOI:Google ScholarGoogle ScholarDigital LibraryDigital Library
  15. [15] Brandts Alex, Wrochna Marcin, and Živný Stanislav. 2021. The complexity of promise SAT on non-boolean domains. ACM Trans. Comput. Theory 13, 4, Article 26 (2021). DOI:Google ScholarGoogle ScholarDigital LibraryDigital Library
  16. [16] Cheilaris Panagiotis, Keszegh Balázs, and Pálvölgyi Dömötör. 2013. Unique-maximum and conflict-free coloring for hypergraphs and tree graphs. SIAM J. Discr. Math. 27, 4 (2013), 17751787. DOI:Google ScholarGoogle ScholarDigital LibraryDigital Library
  17. [17] Cheilaris Panagiotis and Tóth Géza. 2011. Graph unique-maximum and conflict-free colorings. J. Discr. Algor. 9, 3 (2011), 241251. DOI:Google ScholarGoogle ScholarDigital LibraryDigital Library
  18. [18] Chlamtac Eden and Singh Gyanit. 2008. Improved approximation guarantees through higher levels of SDP hierarchies. In Proceedings of the 11th International Workshiop on Approximation, Randomization and Combinatorial Optimization (APPROX’08), Lecture Notes in Computer Science, Vol. 5171. Springer, 4962. DOI:Google ScholarGoogle ScholarDigital LibraryDigital Library
  19. [19] Cormen Thomas H., Leiserson Charles E., Rivest Ronald L., and Stein Clifford. 2009. Introduction to Algorithms, 3rd Edition. MIT Press. Google ScholarGoogle ScholarDigital LibraryDigital Library
  20. [20] Dinur Irit, Mossel Elchanan, and Regev Oded. 2009. Conditional hardness for approximate coloring. SIAM J. Comput. 39, 3 (2009), 843873. DOI:Google ScholarGoogle ScholarDigital LibraryDigital Library
  21. [21] Dinur Irit, Regev Oded, and Smyth Clifford. 2005. The hardness of 3-uniform hypergraph coloring. Comb. 25, 5 (September2005), 519535. DOI:Google ScholarGoogle ScholarDigital LibraryDigital Library
  22. [22] Dinur Irit and Shinkar Igor. 2010. On the conditional hardness of coloring a 4-colorable graph with super-constant number of colors. In Proceedings of the 13th International Workshop on Approximation Algorithms for Combinatorial Optimization (APPROX’10), Lecture Notes in Computer Science, Vol. 6302. Springer, 138151. DOI:Google ScholarGoogle ScholarCross RefCross Ref
  23. [23] Garey M. R. and Johnson D. S.. 1976. The complexity of near-optimal graph coloring. J. ACM 23, 1 (1976), 4349. DOI:Google ScholarGoogle ScholarDigital LibraryDigital Library
  24. [24] Guruswami Venkatesan and Khanna Sanjeev. 2004. On the hardness of 4-coloring a 3-colorable graph. SIAM J. Discr. Math. 18, 1 (2004), 3040. DOI:Google ScholarGoogle ScholarDigital LibraryDigital Library
  25. [25] Guruswami Venkatesan and Lee Euiwoong. 2018. Strong inapproximability results on balanced rainbow-colorable hypergraphs. Combinatorica 38, 3 (2018), 547599. DOI:Google ScholarGoogle ScholarDigital LibraryDigital Library
  26. [26] Guruswami Venkatesan and Sandeep Sai. 2020. d-To-1 hardness of coloring 3-colorable graphs with O(1) colors. In Proceedings of the 47th International Colloquium on Automata, Languages, and Programming (ICALP’20), LIPIcs Vol. 168. Schloss Dagstuhl–Leibniz-Zentrum für Informatik, 62:1–62:12. DOI:Google ScholarGoogle ScholarCross RefCross Ref
  27. [27] Guruswami Venkatesan and Sandeep Sai. 2020. Rainbow coloring hardness via low sensitivity polymorphisms. SIAM J. Discr. Math. 34, 1 (2020), 520537. DOI:Google ScholarGoogle ScholarDigital LibraryDigital Library
  28. [28] Halldórsson Magnús M.. 2000. Approximations of weighted independent set and hereditary subset problems. J. Graph Algor. Appl. 4, 1 (2000), 116. DOI:Google ScholarGoogle ScholarCross RefCross Ref
  29. [29] Huang Sangxia. 2013. Improved hardness of approximating chromatic number. In Proceedings of the 16th International Workshop on Approximation Algorithms for Combinatorial Optimization (APPROX’13). Springer, 233243. DOI:. arXiv:1301.5216.Google ScholarGoogle ScholarCross RefCross Ref
  30. [30] Karger David R., Motwani Rajeev, and Sudan Madhu. 1998. Approximate graph coloring by semidefinite programming. J. ACM 45, 2 (1998), 246265. DOI:Google ScholarGoogle ScholarDigital LibraryDigital Library
  31. [31] Kawarabayashi Ken-ichi and Thorup Mikkel. 2017. Coloring 3-colorable graphs with less than \(n^{1/5}\) colors. J. ACM 64, 1 (2017), 4:1–4:23. DOI:Google ScholarGoogle ScholarDigital LibraryDigital Library
  32. [32] Khanna Sanjeev, Linial Nathan, and Safra Shmuel. 2000. On the hardness of approximating the chromatic number. Combinatorica 20, 3 (March2000), 393415. DOI:Google ScholarGoogle ScholarCross RefCross Ref
  33. [33] Khanna Sanjeev, Sudan Madhu, Trevisan Luca, and Williamson David P.. 2000. The approximability of constraint satisfaction problems. SIAM J. Comput. 30, 6 (2000), 18631920. DOI:Google ScholarGoogle ScholarDigital LibraryDigital Library
  34. [34] Khot Subhash. 2002. On the power of unique 2-prover 1-round games. In Proceedings of the 34th Annual ACM Symposium on Theory of Computing (STOC’02). ACM, 767775. DOI:Google ScholarGoogle ScholarDigital LibraryDigital Library
  35. [35] Krivelevich Michael, Nathaniel Ram, and Sudakov Benny. 2001. Approximating coloring and maximum independent sets in 3-uniform hypergraphs. J. Algor. 41, 1 (2001), 99113. DOI:Google ScholarGoogle ScholarDigital LibraryDigital Library
  36. [36] Krivelevich Michael and Sudakov Benny. 2003. Approximate coloring of uniform hypergraphs. J. Algor. 49, 1 (2003), 212. DOI:Google ScholarGoogle ScholarDigital LibraryDigital Library
  37. [37] Nakajima Tamio-Vesa and Živný Stanislav. 2022. Linearly ordered colourings of hypergraphs. In 49th International Colloquium on Automata, Languages, and Programming (ICALP’22), LIPIcs Vol. 229. 128:1–128:18. DOI:Google ScholarGoogle ScholarCross RefCross Ref
  38. [38] Nakajima Tamio-Vesa and Živný Stanislav. 2022. Linearly Ordered Colourings of Hypergraphs. Technical Report. Google ScholarGoogle ScholarDigital LibraryDigital Library
  39. [39] Wigderson Avi. 1983. Improving the performance guarantee for approximate graph coloring. J. ACM 30, 4 (1983), 729735. DOI:Google ScholarGoogle ScholarDigital LibraryDigital Library
  40. [40] Wrochna Marcin. 2022. A Note on Hardness of Promise Hypergraph Colouring. Technical Report.Google ScholarGoogle Scholar
  41. [41] Wrochna Marcin and Živný Stanislav. 2020. Improved hardness for \(H\)-colourings of \(G\)-colourable graphs. In Proceedings of the 31st Annual ACM-SIAM Symposium on Discrete Algorithms (SODA’20). 14261435. DOI:Google ScholarGoogle ScholarCross RefCross Ref

Index Terms

  1. Linearly Ordered Colourings of Hypergraphs

        Recommendations

        Comments

        Login options

        Check if you have access through your login credentials or your institution to get full access on this article.

        Sign in

        Full Access

        • Published in

          cover image ACM Transactions on Computation Theory
          ACM Transactions on Computation Theory  Volume 14, Issue 3-4
          December 2022
          122 pages
          ISSN:1942-3454
          EISSN:1942-3462
          DOI:10.1145/3582881
          Issue’s Table of Contents

          Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected].

          Publisher

          Association for Computing Machinery

          New York, NY, United States

          Publication History

          • Published: 1 February 2023
          • Online AM: 23 November 2022
          • Accepted: 4 November 2022
          • Revised: 2 November 2022
          • Received: 14 April 2022
          Published in toct Volume 14, Issue 3-4

          Permissions

          Request permissions about this article.

          Request Permissions

          Check for updates

          Qualifiers

          • research-article
          • Refereed

        PDF Format

        View or Download as a PDF file.

        PDF

        eReader

        View online with eReader.

        eReader

        Full Text

        View this article in Full Text.

        View Full Text

        HTML Format

        View this article in HTML Format .

        View HTML Format
        About Cookies On This Site

        We use cookies to ensure that we give you the best experience on our website.

        Learn more

        Got it!