Limits of Sequential Local Algorithms on the Random k-XORSAT Problem

Author Kingsley Yung

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Kingsley Yung
  • The Chinese University of Hong Kong, Hong Kong, China


The author thanks Andrej Bogdanov for his guidance and many insightful discussions.

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Kingsley Yung. Limits of Sequential Local Algorithms on the Random k-XORSAT Problem. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 123:1-123:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


The random k-XORSAT problem is a random constraint satisfaction problem of n Boolean variables and m = rn clauses, which a random instance can be expressed as a G𝔽(2) linear system of the form Ax = b, where A is a random m × n matrix with k ones per row, and b is a random vector. It is known that there exist two distinct thresholds r_{core}(k) < r_{sat}(k) such that as n → ∞ for r < r_{sat}(k) the random instance has solutions with high probability, while for r_{core} < r < r_{sat}(k) the solution space shatters into an exponential number of clusters. Sequential local algorithms are a natural class of algorithms which assign values to variables one by one iteratively. In each iteration, the algorithm runs some heuristics, called local rules, to decide the value assigned, based on the local neighborhood of the selected variables under the factor graph representation of the instance. We prove that for any r > r_{core}(k) the sequential local algorithms with certain local rules fail to solve the random k-XORSAT with high probability. They include (1) the algorithm using the Unit Clause Propagation as local rule for k ≥ 9, and (2) the algorithms using any local rule that can calculate the exact marginal probabilities of variables in instances with factor graphs that are trees, for k ≥ 13. The well-known Belief Propagation and Survey Propagation are included in (2). Meanwhile, the best known linear-time algorithm succeeds with high probability for r < r_{core}(k). Our results support the intuition that r_{core}(k) is the sharp threshold for the existence of a linear-time algorithm for random k-XORSAT. Our approach is to apply the Overlap Gap Property OGP framework to the sub-instance induced by the core of the instance, instead of the whole instance. By doing so, the sequential local algorithms can be ruled out at density as low as r_{core}(k), since the sub-instance exhibits OGP at much lower clause density, compared with the whole instance.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatorial algorithms
  • Random k-XORSAT
  • Sequential local algorithms
  • Average-case complexity
  • Phase transition
  • Overlap gap property


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  1. Emmanuel Abbe, Shuangping Li, and Allan Sly. Binary perceptron: Efficient algorithms can find solutions in a rare well-connected cluster. In Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2022, pages 860-873, New York, NY, USA, June 2022. Association for Computing Machinery. URL:
  2. Emmanuel Abbe, Shuangping Li, and Allan Sly. Proof of the Contiguity Conjecture and Lognormal Limit for the Symmetric Perceptron. In 2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS), pages 327-338, February 2022. URL:
  3. Dimitris Achlioptas and Amin Coja-Oghlan. Algorithmic Barriers from Phase Transitions. In 2008 49th Annual IEEE Symposium on Foundations of Computer Science, pages 793-802, Philadelphia, PA, USA, October 2008. IEEE. URL:
  4. Dimitris Achlioptas, Amin Coja-Oghlan, and Federico Ricci-Tersenghi. On the solution-space geometry of random constraint satisfaction problems. Random Structures & Algorithms, 38(3):251-268, May 2011. URL:
  5. Dimitris Achlioptas, Jeong Han Kim, Michael Krivelevich, and Prasad Tetali. Two-coloring random hypergraphs. Random Structures and Algorithms, 20(2):249-259, March 2002. URL:
  6. Dimitris Achlioptas and Michael Molloy. The solution space geometry of random linear equations. Random Structures & Algorithms, 46(2):197-231, March 2015. URL:
  7. A. Braunstein, M. Mézard, and R. Zecchina. Survey propagation: An algorithm for satisfiability. Random Structures and Algorithms, 27(2):201-226, September 2005. URL:
  8. Alfredo Braunstein and Riccardo Zecchina. Survey propagation as local equilibrium equations. Journal of Statistical Mechanics: Theory and Experiment, 2004(06):P06007, June 2004. URL:
  9. Guy Bresler and Brice Huang. The Algorithmic Phase Transition of Random k-SAT for Low Degree Polynomials. In 2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS), pages 298-309, Denver, CO, USA, February 2022. IEEE. URL:
  10. S. Cocco, O. Dubois, J. Mandler, and R. Monasson. Rigorous Decimation-Based Construction of Ground Pure States for Spin-Glass Models on Random Lattices. Physical Review Letters, 90(4):047205, January 2003. URL:
  11. Amin Coja-Oghlan. A Better Algorithm for Random k -SAT. SIAM Journal on Computing, 39(7):2823-2864, January 2010. URL:
  12. Amin Coja-Oghlan. Belief Propagation Guided Decimation Fails on Random Formulas. Journal of the ACM, 63(6):1-55, February 2017. URL:
  13. Martin Dietzfelbinger, Andreas Goerdt, Michael Mitzenmacher, Andrea Montanari, Rasmus Pagh, and Michael Rink. Tight Thresholds for Cuckoo Hashing via XORSAT. In Automata, Languages and Programming, volume 6198, pages 213-225. Springer Berlin Heidelberg, Berlin, Heidelberg, 2010. URL:
  14. Olivier Dubois and Jacques Mandler. The 3-XORSAT threshold. Comptes Rendus Mathematique, 335(11):963-966, December 2002. URL:
  15. David Gamarnik. The overlap gap property: A topological barrier to optimizing over random structures. Proceedings of the National Academy of Sciences, 118(41):e2108492118, October 2021. URL:
  16. David Gamarnik and Eren C. Kızıldağ. Algorithmic obstructions in the random number partitioning problem. The Annals of Applied Probability, 33(6B), December 2023. URL:
  17. David Gamarnik, Eren C. Kizildağ, Will Perkins, and Changji Xu. Geometric Barriers for Stable and Online Algorithms for Discrepancy Minimization. In Proceedings of Thirty Sixth Conference on Learning Theory, pages 3231-3263. PMLR, July 2023. Google Scholar
  18. David Gamarnik and Quan Li. Finding a large submatrix of a Gaussian random matrix. The Annals of Statistics, 46(6A), December 2018. URL:
  19. David Gamarnik and Madhu Sudan. Limits of local algorithms over sparse random graphs. The Annals of Probability, 45(4), July 2017. URL:
  20. David Gamarnik and Madhu Sudan. Performance of {Sequential Local Algorithms for the {Random NAE-K-SAT Problem. SIAM Journal on Computing, 46(2):590-619, January 2017. URL:
  21. Carla P. Gomes and Bart Selman. Satisfied with Physics. Science, 297(5582):784-785, August 2002. URL:
  22. Marco Guidetti and A. P. Young. Complexity of several constraint-satisfaction problems using the heuristic classical algorithm WalkSAT. Physical Review E, 84(1):011102, July 2011. URL:
  23. Samuel Hetterich. Analysing Survey Propagation Guided Decimation on Random Formulas, February 2016. URL:
  24. Brice Huang and Mark Sellke. Tight Lipschitz Hardness for optimizing Mean Field Spin Glasses. In 2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS), pages 312-322, Denver, CO, USA, October 2022. IEEE. URL:
  25. Morteza Ibrahimi, Yashodhan Kanoria, Matt Kraning, and Andrea Montanari. The Set of Solutions of Random XORSAT Formulae. In Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms, pages 760-779. Society for Industrial and Applied Mathematics, January 2012. URL:
  26. Jeong Han Kim. The Poisson Cloning Model for Random Graphs, Random Directed Graphs and Random k-SAT Problems. In Computing and Combinatorics, volume 3106, pages 2-2. Springer Berlin Heidelberg, Berlin, Heidelberg, 2004. URL:
  27. Lukas Kroc, Ashish Sabharwal, and Bart Selman. Message-passing and local heuristics as decimation strategies for satisfiability. In Proceedings of the 2009 ACM Symposium on Applied Computing, pages 1408-1414, Honolulu Hawaii, March 2009. ACM. URL:
  28. Elitza Maneva, Elchanan Mossel, and Martin J. Wainwright. A new look at survey propagation and its generalizations. Journal of the ACM, 54(4):17, July 2007. URL:
  29. M. Mézard, G. Parisi, and R. Zecchina. Analytic and Algorithmic Solution of Random Satisfiability Problems. Science, 297(5582):812-815, August 2002. URL:
  30. M. Mézard, F. Ricci-Tersenghi, and R. Zecchina. Two Solutions to Diluted p-Spin Models and XORSAT Problems. Journal of Statistical Physics, 111(3/4):505-533, 2003. URL:
  31. Marc Mézard and Andrea Montanari. Information, Physics, and Computation. Oxford Graduate Texts. Oxford University Press, Oxford ; New York, 2009. Google Scholar
  32. Michael Molloy. Cores in random hypergraphs and Boolean formulas. Random Structures and Algorithms, 27(1):124-135, August 2005. URL:
  33. Will Perkins and Changji Xu. Frozen 1-RSB structure of the symmetric Ising perceptron. In Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2021, pages 1579-1588, New York, NY, USA, June 2021. Association for Computing Machinery. URL:
  34. Boris Pittel and Gregory B. Sorkin. The Satisfiability Threshold for k -XORSAT. Combinatorics, Probability and Computing, 25(2):236-268, March 2016. URL:
  35. Boris Pittel, Joel Spencer, and Nicholas Wormald. Sudden Emergence of a Giant k-Core in a Random Graph. Journal of Combinatorial Theory, Series B, 67(1):111-151, May 1996. URL:
  36. Federico Ricci-Tersenghi and Guilhem Semerjian. On the cavity method for decimated random constraint satisfaction problems and the analysis of belief propagation guided decimation algorithms. Journal of Statistical Mechanics: Theory and Experiment, 2009(09):P09001, September 2009. URL:
  37. Alexander S. Wein. Optimal low-degree hardness of maximum independent set. Mathematical Statistics and Learning, 4(3):221-251, January 2022. URL: