Maximal Falsifiability

Definitions, Algorithms, and Applications
  • Alexey Ignatiev
  • Antonio Morgado
  • Jordi Planes
  • Joao Marques-Silva
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8312)


Similarly to Maximum Satisfiability (MaxSAT), Minimum Satisfiability (MinSAT) is an optimization extension of the Boolean Satisfiability (SAT) decision problem. In recent years, both problems have been studied in terms of exact and approximation algorithms. In addition, the MaxSAT problem has been characterized in terms ofMaximal Satisfiable Subsets (MSSes) andMinimal Correction Subsets (MCSes), as well as Minimal Unsatisfiable Subsets (MUSes) and minimal hitting set dualization. However, and in contrast with MaxSAT, no such characterizations exist for MinSAT. This paper addresses this issue by casting the MinSAT problem in a more general framework. The paper studies Maximal Falsifiability, the problem of computing a subset-maximal set of clauses that can be simultaneously falsified, and shows that MinSAT corresponds to the complement of a largest subset-maximal set of simultaneously falsifiable clauses, i.e. the solution of the Maximum Falsifiability (MaxFalse) problem. Additional contributions of the paper include novel algorithms for Maximum and Maximal Falsifiability, as well as minimal hitting set dualization results for the MaxFalse problem. Moreover, the proposed algorithms are validated on practical instances.


Vertex Cover Linear Search Minimum Vertex Cover Relaxation Variable Soft Clause 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Akkoyunlu, E.A.: The enumeration of maximal cliques of large graphs. SIAM J. Comput. 2(1), 1–6 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Angel, E., Bampis, E., Gourvès, L.: On the minimum hitting set of bundles problem. Theor. Comput. Sci. 410(45), 4534–4542 (2009)CrossRefzbMATHGoogle Scholar
  3. 3.
    Ansótegui, C., Bonet, M.L., Levy, J.: A new algorithm for weighted partial maxsat. In: AAAI (2010)Google Scholar
  4. 4.
    Ansótegui, C., Bonet, M.L., Levy, J.: Sat-based maxsat algorithms. Artif. Intell. 196, 77–105 (2013)CrossRefzbMATHGoogle Scholar
  5. 5.
    Ansotegui, C., Li, C.M., Manya, F., Zhu, Z.: A SAT-based approach to MinSAT. In: CCIA, pp. 185–189 (2012)Google Scholar
  6. 6.
    Argelich, J., Li, C.-M., Manyà, F., Zhu, Z.: MinSAT versus MaxSAT for optimization problems. In: Schulte, C. (ed.) CP 2013. LNCS, vol. 8124, pp. 133–142. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  7. 7.
    Audemard, G., Simon, L.: Predicting learnt clauses quality in modern sat solvers. In: IJCAI, pp. 399–404 (2009)Google Scholar
  8. 8.
    Avidor, A., Zwick, U.: Approximating MIN k-SAT. In: Bose, P., Morin, P. (eds.) ISAAC 2002. LNCS, vol. 2518, pp. 465–475. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  9. 9.
    Avidor, A., Zwick, U.: Approximating MIN 2-SAT and MIN 3-SAT. Theory Comput. Syst. 38(3), 329–345 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Bailey, J., Stuckey, P.J.: Discovery of minimal unsatisfiable subsets of constraints using hitting set dualization. In: Hermenegildo, M.V., Cabeza, D. (eds.) PADL 2004. LNCS, vol. 3350, pp. 174–186. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  11. 11.
    Biere, A., Heule, M., van Maaren, H., Walsh, T. (eds.): Handbook of Satisfiability, Frontiers in Artificial Intelligence and Applications, vol. 185. IOS Press (2009)Google Scholar
  12. 12.
    Birnbaum, E., Lozinskii, E.L.: Consistent subsets of inconsistent systems: structure and behaviour. J. Exp. Theor. Artif. Intell. 15(1), 25–46 (2003)CrossRefzbMATHGoogle Scholar
  13. 13.
    Bourke, C., Deng, K., Scott, S.D., Schapire, R.E., Vinodchandran, N.V.: On reoptimizing multi-class classifiers. Machine Learning 71(2-3), 219–242 (2008)CrossRefGoogle Scholar
  14. 14.
    Brihaye, T., Bruyère, V., Doyen, L., Ducobu, M., Raskin, J.-F.: Antichain-based QBF solving. In: Bultan, T., Hsiung, P.-A. (eds.) ATVA 2011. LNCS, vol. 6996, pp. 183–197. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  15. 15.
    Butman, A., Hermelin, D., Lewenstein, M., Rawitz, D.: Optimization problems in multiple-interval graphs. ACM Transactions on Algorithms 6(2) (2010)Google Scholar
  16. 16.
    Chen, T., Filkov, V., Skiena, S.: Identifying gene regulatory networks from experimental data. Parallel Computing 27(1-2), 141–162 (2001)CrossRefzbMATHGoogle Scholar
  17. 17.
    Eén, N., Sörensson, N.: An Extensible SAT-solver. In: Giunchiglia, E., Tacchella, A. (eds.) SAT 2003. LNCS, vol. 2919, pp. 502–518. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  18. 18.
    Gate, J., Stewart, I.A.: Frameworks for logically classifying polynomial-time optimisation problems. In: Ablayev, F., Mayr, E.W. (eds.) CSR 2010. LNCS, vol. 6072, pp. 120–131. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  19. 19.
    Goldstein, A., Kolman, P., Zheng, J.: Minimum common string partition problem: Hardness and approximations. Electr. J. Comb. 12 (2005)Google Scholar
  20. 20.
    Hassin, R., Monnot, J., Segev, D.: Approximation algorithms and hardness results for labeled connectivity problems. J. Comb. Optim. 14(4), 437–453 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Heras, F., Morgado, A., Planes, J., Marques-Silva, J.: Iterative SAT solving for minimum satisfiability. In: ICTAI, pp. 922–927 (2012)Google Scholar
  22. 22.
    Ignatiev, A., Janota, M., Marques-Silva, J.: Quantified maximum satisfiability: A core-guided approach. In: Järvisalo, M., Van Gelder, A. (eds.) SAT 2013. LNCS, vol. 7962, pp. 250–266. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  23. 23.
    Interian, Y., Corvera, G., Selman, B., Williams, R.: Finding small unsatisfiable cores to prove unsatisfiability of QBFs. In: ISAIM (2006)Google Scholar
  24. 24.
    Johnson, D.S., Papadimitriou, C.H., Yannakakis, M.: On generating all maximal independent sets. Inf. Process. Lett. 27(3), 119–123 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Karp, R.M., Wigderson, A.: A fast parallel algorithm for the maximal independent set problem. J. ACM 32(4), 762–773 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Kohli, R., Krishnamurti, R., Jedidi, K.: Subset-conjunctive rules for breast cancer diagnosis. Discrete Applied Mathematics 154(7), 1100–1112 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Kohli, R., Krishnamurti, R., Mirchandani, P.: The minimum satisfiability problem. SIAM J. Discrete Math. 7(2), 275–283 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Kügel, A.: Natural Max-SAT encoding of Min-SAT. In: Hamadi, Y., Schoenauer, M. (eds.) LION 6 2012. LNCS, vol. 7219, pp. 431–436. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  29. 29.
    Lawler, E.L., Lenstra, J.K., Kan, A.H.G.R.: Generating all maximal independent sets: NP-hardness and polynomial-time algorithms. SIAM J. Comput. 9(3), 558–565 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Li, C.M., Manya, F.: MaxSAT, hard and soft constraints. In: Biere, et al. (eds.) [11], pp. 613–631Google Scholar
  31. 31.
    Li, C.M., Manyà, F., Quan, Z., Zhu, Z.: Exact MinSAT solving. In: Strichman, O., Szeider, S. (eds.) SAT 2010. LNCS, vol. 6175, pp. 363–368. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  32. 32.
    Li, C.M., Quan, Z.: Combining graph structure exploitation and propositional reasoning for the maximum clique problem. In: ICTAI, pp. 344–351 (2010)Google Scholar
  33. 33.
    Li, C.M., Zhu, Z., Manya, F., Simon, L.: Minimum satisfiability and its applications. In: IJCAI, pp. 605–610 (2011)Google Scholar
  34. 34.
    Li, C.M., Zhu, Z., Manya, F., Simon, L.: Optimizing with minimum satisfiability. Artif. Intell. 190, 32–44 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Liffiton, M.H., Mneimneh, M.N., Lynce, I., Andraus, Z.S., Marques-Silva, J., Sakallah, K.A.: A branch and bound algorithm for extracting smallest minimal unsatisfiable subformulas. Constraints 14(4), 415–442 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Liffiton, M.H., Sakallah, K.A.: Algorithms for computing minimal unsatisfiable subsets of constraints. J. Autom. Reasoning 40(1), 1–33 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Marathe, M.V., Ravi, S.S.: On approximation algorithms for the minimum satisfiability problem. Inf. Process. Lett. 58(1), 23–29 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Marques-Silva, J., Heras, F., Janota, M., Previti, A., Belov, A.: On computing minimal correction subsets. In: IJCAI (to appear 2013)Google Scholar
  39. 39.
    Morgado, A., Heras, F., Liffiton, M.H., Planes, J., Marques-Silva, J.: Iterative and core-guided maxsat solving: A survey and assessment. Constraints 18(4), 478–534 (2013)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Morgado, A., Heras, F., Marques-Silva, J.: Improvements to core-guided binary search for maxsat. In: Cimatti, A., Sebastiani, R. (eds.) SAT 2012. LNCS, vol. 7317, pp. 284–297. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  41. 41.
    Morgado, A., Liffiton, M., Marques-Silva, J.: MaxSAT-based MCS enumeration. In: Biere, A., Nahir, A., Vos, T. (eds.) HVC. LNCS, vol. 7857, pp. 86–101. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  42. 42.
    Nöhrer, A., Biere, A., Egyed, A.: Managing SAT inconsistencies with HUMUS. In: VaMoS, pp. 83–91 (2012)Google Scholar
  43. 43.
    Reiter, R.: A theory of diagnosis from first principles. Artif. Intell. 32(1), 57–95 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Tsukiyama, S., Ide, M., Ariyoshi, H., Shirakawa, I.: A new algorithm for generating all the maximal independent sets. SIAM J. Comput. 6(3), 505–517 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Zhu, Z., Li, C.-M., Manyà, F., Argelich, J.: A new encoding from MinSAT into MaxSAT. In: Milano, M. (ed.) CP 2012. LNCS, vol. 7514, pp. 455–463. Springer, Heidelberg (2012)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Alexey Ignatiev
    • 1
    • 4
  • Antonio Morgado
    • 1
  • Jordi Planes
    • 3
  • Joao Marques-Silva
    • 1
    • 2
  1. 1.IST/INESC-IDLisbonPortugal
  2. 2.University College DublinIreland
  3. 3.Universitat de LleidaSpain
  4. 4.ISDCT SB RASIrkutskRussia

Personalised recommendations