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Improved Linearly Ordered Colorings of Hypergraphs via SDP Rounding

Authors Anand Louis , Alantha Newman, Arka Ray

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  • Theory of computation → Approximation algorithms analysis
Keywords
  • hypergraph coloring
  • SDP rounding
  • promise constraint satisfaction problems

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Abstract

We consider the problem of linearly ordered (LO) coloring of hypergraphs. A hypergraph has an LO coloring if there is a vertex coloring, using a set of ordered colors, so that (i) no edge is monochromatic, and (ii) each edge has a unique maximum color. It is an open question as to whether or not a 2-LO colorable 3-uniform hypergraph can be LO colored with 3 colors in polynomial time. Nakajima and Živný recently gave a polynomial-time algorithm to color such hypergraphs with Õ(n^{1/3}) colors and asked if SDP methods can be used directly to obtain improved bounds. Our main result is to show how to use SDP-based rounding methods to produce an LO coloring with Õ(n^{1/5}) colors for such hypergraphs. We show how to reduce the problem to cases with highly structured SDP solutions, which we call balanced hypergraphs. Then we discuss how to apply classic SDP-rounding tools in this case to obtain improved bounds.

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Anand Louis, Alantha Newman, and Arka Ray. Improved Linearly Ordered Colorings of Hypergraphs via SDP Rounding. In 44th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 323, pp. 30:1-30:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.FSTTCS.2024.30

Author Details

Anand Louis
  • Indian Institute of Science, Bengaluru, India
Alantha Newman
  • Université Grenoble Alpes, France
Arka Ray
  • Indian Institute of Science, Bengaluru, India

Funding

  • Louis, Anand: Supported in part by SERB Award CRG/2023/002896 and the Walmart Center for Tech Excellence at IISc (CSR Grant WMGT-23-0001).
  • Ray, Arka: Supported in part by the Walmart Center for Tech Excellence at IISc (CSR Grant WMGT-23-0001).

References

  1. Noga Alon, Pierre Kelsen, Sanjeev Mahajan, and Hariharan Ramesh. Coloring 2-colorable hypergraphs with a sublinear number of colors. Nordic Journal of Computing, 3:425-439, 1996. Google Scholar
  2. Sanjeev Arora, Eden Chlamtac, and Moses Charikar. New approximation guarantee for chromatic number. In Proceedings of the 38th Annual ACM Symposium on Theory of Computing (STOC), pages 215-224, 2006. Google Scholar
  3. Libor Barto, Diego Battistelli, and Kevin M. Berg. Symmetric promise constraint satisfaction problems: Beyond the Boolean case. In 38th International Symposium on Theoretical Aspects of Computer Science (STACS), volume 187 of LIPIcs, pages 10:1-10:16, 2021. URL: https://doi.org/10.4230/LIPICS.STACS.2021.10.
  4. Libor Barto, Jakub Bulín, Andrei A. Krokhin, and Jakub Oprsal. Algebraic approach to promise constraint satisfaction. Journal of the ACM, 68(4):28:1-28:66, 2021. URL: https://doi.org/10.1145/3457606.
  5. Avrim Blum. New approximation algorithms for graph coloring. Journal of the ACM, 41(3):470-516, 1994. URL: https://doi.org/10.1145/176584.176586.
  6. Avrim Blum and David R. Karger. An O(n^3/14)-coloring algorithm for 3-colorable graphs. Information Processing Letters, 61(1):49-53, 1997. URL: https://doi.org/10.1016/S0020-0190(96)00190-1.
  7. Joshua Brakensiek and Venkatesan Guruswami. Promise constraint satisfaction: Algebraic structure and a symmetric Boolean dichotomy. SIAM Journal on Computing, 50(6):1663-1700, 2021. URL: https://doi.org/10.1137/19M128212X.
  8. Joshua Brakensiek, Venkatesan Guruswami, and Sai Sandeep. SDPs and robust satisfiability of promise CSP. In Proceedings of the 55th Annual ACM Symposium on Theory of Computing (STOC), pages 609-622, 2023. URL: https://doi.org/10.1145/3564246.3585180.
  9. Panagiotis Cheilaris and Géza Tóth. Graph unique-maximum and conflict-free colorings. Journal of Discrete Algorithms, 9(3):241-251, 2011. URL: https://doi.org/10.1016/J.JDA.2011.03.005.
  10. Hui Chen and Alan Frieze. Coloring bipartite hypergraphs. In International Conference on Integer Programming and Combinatorial Optimization (IPCO), pages 345-358, 1996. URL: https://doi.org/10.1007/3-540-61310-2_26.
  11. Irit Dinur, Elchanan Mossel, and Oded Regev. Conditional hardness for approximate coloring. In Proceedings of the 38th Annual ACM Symposium on Theory of Computing (STOC), pages 344-353, 2006. URL: https://doi.org/10.1145/1132516.1132567.
  12. Irit Dinur, Oded Regev, and Clifford D. Smyth. The hardness of 3-uniform hypergraph coloring. Combinatorica, 25(5):519-535, 2005. URL: https://doi.org/10.1007/S00493-005-0032-4.
  13. Miron Ficak, Marcin Kozik, Miroslav Olsák, and Szymon Stankiewicz. Dichotomy for symmetric Boolean PCSPs. In 46th International Colloquium on Automata, Languages, and Programming (ICALP), volume 132 of LIPIcs, pages 57:1-57:12, 2019. URL: https://doi.org/10.4230/LIPICS.ICALP.2019.57.
  14. 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 41st International Symposium on Theoretical Aspects of Computer Science, (STACS), volume 289 of LIPIcs, pages 34:1-34:19, 2024. URL: https://doi.org/10.4230/LIPICS.STACS.2024.34.
  15. Magnús M. Halldórsson. Approximations of weighted independent set and hereditary subset problems. J. Graph Algorithms Appl., 4(1):1-16, 2000. URL: https://doi.org/10.7155/JGAA.00020.
  16. Johan Håstad, Björn Martinsson, Tamio-Vesa Nakajima, and Stanislav Živný. A logarithmic approximation of linearly-ordered colourings. In Amit Kumar and Noga Ron-Zewi, editors, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2024, August 28-30, 2024, London School of Economics, London, UK, volume 317 of LIPIcs, pages 7:1-7:6. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. URL: https://doi.org/10.4230/LIPICS.APPROX/RANDOM.2024.7.
  17. David R. Karger, Rajeev Motwani, and Madhu Sudan. Approximate graph coloring by semidefinite programming. Journal of the ACM, 45(2):246-265, 1998. URL: https://doi.org/10.1145/274787.274791.
  18. Meir Katchalski, William McCuaig, and Suzanne Seager. Ordered colourings. Discrete Mathematics, 1(142):141-154, 1995. URL: https://doi.org/10.1016/0012-365X(93)E0216-Q.
  19. Ken-ichi Kawarabayashi and Mikkel Thorup. Coloring 3-colorable graphs with less than n^1/5 colors. Journal of the ACM, 64(1):4:1-4:23, 2017. URL: https://doi.org/10.1145/3001582.
  20. Michael Krivelevich, Ram Nathaniel, and Benny Sudakov. Approximating coloring and maximum independent sets in 3-uniform hypergraphs. Journal of Algorithms, 41(1):99-113, 2001. URL: https://doi.org/10.1006/jagm.2001.1173.
  21. Anand Louis, Alantha Newman, and Arka Ray. Improved linearly ordered colorings of hypergraphs via SDP rounding. CoRR, abs/2405.00427, 2024. URL: https://doi.org/10.48550/arXiv.2405.00427.
  22. Tamio-Vesa Nakajima and Stanislav Živný. Linearly ordered colourings of hypergraphs. ACM Trans. Comput. Theory, 14(3-4):1-19, 2022. URL: https://doi.org/10.1145/3570909.
  23. Avi Wigderson. Improving the performance guarantee for approximate graph coloring. Journal of the ACM, 30(4):729-735, 1983. URL: https://doi.org/10.1145/2157.2158.
  24. David P. Williamson and David B. Shmoys. The Design of Approximation Algorithms. Cambridge University Press, 2011. Google Scholar
  25. Marcin Wrochna and Stanislav Živný. Improved hardness for H-colourings of G-colourable graphs. In Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms, (SODA), pages 1426-1435, 2020. URL: https://doi.org/10.1137/1.9781611975994.86.
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