Implicit Hitting Set Algorithms for Maximum Satisfiability Modulo Theories

  • Katalin Fazekas
  • Fahiem Bacchus
  • Armin Biere
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10900)


Solving optimization problems with SAT has a long tradition in the form of MaxSAT, which maximizes the weight of satisfied clauses in a propositional formula. The extension to maximum satisfiability modulo theories (MaxSMT) is less mature but allows problems to be formulated in a higher-level language closer to actual applications. In this paper we describe a new approach for solving MaxSMT based on lifting one of the currently most successful approaches for MaxSAT, the implicit hitting set approach, from the propositional level to SMT. We also provide a unifying view of how optimization, propositional reasoning, and theory reasoning can be combined in a MaxSMT solver. This leads to a generic framework that can be instantiated in different ways, subsuming existing work and supporting new approaches. Experiments with two instantiations clearly show the benefit of our generic framework.



This research has been supported by the Austrian Science Fund (FWF) under projects W1255-N23 and S11408-N23.


  1. 1.
    Li, Y., Albarghouthi, A., Kincaid, Z., Gurfinkel, A., Chechik, M.: Symbolic optimization with SMT solvers. In: POPL, pp. 607–618. ACM (2014)Google Scholar
  2. 2.
    Nieuwenhuis, R., Oliveras, A.: On SAT modulo theories and optimization problems. In: Biere, A., Gomes, C.P. (eds.) SAT 2006. LNCS, vol. 4121, pp. 156–169. Springer, Heidelberg (2006). Scholar
  3. 3.
    Cimatti, A., Griggio, A., Schaafsma, B.J., Sebastiani, R.: A modular approach to MaxSAT modulo theories. In: Järvisalo, M., Van Gelder, A. (eds.) SAT 2013. LNCS, vol. 7962, pp. 150–165. Springer, Heidelberg (2013). Scholar
  4. 4.
    Sebastiani, R., Tomasi, S.: Optimization modulo theories with linear rational costs. ACM Trans. Comput. Log. 16(2), 12:1–12:43 (2015)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bjørner, N., Phan, A.-D., Fleckenstein, L.: vZ - an optimizing SMT solver. In: Baier, C., Tinelli, C. (eds.) TACAS 2015. LNCS, vol. 9035, pp. 194–199. Springer, Heidelberg (2015). Scholar
  6. 6.
    Sebastiani, R., Trentin, P.: OptiMathSAT: a tool for optimization modulo theories. In: Kroening, D., Păsăreanu, C.S. (eds.) CAV 2015. LNCS, vol. 9206, pp. 447–454. Springer, Cham (2015). Scholar
  7. 7.
    Manolios, P., Pais, J., Papavasileiou, V.: The Inez mathematical programming modulo theories framework. In: Kroening, D., Păsăreanu, C.S. (eds.) CAV 2015. LNCS, vol. 9207, pp. 53–69. Springer, Cham (2015). Scholar
  8. 8.
    Sebastiani, R., Trentin, P.: On optimization modulo theories, MaxSMT and sorting networks. In: Legay, A., Margaria, T. (eds.) TACAS 2017. LNCS, vol. 10206, pp. 231–248. Springer, Heidelberg (2017). Scholar
  9. 9.
    Ansótegui, C., Bacchus, F., Järvisalo, M., Martins, R.: MaxSAT evaluation 2017 (2017).
  10. 10.
    Bacchus, F., Järvisalo, M.: Algorithms for maximum satisfiability with applications to AI. In: AAAI-2016 Tutoral (2016).
  11. 11.
    Davies, J., Bacchus, F.: Solving MAXSAT by solving a sequence of simpler SAT instances. In: Lee, J. (ed.) CP 2011. LNCS, vol. 6876, pp. 225–239. Springer, Heidelberg (2011). Scholar
  12. 12.
    Nieuwenhuis, R., Oliveras, A., Tinelli, C.: Solving SAT and SAT modulo theories: from an abstract Davis-Putnam-Logemann-Lovel and procedure to DPLL(T). J. ACM 53(6), 937–977 (2006)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Sebastiani, R.: Lazy satisability modulo theories. JSAT 3(3–4), 141–224 (2007)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Narodytska, N., Bacchus, F.: Maximum satisfiability using core-guided MaxSAT resolution. In: AAAI, pp. 2717–2723. AAAI Press (2014)Google Scholar
  15. 15.
    Martins, R., Joshi, S., Manquinho, V., Lynce, I.: Incremental cardinality constraints for MaxSAT. In: O’Sullivan, B. (ed.) CP 2014. LNCS, vol. 8656, pp. 531–548. Springer, Cham (2014). Scholar
  16. 16.
    Ansótegui, C., Bonet, M.L., Levy, J.: SAT-based MaxSAT algorithms. Artif. Intell. 196, 77–105 (2013)MathSciNetCrossRefGoogle 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). Scholar
  18. 18.
    Chandrasekaran, K., Karp, R.M., Moreno-Centeno, E., Vempala, S.: Algorithms for implicit hitting set problems. In: SODA, SIAM, pp. 614–629 (2011)CrossRefGoogle Scholar
  19. 19.
    Saikko, P., Wallner, J.P., Järvisalo, M.: Implicit hitting set algorithms for reasoning beyond NP. In: KR, pp. 104–113. AAAI Press (2016)Google Scholar
  20. 20.
    Davies, J., Bacchus, F.: Postponing optimization to speed up MAXSAT solving. In: Schulte, C. (ed.) CP 2013. LNCS, vol. 8124, pp. 247–262. Springer, Heidelberg (2013). Scholar
  21. 21.
    Lagniez, J.-M., Biere, A.: Factoring out assumptions to speed Up MUS extraction. In: Järvisalo, M., Van Gelder, A. (eds.) SAT 2013. LNCS, vol. 7962, pp. 276–292. Springer, Heidelberg (2013). Scholar
  22. 22.
    Ansótegui, C., Gabàs, J., Levy, J.: Exploiting subproblem optimization in SAT-based MaxSAT algorithms. J. Heuristics 22(1), 1–53 (2016)CrossRefGoogle Scholar
  23. 23.
    Cimatti, A., Griggio, A., Schaafsma, B.J., Sebastiani, R.: The MathSAT5 SMT solver. In: Piterman, N., Smolka, S.A. (eds.) TACAS 2013. LNCS, vol. 7795, pp. 93–107. Springer, Heidelberg (2013). Scholar
  24. 24.
    Bofill, M., Muñoz, V., Murillo, J.: Solving the wastewater treatment plant problem with SMT. CoRR abs/1609.05367 (2016)Google Scholar
  25. 25.
    Marques-Silva, J., Argelich, J., Graça, A., Lynce, I.: Boolean lexicographic optimization: algorithms & applications. Ann. Math. AI 62(3–4), 317–343 (2011)MathSciNetzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Johannes Kepler UniversityLinzAustria
  2. 2.University of TorontoTorontoCanada

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