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Optimal Inapproximability of Promise Equations over Finite Groups

Authors Silvia Butti , Alberto Larrauri , Stanislav Živný

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ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
  • Theory of computation → Problems, reductions and completeness
Keywords
  • promise constraint satisfaction
  • approximation
  • linear equations

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Abstract

A celebrated result of Håstad established that, for any constant ε > 0, it is NP-hard to find an assignment satisfying a (1/|G|+ε)-fraction of the constraints of a given 3-LIN instance over an Abelian group G even if one is promised that an assignment satisfying a (1-ε)-fraction of the constraints exists. Engebretsen, Holmerin, and Russell showed the same result for 3-LIN instances over any finite (not necessarily Abelian) group. In other words, for almost-satisfiable instances of 3-LIN the random assignment achieves an optimal approximation guarantee. We prove that the random assignment algorithm is still best possible under a stronger promise that the 3-LIN instance is almost satisfiable over an arbitrarily more restrictive group.

Cite As Get BibTex

Silvia Butti, Alberto Larrauri, and Stanislav Živný. Optimal Inapproximability of Promise Equations over Finite Groups. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 38:1-38:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ICALP.2025.38

Author Details

Silvia Butti
  • Department of Computer Science, University of Oxford, UK
Alberto Larrauri
  • Department of Computer Science, University of Oxford, UK
Stanislav Živný
  • Department of Computer Science, University of Oxford, UK

Funding

  • Butti, Silvia: UKRI EP/X024431/1.
  • Larrauri, Alberto: UKRI EP/X024431/1.
  • Živný, Stanislav: UKRI EP/X024431/1.

Acknowledgements

We thank the anonymous reviewers for their feedback.

References

  1. Sanjeev Arora and Boaz Barak. Computational complexity: a modern approach. Cambridge University Press, 2009. Google Scholar
  2. Sanjeev Arora, Carsten Lund, Rajeev Motwani, Madhu Sudan, and Mario Szegedy. Proof verification and the hardness of approximation problems. J. ACM, 45(3):501-555, 1998. URL: https://doi.org/10.1145/278298.278306.
  3. Sanjeev Arora and Shmuel Safra. Probabilistic checking of proofs: A new characterization of NP. J. ACM, 45(1):70-122, 1998. URL: https://doi.org/10.1145/273865.273901.
  4. Per Austrin, Jonah Brown-Cohen, and Johan Håstad. Optimal inapproximability with universal factor graphs. ACM Trans. Algorithms, 2023. URL: https://doi.org/10.1145/3631119.
  5. Per Austrin, Venkatesan Guruswami, and Johan Håstad. (2+ε)-Sat is NP-hard. SIAM J. Comput., 46(5):1554-1573, 2017. URL: https://doi.org/10.1137/15M1006507.
  6. Per Austrin and Johan Håstad. On the usefulness of predicates. ACM Trans. Comput. Theory, 5(1):1:1-1:24, 2013. URL: https://doi.org/10.1145/2462896.2462897.
  7. Per Austrin and Elchanan Mossel. Approximation resistant predicates from pairwise independence. Computational Complexity, 18(2):249-271, 2009. URL: https://doi.org/10.1007/s00037-009-0272-6.
  8. 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.
  9. Libor Barto, Silvia Butti, Alexandr Kazda, Caterina Viola, and Stanislav Živný. Algebraic approach to approximation. In Proc. 39th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS'24), pages 10:1-10:14. ACM, 2024. URL: https://doi.org/10.1145/3661814.3662076.
  10. Amey Bhangale and Subhash Khot. Optimal Inapproximability of Satisfiable k-LIN over Non-Abelian Groups. In Proc. 53rd Annual ACM SIGACT Symposium on Theory of Computing (STOC'21), pages 1615-1628. ACM, 2021. URL: https://doi.org/10.1145/3406325.3451003.
  11. Amey Bhangale, Subhash Khot, and Dor Minzer. On Approximability of Satisfiable k-CSPs: I. In Proc. 54th Annual ACM Symposium on Theory of Computing (STOC'22), pages 976-988. ACM, 2022. URL: https://doi.org/10.1145/3519935.3520028.
  12. Amey Bhangale, Subhash Khot, and Dor Minzer. On Approximability of Satisfiable k-CSPs: II. In Proc. 55th Annual ACM Symposium on Theory of Computing (STOC'23), pages 632-642. ACM, 2023. URL: https://doi.org/10.1145/3564246.3585120.
  13. Amey Bhangale, Subhash Khot, and Dor Minzer. On Approximability of Satisfiable k-CSPs: III. In Proc. 55th Annual ACM Symposium on Theory of Computing (STOC'23), pages 643-655. ACM, 2023. URL: https://doi.org/10.1145/3564246.3585121.
  14. Amey Bhangale and Aleksa Stankovic. Max-3-Lin Over Non-abelian Groups with Universal Factor Graphs. Algorithmica, 85(9):2693-2734, 2023. URL: https://doi.org/10.1007/S00453-023-01115-1.
  15. 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.
  16. Joshua Brakensiek, Venkatesan Guruswami, and Sai Sandeep. SDPs and robust satisfiability of promise CSP. In Proc. 55th Annual ACM Symposium on Theory of Computing (STOC'23), pages 609-622, 2023. URL: https://doi.org/10.1145/3564246.3585180.
  17. Jonah Brown-Cohen and Prasad Raghavendra. Combinatorial optimization algorithms via polymorphisms. CoRR, 2015. URL: https://arxiv.org/abs/1501.01598.
  18. Silvia Butti, Alberto Larrauri, and Stanislav Živný. Optimal inapproximability of promise equations over finite groups. CoRR, 2024. URL: https://doi.org/10.48550/arXiv.2411.01630.
  19. Siu On Chan. Approximation resistance from pairwise-independent subgroups. J. ACM, 63(3):27:1-27:32, 2016. URL: https://doi.org/10.1145/2873054.
  20. Irit Dinur. The PCP theorem by gap amplification. J. ACM, 54(3):12, 2007. URL: https://doi.org/10.1145/1236457.1236459.
  21. Irit Dinur, Subhash Khot, Will Perkins, and Muli Safra. Hardness of finding independent sets in almost 3-colorable graphs. In Proc. 51th Annual IEEE Symposium on Foundations of Computer Science (FOCS'10), pages 212-221. IEEE Computer Society, 2010. URL: https://doi.org/10.1109/FOCS.2010.84.
  22. Lars Engebretsen and Jonas Holmerin. More efficient queries in PCPs for NP and improved approximation hardness of maximum CSP. Random Struct. Algorithms, 33(4):497-514, 2008. URL: https://doi.org/10.1002/RSA.20226.
  23. Lars Engebretsen, Jonas Holmerin, and Alexander Russell. Inapproximability results for equations over finite groups. Theor. Comput. Sci., 312(1):17-45, 2004. URL: https://doi.org/10.1016/S0304-3975(03)00401-8.
  24. Venkatesan Guruswami and Prasad Raghavendra. Hardness of Solving Sparse Overdetermined Linear Systems: A 3-Query PCP over Integers. ACM Trans. Comput. Theory, 1(2):6:1-6:20, 2009. URL: https://doi.org/10.1145/1595391.1595393.
  25. Johan Håstad. Some optimal inapproximability results. J. ACM, 48(4):798-859, 2001. URL: https://doi.org/10.1145/502090.502098.
  26. Yahli Hecht, Dor Minzer, and Muli Safra. NP-Hardness of Almost Coloring Almost 3-Colorable Graphs. In Proc. Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM'23), volume 275 of LIPIcs, pages 51:1-51:12. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2023.51.
  27. Subhash Khot. On the power of unique 2-prover 1-round games. In Proc. 34th Annual ACM Symposium on Theory of Computing (STOC'02), pages 767-775, 2002. URL: https://doi.org/10.1145/509907.510017.
  28. Subhash Khot and Dana Moshkovitz. NP-Hardness of Approximately Solving Linear Equations over Reals. SIAM J. Comput., 42(3):752-791, 2013. URL: https://doi.org/10.1137/110846415.
  29. Subhash Khot and Rishi Saket. Hardness of finding independent sets in almost q-colorable graphs. In Proc. 53rd Annual IEEE Symposium on Foundations of Computer Science (FOCS'12), pages 380-389. IEEE Computer Society, 2012. URL: https://doi.org/10.1109/FOCS.2012.75.
  30. Subhash Khot, Madhur Tulsiani, and Pratik Worah. A characterization of strong approximation resistance. In Proc. 46th Symposium on Theory of Computing (STOC'14), pages 634-643. ACM, 2014. URL: https://doi.org/10.1145/2591796.2591817.
  31. Subhash Khot, Madhur Tulsiani, and Pratik Worah. The complexity of somewhat approximation resistant predicates. In Proc. 41st International Colloquium on Automata, Languages, and Programming (ICALP'14), volume 8572 of Lecture Notes in Computer Science, pages 689-700. Springer, 2014. URL: https://doi.org/10.1007/978-3-662-43948-7_57.
  32. Alberto Larrauri and Stanislav Živný. Solving promise equations over monoids and groups. In Proc. 51st International Colloquium on Automata, Languages, and Programming (ICALP'24), volume 297 of LIPIcs, pages 146:1-146:18, 2024. URL: https://doi.org/10.4230/LIPIcs.ICALP.2024.146.
  33. Tamio-Vesa Nakajima and Stanislav Živný. Maximum k- vs. 𝓁-colourings of graphs. CoRR, 2023. URL: https://arxiv.org/abs/2311.00440.
  34. Tamio-Vesa Nakajima and Stanislav Živný. Maximum bipartite vs. triangle-free subgraph. In Proc. 52nd International Colloquium on Automata, Languages, and Programming (ICALP'25), 2025. URL: https://arxiv.org/abs/2406.20069.
  35. Ryan O'Donnell, Yi Wu, and Yuan Zhou. Hardness of Max-2Lin and Max-3Lin over Integers, Reals, and Large Cyclic Groups. ACM Trans. Comput. Theory, 7(2):9:1-9:16, 2015. URL: https://doi.org/10.1145/2751322.
  36. Prasad Raghavendra. Optimal algorithms and inapproximability results for every CSP? In Proc. 40th Annual ACM Symposium on Theory of Computing (STOC'08), pages 245-254, 2008. URL: https://doi.org/10.1145/1374376.1374414.
  37. Ran Raz. A parallel repetition theorem. SIAM J. Comput., 27(3):763-803, 1998. URL: https://doi.org/10.1137/S0097539795280895.
  38. Thomas Schaefer. The complexity of satisfiability problems. In Proc. 10th Annual ACM Symposium on the Theory of Computing (STOC'78), pages 216-226, 1978. URL: https://doi.org/10.1145/800133.804350.
  39. Audrey Terras. Fourier Transform and Representations of Finite Groups, pages 240-266. London Mathematical Society Student Texts. Cambridge University Press, 1999. Google Scholar
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