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Approximating 1-In-3 SAT by Linearly Ordered Hypergraph 3-Colouring Is NP-Hard

Authors Andrei Krokhin , Danny Vagnozzi

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LIPIcs.ICALP.2026.184.pdf
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  • Theory of computation
Keywords
  • Constraint satisfaction
  • complexity theory

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Abstract

1-in-3 SAT is a classical NP-hard constraint satisfaction problem (CSP). Given a satisfiable instance of 1-in-3 SAT, it is NP-hard to find a satisfying assignment for it, but it may be possible to efficiently find a solution subject to a weaker (not necessarily Boolean) predicate than "1-in-3". There is a conjecture, which we call the Approximate 1-in-3 SAT conjecture, made independently by several researchers, that predicts a dichotomy: for certain choices of weaker predicates the problem becomes tractable and for the remaining choices the task remains NP-hard. Such problems belong to the Promise CSP (PCSP) framework, which studies how one CSP can be approximated by another, in a specific qualitative sense. The Approximate 1-in-3 SAT conjecture is notable because there is no P versus NP-hard dichotomy conjecture for general PCSPs yet (due to insufficient evidence). One specific predicate, corresponding to the problem of linearly ordered 3-colouring of 3-uniform hypergraphs, has been mentioned in several recent papers as an obstacle to further progress in proving the Approximate 1-in-3 SAT conjecture. We prove that the problem for this predicate is NP-hard, as predicted by the conjecture. This completes the proof of the conjecture for predicates on a 3-element domain.

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Andrei Krokhin and Danny Vagnozzi. Approximating 1-In-3 SAT by Linearly Ordered Hypergraph 3-Colouring Is NP-Hard. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 184:1-184:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ICALP.2026.184

Author Details

Andrei Krokhin
  • Durham University, UK
Danny Vagnozzi
  • Durham University, UK

Funding

Both authors are supported by the UK EPSRC grant EP/X033201/1.

Acknowledgements

We are grateful to Jakub Opršal, Tamio-Vesa Nakajima and Marek Filakovský for many a helpful discussion. We would also like to thank Joshua Brakensiek and the anonymous reviewers for useful feedback which improved the presentation.

References

  1. Per Austrin, Amey Bhangale, and Aditya Potukuchi. Improved inapproximability of rainbow coloring. In Shuchi Chawla, editor, Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms, SODA 2020, Salt Lake City, UT, USA, January 5-8, 2020, pages 1479-1495. SIAM, 2020. URL: https://doi.org/10.1137/1.9781611975994.90.
  2. Per Austrin, Venkatesan Guruswami, and Johan Håstad. (2+ε)-SAT is NP-hard. SIAM Journal on Computing, 46(5):1554-1573, 2017. URL: https://doi.org/10.1137/15M1006507.
  3. Sergey Avvakumov, Marek Filakovský, Jakub Opršal, Gianluca Tasinato, and Uli Wagner. Hardness of 4-colouring k-colourable graphs. In Michal Koucký and Nikhil Bansal, editors, Proceedings of the 57th Annual ACM Symposium on Theory of Computing, STOC 2025, Prague, Czechia, June 23-27, 2025, pages 72-83. ACM, 2025. URL: https://doi.org/10.1145/3717823.3718154.
  4. Demian Banakh and Marcin Kozik. Injective hardness condition for PCSPs. In Pawel Sobocinski, Ugo Dal Lago, and Javier Esparza, editors, Proceedings of the 39th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2024, Tallinn, Estonia, July 8-11, 2024, pages 8:1-8:10. ACM, 2024. URL: https://doi.org/10.1145/3661814.3662072.
  5. Libor Barto, Diego Battistelli, and Kevin M. Berg. Symmetric Promise Constraint Satisfaction Problems: Beyond the Boolean Case. In Markus Bläser and Benjamin Monmege, editors, 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021), volume 187 of Leibniz International Proceedings in Informatics (LIPIcs), pages 10:1-10:16, Dagstuhl, Germany, 2021. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.STACS.2021.10.
  6. Libor Barto, Jakub Bulín, Andrei 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.
  7. Libor Barto and Marcin Kozik. Combinatorial gap theorem and reductions between promise CSPs. In Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1204-1220, 2022. URL: https://doi.org/10.1137/1.9781611977073.50.
  8. Libor Barto, Andrei Krokhin, and Ross Willard. Polymorphisms, and how to use them. In Andrei A. Krokhin and Stanislav Zivný, editors, The Constraint Satisfaction Problem: Complexity and Approximability, volume 7 of Dagstuhl Follow-Ups, pages 1-44. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2017. URL: https://doi.org/10.4230/DFU.VOL7.15301.1.
  9. Libor Barto, Jakub Opršal, and Michael Pinsker. The wonderland of reflections. Israel J. Math., 223(1):363-398, 2018. Google Scholar
  10. Benjamin Bedert, Tamio-Vesa Nakajima, Karolina Okrasa, and Stanislav Živnỳ. Strong sparsification for 1-in-3-SAT via polynomial Freiman-Ruzsa, 2025. to appear in FOCS'25. URL: https://arxiv.org/abs/2507.17878.
  11. Joshua Brakensiek and Venkatesan Guruswami. New hardness results for graph and hypergraph colorings. In Ran Raz, editor, 31st Conference on Computational Complexity, CCC 2016, May 29 to June 1, 2016, Tokyo, Japan, volume 50 of LIPIcs, pages 14:1-14:27. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2016. URL: https://doi.org/10.4230/LIPIcs.CCC.2016.14.
  12. 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.
  13. Alex Brandts, Marcin Wrochna, and Stanislav Živný. The complexity of promise SAT on non-Boolean domains. ACM Trans. Comput. Theory, 13(4), September 2021. URL: https://doi.org/10.1145/3470867.
  14. Mark Braverman, Subhash Khot, Noam Lifshitz, and Dor Minzer. An invariance principle for the multi-slice, with applications. In 2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS), pages 228-236. IEEE, 2022. Google Scholar
  15. Andrei Bulatov. A dichotomy theorem for nonuniform CSPs. In 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS), pages 319-330. IEEE, 2017. URL: https://doi.org/10.1109/FOCS.2017.37.
  16. Andrei Bulatov, Peter Jeavons, and Andrei Krokhin. Classifying the complexity of constraints using finite algebras. SIAM J. Comput., 34(3):720-742, 2005. URL: https://doi.org/10.1137/S0097539700376676.
  17. Panagiotis Cheilaris, Balázs Keszegh, and Dömötör Pálvölgyi. Unique-maximum and conflict-free coloring for hypergraphs and tree graphs. SIAM J. Discret. Math., 27(4):1775-1787, 2013. URL: https://doi.org/10.1137/120880471.
  18. Lorenzo Ciardo, Marcin Kozik, Andrei Krokhin, Tamio-Vesa Nakajima, and Stanislav Živný. 1-in-3 vs. not-all-equal: Dichotomy of a broken promise. ACM Trans. Comput. Log., 26(2):10:1-10:22, 2025. URL: https://doi.org/10.1145/3719007.
  19. Víctor Dalmau and Jakub Opršal. Local consistency as a reduction between constraint satisfaction problems. In Pawel Sobocinski, Ugo Dal Lago, and Javier Esparza, editors, Proceedings of the 39th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2024, Tallinn, Estonia, July 8-11, 2024, pages 29:1-29:15. ACM, 2024. URL: https://doi.org/10.1145/3661814.3662068.
  20. Irit Dinur, Elchanan Mossel, and Oded Regev. Conditional hardness for approximate coloring. SIAM J. Comput., 39(3):843-873, 2009. URL: https://doi.org/10.1137/07068062X.
  21. Irit Dinur, Oded Regev, and Clifford D. Smyth. The hardness of 3-uniform hypergraph coloring. Comb., 25(5):519-535, 2005. URL: https://doi.org/10.1007/S00493-005-0032-4.
  22. David Ellis. Intersection problems in extremal combinatorics: Theorems, techniques and questions old and new, 2021. URL: https://arxiv.org/abs/2107.06371.
  23. Marek Filakovský, Tamio-Vesa Nakajima, Jakub Opršal, Gianluca Tasinato, and Uli Wagner. Hardness of linearly ordered 4-colouring of 3-colourable 3-uniform hypergraphs. ACM Trans. Comput. Theory, 18(2), May 2026. URL: https://doi.org/10.1145/3779121.
  24. Venkatesan Guruswami and Sai Sandeep. d-To-1 Hardness of Coloring 3-Colorable Graphs with O(1) Colors. In Artur Czumaj, Anuj Dawar, and Emanuela Merelli, editors, 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020), volume 168 of Leibniz International Proceedings in Informatics (LIPIcs), pages 62:1-62:12, Dagstuhl, Germany, 2020. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.ICALP.2020.62.
  25. Venkatesan Guruswami and Sai Sandeep. Rainbow coloring hardness via low sensitivity polymorphisms. SIAM J. Discret. Math., 34(1):520-537, 2020. URL: https://doi.org/10.1137/19M127731X.
  26. Johan Håstad. Some optimal inapproximability results. J. ACM, 48(4):798-859, 2001. URL: https://doi.org/10.1145/502090.502098.
  27. 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), volume 317 of Leibniz International Proceedings in Informatics (LIPIcs), pages 7:1-7:6, Dagstuhl, Germany, 2024. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2024.7.
  28. Ravindran Kannan and Achim Bachem. Polynomial algorithms for computing the smith and hermite normal forms of an integer matrix. SIAM J. Comput., 8(4):499-507, 1979. URL: https://doi.org/10.1137/0208040.
  29. Ken-ichi Kawarabayashi, Mikkel Thorup, and Hirotaka Yoneda. Better coloring of 3-colorable graphs. In Bojan Mohar, Igor Shinkar, and Ryan O'Donnell, editors, Proceedings of the 56th Annual ACM Symposium on Theory of Computing, STOC 2024, Vancouver, BC, Canada, June 24-28, 2024, pages 331-339. ACM, 2024. URL: https://doi.org/10.1145/3618260.3649768.
  30. Andrei Krokhin and Jakub Opršal. An invitation to the promise constraint satisfaction problem. ACM SIGLOG News, 9(3):30-59, 2022. URL: https://doi.org/10.1145/3559736.3559740.
  31. Andrei Krokhin, Jakub Opršal, Marcin Wrochna, and Stanislav Živný. Topology and adjunction in promise constraint satisfaction. SIAM Journal on Computing, 52(1):38-79, 2023. URL: https://doi.org/10.1137/20M1378223.
  32. Andrei Krokhin and Danny Vagnozzi. Approximating 1-in-3 SAT by linearly ordered hypergraph 3-colouring is NP-hard. arXiv preprint, 2025. URL: https://doi.org/10.48550/arXiv.2508.14606.
  33. Andrei Krokhin and Stanislav Živný, editors. The Constraint Satisfaction Problem: Complexity and Approximability, volume 7 of Dagstuhl Follow-Ups. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2017. URL: http://www.dagstuhl.de/dagpub/978-3-95977-003-3.
  34. Alberto Larrauri. Ineffectiveness for search and undecidability of PCSP meta-problems. CoRR, abs/2504.04639, 2025. to appear in FOCS'25. URL: https://doi.org/10.48550/arXiv.2504.04639.
  35. Konstantin Makarychev and Yury Makarychev. Approximation algorithms for CSPs. In Andrei A. Krokhin and Stanislav Zivný, editors, The Constraint Satisfaction Problem: Complexity and Approximability, volume 7 of Dagstuhl Follow-Ups, pages 287-325. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2017. URL: https://doi.org/10.4230/DFU.VOL7.15301.11.
  36. Jiří Matoušek. Using the Borsuk-Ulam Theorem. Universitext. Springer, 2003. Google Scholar
  37. Sebastian Meyer and Jakub Opršal. A topological proof of the Hell-Nešetřil dichotomy. In Proceedings of the 2025 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 4507-4519, 2025. URL: https://doi.org/10.1137/1.9781611978322.154.
  38. Aaron Meyerowitz. Maximal intersecting families. European Journal of Combinatorics, 16(5):491-501, 1995. URL: https://doi.org/10.1016/0195-6698(95)90004-7.
  39. Tamio-Vesa Nakajima, Zephyr Verwimp, Marcin Wrochna, and Stanislav Živný. Complexity of approximate conflict-free, linearly-ordered, and nonmonochromatic hypergraph colourings. In Keren Censor-Hillel, Fabrizio Grandoni, Joël Ouaknine, and Gabriele Puppis, editors, 52nd International Colloquium on Automata, Languages, and Programming, ICALP 2025, July 8-11, 2025, Aarhus, Denmark, volume 334 of LIPIcs, pages 169:1-169:10. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2025. to appear in ACM Transactions on Computation Theory. URL: https://doi.org/10.4230/LIPIcs.ICALP.2025.169.
  40. Tamio-Vesa Nakajima and Stanislav Živný. Linearly ordered colourings of hypergraphs. ACM Trans. Comput. Theory, 14(3–4), February 2023. URL: https://doi.org/10.1145/3570909.
  41. Tamio-Vesa Nakajima and Stanislav Živný. On the complexity of symmetric vs. functional PCSPs. ACM Trans. Algorithms, 20(4):33:1-33:29, 2024. URL: https://doi.org/10.1145/3673655.
  42. Dmitriy Zhuk. A proof of the CSP dichotomy conjecture. Journal of the ACM (JACM), 67(5):1-78, 2020. URL: https://doi.org/10.1145/3402029.
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