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Hybrid classical and quantum-inspired features framework for MAX-3-SAT difficulty classification using machine learning

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Abstract

This study presents a hybrid classification framework that leverages both classical and quantum-inspired features to predict the complexity levels of MAX-3-SAT problem instances using supervised and unsupervised machine learning. The MAX-3-SAT problem, a canonical NP-complete optimization challenge, serves as a key benchmark for evaluating the effectiveness of optimization algorithms. The proposed approach utilizes classical structural features, such as the clause-to-variable ratio, alongside features derived from quantum-native representations of the problem, including interaction means and Hamiltonian coefficient sums. Although these Hamiltonian-based features can be computed classically for small instances using simulators, they are based on quantum encodings employed by variational quantum algorithms. The variational quantum eigensolver (VQE) is employed not to solve the optimization problem directly but to extract quantum-derived features such as ground-state energy and interaction metrics from Hamiltonians. By integrating these feature sets, the framework achieves high classification accuracy across four difficulty levels. The results indicate that combining classical and quantum-inspired features provides deeper insights into problem complexity, supports adaptive decision-making, and enhances the prioritization of quantum resources.

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References

  1. Belova T, Bliznets I (2020) Algorithms for (n,3)-MAXSAT and parameterization above the all-true assignment. Theor Comput Sci 803:222–233. https://doi.org/10.1016/j.tcs.2019.11.033

    Article  MathSciNet  MATH  Google Scholar 

  2. Cook SA (1971) The complexity of theorem-proving procedures. In: Proceedings of the Third Annual ACM Symposium on Theory of Computing, pp. 151–158. Association for Computing Machinery, New York, NY, USA

  3. Trakhtenbrot BA (1984) A survey of russian approaches to perebor (brute-force searches) algorithms. Annal Hist Comput 6(4):384–400. https://doi.org/10.1109/MAHC.1984.10036

    Article  MATH  Google Scholar 

  4. Vu HT (2024) Revisiting maximum satisfiability and related problems in data streams. Theor Comput Sci 982:114271. https://doi.org/10.1016/j.tcs.2023.114271

    Article  MathSciNet  MATH  Google Scholar 

  5. Coll J, Li C-M, Manyà F, Yangin E (2023) MaxSAT resolution for regular propositional logic. Int J Approx Reason 162:109010. https://doi.org/10.1016/j.ijar.2023.109010

    Article  MathSciNet  MATH  Google Scholar 

  6. Wang X, Jiang J (2019) Warning propagation algorithm for the MAX-3-SAT problem. IEEE Trans Emerg Top Comput 7(4):578–584. https://doi.org/10.1109/TETC.2017.2736504

    Article  MATH  Google Scholar 

  7. Fernandez de la Vega W, Karpinski M (2007) 1.0957-Approximation algorithm for random MAX-3SAT. RAIRO - Oper Res 41(1):95–103. https://doi.org/10.1051/ro:2007008

    Article  MathSciNet  MATH  Google Scholar 

  8. Karloff H, Zwick U (1997) A 7/8-approximation algorithm for max 3sat? In: Proceedings of the 38th Annual Symposium on Foundations of Computer Science, pp. 406–415. IEEE Computer Society, Los Alamitos, CA, USA

  9. Ono T, Hirata T, Asano T (1995) An approximation algorithm for MAX 3-SAT. In: Staples J, Eades P, Katoh N, Moffat A (eds) Algorithms and computations. Springer, Berlin, Heidelberg, pp 163–170

    Chapter  MATH  Google Scholar 

  10. Bouhmala N (2014) A variable neighborhood Walksat-based algorithm for MAX-SAT problems. Sci World J 2014:798323. https://doi.org/10.1155/2014/798323

    Article  MATH  Google Scholar 

  11. Luo W, Wang J, Hu Y, Xu P (2020) Solving dynamic 3-sat formula: an empirical study. In: 2020 3rd International Conference on Data Intelligence and Security (ICDIS), pp. 134–140. IEEE, New York, NY, USA https://doi.org/10.1109/ICDIS50059.2020.00024

  12. Darmann A, Döcker J (2021) On simplified NP-complete variants of Monotone 3 -Sat. Discret Appl Math 292:45–58. https://doi.org/10.1016/j.dam.2020.12.010

    Article  MathSciNet  MATH  Google Scholar 

  13. Weiss A (2022) A Polynomial Decision for 3-sat. arXiv preprint arXiv:2208.12598

  14. Herrman R, Treffert L, Ostrowski J, Lotshaw PC, Humble TS, Siopsis G (2021) Globally optimizing QAOA circuit depth for constrained optimization problems. Algorithms 14(10):294. https://doi.org/10.3390/a14100294

    Article  MATH  Google Scholar 

  15. Nüßlein J, Zielinski S, Gabor T, Linnhoff-Popien C, Feld S (2023) Solving (max) 3-SAT via quadratic unconstrained binary optimization. In: Mikyška J, Mulatier C, Paszynski M, Krzhizhanovskaya VV, Dongarra JJ, Sloot PMA (eds) Proceedings of the International Conference on Computational Science, pp. 34–47. Springer, Cham, Switzerland

  16. Yu Y, Cao C, Wang X-B, Shannon N, Joynt R (2023) Solution of SAT problems with the adaptive-bias quantum approximate optimization algorithm. Phys Rev Res 5(2):23147. https://doi.org/10.1103/PhysRevResearch.5.023147

    Article  Google Scholar 

  17. Jeong S, Kim M, Hhan M, Park J, Ahn J (2023) Quantum programming of the satisfiability problem with Rydberg atom graphs. Phys Rev Res 5(4):43037. https://doi.org/10.1103/PhysRevResearch.5.043037

    Article  MATH  Google Scholar 

  18. Paulet JJ, Lana LF, Calvo HI, Mezzini M, Cuartero F, Pelayo FL (2023) Heuristics for quantum computing dealing with 3-SAT. Mathematics 11(8):1888. https://doi.org/10.3390/math11081888

    Article  MATH  Google Scholar 

  19. Varmantchaonala CM, Fendji JLKE, Njafa JPT, Atemkeng M (2023) Quantum hybrid algorithm for solving SAT problem. Eng Appl Artif Intell 121:106058. https://doi.org/10.1016/j.engappai.2023.106058

    Article  Google Scholar 

  20. Zhang Z, Paredes R, Sundar B, Quiroga D, Kyrillidis A, Duenas-Osorio L, Pagano G, Hazzard KRA (2024) Grover-QAOA for 3-SAT: Quadratic speedup, fair-sampling, and parameter clustering. arXiv preprint arXiv:2402.02585

  21. Ebdelrehime E, Younes A, Elkabani I (2024) Quantum answer set programming solver using amplitude amplification. In: 2024 International Conference on Machine Intelligence and Smart Innovation (ICMISI), pp. 168–173. IEEE, New York, NY, USA https://doi.org/10.1109/ICMISI61517.2024.10580520

  22. Rodríguez-Farrés P, Ballester R, Ansótegui C, Levy J, Cerquides J (2024) Implementing 3-sat gadgets for quantum annealers with random instances. In: Franco L, Mulatier C, Paszynski M, et al (eds) Proceedings of the International Conference on High-Performance Computing, pp. 277–291. Springer, Cham, Switzerland

  23. AbuGhanem M, Eleuch H (2022) NISQ computers: a path to quantum supremacy. IEEE Access 12:102941–102961

    Article  MATH  Google Scholar 

  24. Ansótegui C, Bonet ML, Giráldez-Cru J, Levy J (2017) Structure features for SAT instances classification. J Appl Logic 23:27–39. https://doi.org/10.1016/j.jal.2016.11.004

    Article  MathSciNet  MATH  Google Scholar 

  25. Atkari A, Dhargalkar N, Angne, H (2020) Employing machine learning models to solve uniform random 3-SAT. In: Jain LC, Tsihrintzis GA, Balas VE, Sharma DK (eds) Proceedings of the International Conference on Applied Informatics, pp. 255–264. Springer, Singapore

  26. Danisovszky M, Yang Z, Kusper G (2020) Classification of sat problem instances by machine learning methods. In: Proceedings of the 11th International Conference on Applied Informatics. CEUR Workshop Proceedings, Eger, Hungary

  27. Chang W, Zhang H, Luo J (2022) Predicting propositional satisfiability based on graph attention networks. Int J Comput Intell Syst 15(1):84. https://doi.org/10.1007/s44196-022-00139-9

    Article  MATH  Google Scholar 

  28. Yau M, Lu E, Karalias N, Xu J, Jegelka S (2023) Are Graph Neural Networks Optimal Approximation Algorithms? arXiv preprint arXiv:2310.00526

  29. Guo W, Zhen H-L, Li X, Luo W, Yuan M, Jin Y, Yan J (2023) Machine learning methods in solving the boolean satisfiability problem. Mach Intell Res 20(5):640–655. https://doi.org/10.1007/s11633-022-1396-2

    Article  MATH  Google Scholar 

  30. Bergin R, Dalla M, Visentin A, O’Sullivan B, Provan G (2023) Using machine learning classifiers in SAT branching. In: Proceedings of the International Symposium on Combinatorial Search. vol. 16, pp. 169–170. https://doi.org/10.1609/socs.v16i1.27298

  31. Austrin P, Brown-Cohen J, Hastad J (2023) Optimal inapproximability with universal factor graphs. ACM Trans Algorithms. https://doi.org/10.1145/3631119

    Article  MATH  Google Scholar 

  32. Lykov D, Wurtz J, Poole C, Saffman M, Noel T, Alexeev Y (2023) Sampling frequency thresholds for the quantum advantage of the quantum approximate optimization algorithm. NPJ Quant Inf 9(1):73. https://doi.org/10.1038/s41534-023-00718-4

    Article  Google Scholar 

  33. Zielinski S, Zorn M, Gabor T, Feld S, Linnhoff-Popien C (2024) Using an evolutionary algorithm to create (max)-3sat qubos. In: Proceedings of the Genetic and Evolutionary Computation Conference Companion, pp. 1984–1992. Association for Computing Machinery, New York, NY, USA https://doi.org/10.1145/3638530.3664153

  34. Sawaya NPD, Marti-Dafcik D, Ho Y, Tabor DP, Neira DEB, Magann AB, Premaratne S, Dubey P, Matsuura A, Bishop N, Jong WAD, Benjamin S, Parekh OD, Tubman NM, Klymko K, Camps D (2023) Hamlib: a library of hamiltonians for benchmarking quantum algorithms and hardware. In: 2023 IEEE International Conference on Quantum Computing and Engineering (QCE), pp. 389–390. IEEE, New York, NY, USA https://doi.org/10.1109/QCE57702.2023.10296

  35. Ren W, Li W, Xu S, Wang K, Jiang W, Jin F, Zhu X, Chen J, Song Z, Zhang P, Dong H, Zhang X, Deng J, Gao Y, Zhang C, Wu Y, Zhang B, Guo Q, Li H, Wang Z, Biamonte J, Song C, Deng D-L, Wang H (2022) Experimental quantum adversarial learning with programmable superconducting qubits. Nat Comput Sci 2:711. https://doi.org/10.1038/s43588-022-00351-9

    Article  MATH  Google Scholar 

  36. Wang K, Li W, Xu S, Hu M, Chen J, Wu Y, Zhang C, Jin F, Zhu X, Gao Y, Tan Z, Zhang A, Wang N, Zou Y, Li T, Shen F, Zhong J, Bao Z, Zhu Z, Song Z, Deng J, Dong H, Zhang X, Zhang P, Jiang W, Lu Z, Sun Z-Z, Li H, Guo Q, Wang Z, Emonts P, Tura J, Song C, Wang H, Deng D-L (2024) Probing many-body Bell correlation depth with superconducting qubits. arXiv:2406.17841

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  1. College of Natural Sciences, The University of Texas at Austin, 120 Inner Campus Dr, Austin, TX, 78712, USA

    David Andres Garcia-Barrios

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Garcia-Barrios, D.A. Hybrid classical and quantum-inspired features framework for MAX-3-SAT difficulty classification using machine learning. J Supercomput 81, 578 (2025). https://doi.org/10.1007/s11227-025-07092-2

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