Document Open Access Logo

Unbounded-Width CSPs Are Untestable in a Sublinear Number of Queries

Author Yumou Fei

PDF

File

Thumbnail PDF
PDF
LIPIcs.APPROX-RANDOM.2026.31.pdf
  • Filesize: 0.75 MB
  • 9 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational complexity and cryptography
Keywords
  • constraint satisfaction problems
  • property testing

Metrics

Abstract

The bounded-degree query model, introduced by Goldreich and Ron (Algorithmica, 2002), is a standard framework in graph property testing and sublinear-time algorithms. Many properties studied in this model, such as bipartiteness and 3-colorability of graphs, can be expressed as satisfiability of constraint satisfaction problems (CSPs). We prove that for the entire class of unbounded-width CSPs, testing satisfiability requires Ω(n) queries in the bounded-degree model. This result unifies and generalizes several previous lower bounds. In particular, it applies to all CSPs that are known to be NP-hard to solve, including k-colorability of 𝓁-uniform hypergraphs for any k,𝓁 ⩾ 2 with (k,𝓁) ≠ (2,2). 
Our proof combines the techniques from Bogdanov, Obata, and Trevisan (FOCS, 2002), who established the first Ω(n) query lower bound for CSP testing in the bounded-degree model, with known results from universal algebra.

Cite As Get BibTex

Yumou Fei. Unbounded-Width CSPs Are Untestable in a Sublinear Number of Queries. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 31:1-31:9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2026.31

Author Details

Yumou Fei
  • Massachusetts Institute of Technology, Cambridge, MA, USA

Acknowledgements

The author would like to thank Dor Minzer, Ronitt Rubinfeld, Shuo Wang and Standa Živný for many stimulating discussions during the development of this work. Special thanks to Ronitt Rubinfeld, for her helpful suggestions in improving the presentation of the paper.

References

  1. Hugo Aaronson, Gaia Carenini, and Atreyi Chanda. Property testing in bounded degree hypergraphs. arXiv preprint arXiv:2502.18382, 2025. URL: https://doi.org/10.48550/arXiv.2502.18382.
  2. Noga Alon and Asaf Shapira. Testing satisfiability. Journal of Algorithms, 47(2):87-103, 2003. URL: https://doi.org/10.1016/S0196-6774(03)00019-1.
  3. Sanjeev Arora, David Karger, and Marek Karpinski. Polynomial time approximation schemes for dense instances of np-hard problems. In Proceedings of the twenty-seventh annual ACM symposium on Theory of computing, pages 284-293, 1995. URL: https://doi.org/10.1145/225058.225140.
  4. Libor Barto and Marcin Kozik. Robustly solvable constraint satisfaction problems. SIAM Journal on Computing, 45(4):1646-1669, 2016. URL: https://doi.org/10.1137/130915479.
  5. Itai Benjamini, Oded Schramm, and Asaf Shapira. Every minor-closed property of sparse graphs is testable. Advances in Mathematics, 223:2200-2218, 2010. Google Scholar
  6. Andrej Bogdanov, Kenji Obata, and Luca Trevisan. A lower bound for testing 3-colorability in bounded-degree graphs. In The 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002. Proceedings., pages 93-102. IEEE, 2002. URL: https://doi.org/10.1109/SFCS.2002.1181886.
  7. Zarathustra Brady. Notes on csps and polymorphisms. arXiv preprint arXiv:2210.07383, 2022. URL: https://doi.org/10.48550/arXiv.2210.07383.
  8. Chi-Ning Chou, Alexander Golovnev, Madhu Sudan, Ameya Velingker, and Santhoshini Velusamy. Linear space streaming lower bounds for approximating csps. In Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing, pages 275-288, 2022. URL: https://doi.org/10.1145/3519935.3519983.
  9. Chi-Ning Chou, Alexander Golovnev, Madhu Sudan, and Santhoshini Velusamy. Sketching approximability of all finite csps. Journal of the ACM, 71(2):1-74, 2024. URL: https://doi.org/10.1145/3649435.
  10. Artur Czumaj, Morteza Monemizadeh, Krzysztof Onak, and Christian Sohler. Planar graphs: Random walks and bipartiteness testing. Random Structures & Algorithms, 55(1):104-124, 2019. URL: https://doi.org/10.1002/RSA.20826.
  11. Víctor Dalmau and Andrei Krokhin. Robust satisfiability for csps: Hardness and algorithmic results. ACM Transactions on Computation Theory (TOCT), 5(4):1-25, 2013. URL: https://doi.org/10.1145/2540090.
  12. Tomás Feder and Moshe Y Vardi. The computational structure of monotone monadic snp and constraint satisfaction: A study through datalog and group theory. SIAM Journal on Computing, 28(1):57-104, 1998. URL: https://doi.org/10.1137/S0097539794266766.
  13. Yumou Fei, Dor Minzer, and Shuo Wang. A dichotomy theorem for multi-pass streaming csps. arXiv preprint arXiv:2509.11399, 2025. URL: https://doi.org/10.48550/arXiv.2509.11399.
  14. Yumou Fei, Dor Minzer, and Shuo Wang. Multi-pass streaming lower bounds for approximating max-cut. arXiv preprint arXiv:2503.23404, 2025. URL: https://doi.org/10.48550/arXiv.2503.23404.
  15. Oded Goldreich, Shari Goldwasser, and Dana Ron. Property testing and its connection to learning and approximation. Journal of the ACM (JACM), 45(4):653-750, 1998. URL: https://doi.org/10.1145/285055.285060.
  16. Oded Goldreich and Dana Ron. A sublinear bipartiteness tester for bounded degree graphs. Combinatorica, 19(3):335-373, 1999. URL: https://doi.org/10.1007/S004930050060.
  17. Oded Goldreich and Dana Ron. Property testing in bounded degree graphs. Algorithmica, 32(2):302-343, 2002. URL: https://doi.org/10.1007/S00453-001-0078-7.
  18. Oded Goldreich and Dana Ron. On testing expansion in bounded-degree graphs. In Studies in Complexity and Cryptography: Miscellanea on the Interplay between Randomness and Computation, pages 68-75. Springer, 2011. URL: https://doi.org/10.1007/978-3-642-22670-0_9.
  19. Venkatesan Guruswami and Yuan Zhou. Tight bounds on the approximability of almost-satisfiable horn sat and exact hitting set. Theory of Computing, 8(1):239-267, 2012. Google Scholar
  20. Johan Håstad. Some optimal inapproximability results. Journal of the ACM (JACM), 48(4):798-859, 2001. URL: https://doi.org/10.1145/502090.502098.
  21. Michael Kapralov, Sanjeev Khanna, and Madhu Sudan. Streaming lower bounds for approximating max-cut. In Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1263-1282. SIAM, 2014. Google Scholar
  22. Michael Kapralov and Dmitry Krachun. An optimal space lower bound for approximating max-cut. In Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, pages 277-288, 2019. URL: https://doi.org/10.1145/3313276.3316364.
  23. Subhash Khot. On the power of unique 2-prover 1-round games. In Proceedings of the thiry-fourth annual ACM symposium on Theory of computing, pages 767-775, 2002. URL: https://doi.org/10.1145/509907.510017.
  24. Prasad Raghavendra. Optimal algorithms and inapproximability results for every csp? In Proceedings of the fortieth annual ACM symposium on Theory of computing, pages 245-254, 2008. URL: https://doi.org/10.1145/1374376.1374414.
  25. Raghuvansh R Saxena, Noah G Singer, Madhu Sudan, and Santhoshini Velusamy. Streaming algorithms via local algorithms for maximum directed cut. In Proceedings of the 2025 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 3392-3408. SIAM, 2025. URL: https://doi.org/10.1137/1.9781611978322.111.
  26. Luca Trevisan. Non-approximability results for optimization problems on bounded degree instances. In Proceedings of the thirty-third annual ACM symposium on Theory of computing, pages 453-461, 2001. URL: https://doi.org/10.1145/380752.380839.
  27. Yuichi Yoshida. Optimal constant-time approximation algorithms and (unconditional) inapproximability results for every bounded-degree csp. In Proceedings of the forty-third annual ACM symposium on Theory of computing, pages 665-674, 2011. URL: https://doi.org/10.1145/1993636.1993725.
  28. Yuichi Yoshida and Hiro Ito. Query-number preserving reductions and linear lower bounds for testing. IEICE transactions on information and systems, 93(2):233-240, 2010. URL: https://doi.org/10.1587/TRANSINF.E93.D.233.
  29. Yuichi Yoshida and Hiro Ito. Property testing on k-vertex-connectivity of graphs. Algorithmica, 62(3):701-712, 2012. URL: https://doi.org/10.1007/S00453-010-9477-Y.
Any Issues?
X

Feedback on the Current Page

CAPTCHA

Thanks for your feedback!

Feedback submitted to Dagstuhl Publishing

Could not send message

Please try again later or send an E-mail
© 2023-2026 Schloss Dagstuhl – LZI GmbH Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact