Soft Computing

, Volume 23, Issue 9, pp 3095–3112 | Cite as

Bee swarm optimization for solving the MAXSAT problem using prior knowledge

  • Youcef DjenouriEmail author
  • Zineb Habbas
  • Djamel Djenouri
  • Philippe Fournier-Viger
Methodologies and Application


This paper explores rule decomposition for solving the MAXSAT problem. Four approaches are proposed to steer a bee swarm optimization metaheuristic. Two decomposition methods are proposed: direct and indirect. The first one applies the Kmeans algorithm, while the second one transforms a MAXSAT instance into a transactional database before performing decomposition using the Apriori algorithm. Several experiments conducted on DIMACS benchmark instances, and some other hard and large SAT instances have been carried out. Results show clear improvement compared to the state-of-the-art MAXSAT algorithms in terms of the quality of the obtained solutions. They show that the proposed approaches are stable when dealing with hard instances such as Parity8 from DIMACS. Results also demonstrate the superiority of the proposed approaches for medium and large instances. The proposed approaches could be applied to other optimization problems such as the weighted MAXSAT problem, the MAXCSP and coloring problems. They may also be adapted for other metaheuristics and decomposition methods.


Decomposition Bee swarm optimization Kmeans Apriori MAXSAT 


Compliance with ethical standards

Conflict of interest

The authors declare that there is no conflict of interests.

Human participants

This article does not involve studies with human participants.


  1. Abrame A, Habet D (2014) Local max-resolution in branch and bound solvers for max-sat. In: Tools with artificial intelligence (ICTAI), 2014 IEEE 26th international conference on IEEE, pp 336–343Google Scholar
  2. Abrame A, Habet D (2014) On the extension of learning for Max-SAT. In: STAIRS, pp 1–10Google Scholar
  3. Agrawal R, Imielinski T, Swami A (1993) Mining association rules between sets of items in large databases. In: Acm sigmod record, vol 22(2). ACM, pp 207–216Google Scholar
  4. Akbari R, Mohammadi A, Ziarati K (2010) A novel bee swarm optimization algorithm for numerical function optimization. Commun Nonlinear Sci Numer Simul 15(10):3142–3155MathSciNetCrossRefzbMATHGoogle Scholar
  5. Ali HM, Ashrafinia S, Liu J, Lee D (2013) Broadband wireless network planning using evolutionary algorithms. In: Evolutionary computation (CEC), 2013 IEEE congress on IEEE, pp 1045–1052Google Scholar
  6. Ali HM, Mitchell D, Lee DC (2014) MAX-SAT problem using evolutionary algorithms. In: Swarm intelligence (SIS), 2014 IEEE symposium on IEEE, pp 1–8Google Scholar
  7. Ansetegui C, Gireldez-Cru J, Levy J (2012) The community structure of SAT formulas. In: International conference on theory and applications of satisfiability testing. Springer, Berlin, pp 410–423Google Scholar
  8. Bouhmala N (2014) A variable neighborhood Walksat-based algorithm for MAX-SAT problems. Sci World J 2014.
  9. Bouhmala N (2015) A multilevel learning automata for MAX-SAT. Int J Mach Learn Cybernet 6(6):911–921CrossRefGoogle Scholar
  10. Cai S, Jie Z, Su K (2015) An effective variable selection heuristic in SLS for weighted Max-2-SAT. J Heuristics 21(3):433–456CrossRefGoogle Scholar
  11. Chen R, Santhanam R (2015) Improved algorithms for sparse MAX-SAT and MAX-k-CSP. In: International conference on theory and applications of satisfiability testing. Springer, pp 33–45Google Scholar
  12. Chicano F, Sutton AM, Whitley LD, Alba E (2015) Fitness probability distribution of bit-flip mutation. Evol Comput 23(2):217–248CrossRefGoogle Scholar
  13. Davis M, Putnam H (1960) A computing procedure for quantification theory. J ACM (JACM) 7(3):201–215MathSciNetCrossRefzbMATHGoogle Scholar
  14. Djeffal M, Drias H (2013) Multilevel bee swarm optimization for large satisfiability problem instances. In: Intelligent data engineering and automated learning IDEAL 2013. Springer, Berlin, pp 594–602Google Scholar
  15. Djenouri Y, Drias H, Habbas Z (2014) Bees swarm optimisation using multiple strategies for association rule mining. Int J Bio-Inspired Comput 6(4):239–249CrossRefGoogle Scholar
  16. Djenouri Y, Drias H, Bendjoudi A (2014) Pruning irrelevant association rules using knowledge mining. Int J Bus Intelli Data Min 9(2):112–144CrossRefGoogle Scholar
  17. Djenouri Y, Habbas Z, Aggoune-Mtalaa W (2016) Bees swarm optimization metaheuristic guided by decomposition for solving MAX-SAT. In: Proceedings of the 8th international conference on agents and artificial intelligence, pp. 472–479Google Scholar
  18. Djenouri Y, Habbas Z, Djenouri D (2017) Data mining-based decomposition for solving the MAXSAT problem: toward a new approach. IEEE Intell Syst 32(4):48–58CrossRefGoogle Scholar
  19. Drias H, Hireche C, Douib A (2013) Datamining techniques and swarm intelligence for problem solving: application to SAT. In: Nature and biologically inspired computing (NaBIC), 2013 World congress on IEEE, pp 200–206Google Scholar
  20. Drias H, Sadeg S, Yahi S (2005) Cooperative bees swarm for solving the maximum weighted satisfiability problem. In: Computational intelligence and bioinspired systems, vol 3512. Springer, Heidelberg, pp 417–448Google Scholar
  21. Escoffier B, Paschos VT, Tourniaire E (2014) Approximating Max Sat by moderately exponential and parameterized algorithms. Theoret Comput Sci 560:147–157MathSciNetCrossRefzbMATHGoogle Scholar
  22. Folino G, Pizzuti C, Spezzano G (2001) Parallel hybrid method for SAT that couples genetic algorithms and local search. Evolut Comput IEEE Trans 5(4):323–334CrossRefzbMATHGoogle Scholar
  23. Fontaine M, Loudni S, Boizumault P (2011) Guiding VNS with tree decomposition. In: Tools with artificial intelligence (ICTAI), 2011 23rd IEEE international conference on IEEE, pp 505–512Google Scholar
  24. Fukunaga AS (2004) Evolving local search heuristics for SAT using genetic programming. In: Genetic and evolutionary computation conference. Springer, Berlin, pp 483–494Google Scholar
  25. Jabbour S, Sais L, Salhi Y (2013) Boolean satisfiability for sequence mining. In: Proceedings of the 22nd ACM international conference on conference on information and knowledge management. ACM, pp 649–658Google Scholar
  26. Jabbour S, Sais L, Salhi Y (2015) Decomposition based SAT encodings for itemset mining problems. In: Advances in knowledge discovery and data mining. Springer, Berlin, pp 662–674Google Scholar
  27. Kanazawa K, Maruyama T (2014) FPGA acceleration of SAT/Max-SAT solving using variable-way cache. In: Field programmable logic and applications (FPL), 2014 24th international conference on IEEE, pp 1–4Google Scholar
  28. Kashan MH, Nahavandi N, Kashan AH (2012) DisABC: a new artificial bee colony algorithm for binary optimization. Appl Soft Comput 12(1):342–352CrossRefGoogle Scholar
  29. Kolokolov A, Adelshin A, Yagofarova D (2013) Analysis and solving SAT and MAX-SAT problems using an L-partition approach. J Math Modell Algorithms Oper Res 12(2):201–212MathSciNetzbMATHGoogle Scholar
  30. Kumar V (1992) Algorithms for constraint-satisfaction problems: a survey. AI Mag 13(1):32MathSciNetGoogle Scholar
  31. Lardeux F, Saubion F, Hao JK (2006) GASAT: a genetic local search algorithm for the satisfiability problem. Evol Comput 14(2):223–253CrossRefGoogle Scholar
  32. Li CM, Anbulagan A (1997) Heuristics based on unit propagation for satisfiability problems. In: Proceedings of the 15th international joint conference on artificial intelligence, vol 1. Morgan Kaufmann Publishers Inc, pp 366–371Google Scholar
  33. Luo C, Cai S, Wu W, Jie Z, Su K (2015) CCLS: an efficient local search algorithm for weighted maximum satisfiability. IEEE Trans Comput 64(7):1830–1843MathSciNetCrossRefzbMATHGoogle Scholar
  34. MacQueen J (1967) Some methods for classification and analysis of multivariate observations. In: Proceedings of the fifth Berkeley symposium on mathematical statistics and probability, vol 1(14), pp 281–297Google Scholar
  35. Molnar B, Ercsey-Ravasz M (2014) Analog dynamics for solving max-SAT problems. In: 2014 IEEE symposium on IEEEGoogle Scholar
  36. Park TJ, Van Gelder A (2000) Partitioning methods for satisfiability testing on large formulas. Inf Comput 162(1–2):179–184MathSciNetCrossRefzbMATHGoogle Scholar
  37. Poli R, Langdon WB, Holland O (2005) Extending particle swarm optimisation via genetic programming. In: European conference on genetic programming. Springer, Berlin, pp 291–300Google Scholar
  38. Poloczek M, Williamson DP, van Zuylen A (2014) On some recent approximation algorithms for MAX SAT. In Latin American symposium on theoretical informatics. Springer, Berlin, pp 598–609Google Scholar
  39. Sabar NR, Kendall G (2015) Population based Monte Carlo tree search hyper-heuristic for combinatorial optimization problems. Inf Sci 314:225–239CrossRefGoogle Scholar
  40. Sadowski KL, Bosman PA, Thierens D (2013) On the usefulness of linkage processing for solving MAX-SAT. In: Proceedings of the 15th annual conference on genetic and evolutionary computation. ACM, pp 853–860Google Scholar
  41. Sakai T, Seto K, Tamaki S (2015) Solving sparse instances of max SAT via width reduction and greedy restriction. Theory Comput Syst 57(2):426–443MathSciNetCrossRefzbMATHGoogle Scholar
  42. Selman B, Kautz H (1993) Domain-independent extensions to GSAT: solving large structured satisfiability problems. In: IJCAI. vol 93, pp 290–295Google Scholar
  43. Tompkins DA, Hoos HH (2004) UBCSAT: an implementation and experimentation environment for SLS algorithms for SAT and MAX-SAT. In: International conference on theory and applications of satisfiability testing. Springer, Berlin, pp 306–320Google Scholar
  44. Wu X, Kumar V, Quinlan JR, Ghosh J, Yang Q, Motoda H, Zhou ZH (2008) Top 10 algorithms in data mining. Knowl Inf Syst 14(1):1–37CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  • Youcef Djenouri
    • 1
    Email author
  • Zineb Habbas
    • 2
  • Djamel Djenouri
    • 3
  • Philippe Fournier-Viger
    • 4
  1. 1.IMADA, Mathematic and Computer Science DepartmentSouthern Denmark UniversityOdenseDenmark
  2. 2.Lorraine UniversityMetzFrance
  3. 3.CERIST Research CenterAlgiersAlgeria
  4. 4.Harbin Institute of Technology (Shenzhen)ShenzhenChina

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