Reduced Cost Fixing in MaxSAT

  • Fahiem Bacchus
  • Antti Hyttinen
  • Matti Järvisalo
  • Paul Saikko
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10416)

Abstract

We investigate utilizing the integer programming (IP) technique of reduced cost fixing to improve maximum satisfiability (MaxSAT) solving. In particular, we show how reduced cost fixing can be used within the implicit hitting set approach (IHS) for solving MaxSAT. Solvers based on IHS have proved to be quite effective for MaxSAT, especially on problems with a variety of clause weights. The unique feature of IHS solvers is that they utilize both SAT and IP techniques. We show how reduced cost fixing can be used in this framework to conclude that some soft clauses can be left falsified or forced to be satisfied without influencing the optimal cost. Applying these forcings simplifies the remaining problem. We provide an extensive empirical study showing that reduced cost fixing employed in this manner can be useful in improving the state-of-the-art in MaxSAT solving especially on hard instances arising from real-world application domains.

References

  1. 1.
    Ansótegui, C., Bonet, M.L., Gabàs, J., Levy, J.: Improving WPM2 for (weighted) partial MaxSAT. In: Schulte, C. (ed.) CP 2013. LNCS, vol. 8124, pp. 117–132. Springer, Heidelberg (2013). doi: 10.1007/978-3-642-40627-0_12 CrossRefGoogle Scholar
  2. 2.
    Argelich, J., Li, C.M., Manyà, F., Planes, J.: MaxSAT evaluation (2016). http://maxsat.ia.udl.cat/introduction/. Accessed 27 Apr 2017
  3. 3.
    Bajgiran, O.S., Cire, A.A., Rousseau, L.-M.: A first look at picking dual variables for maximizing reduced cost fixing. In: Salvagnin, D., Lombardi, M. (eds.) CPAIOR 2017. LNCS, vol. 10335, pp. 221–228. Springer, Cham (2017). doi: 10.1007/978-3-319-59776-8_18 CrossRefGoogle Scholar
  4. 4.
    Biere, A., Heule, M., van Maaren, H., Walsh, T. (eds.): Handbook of Satisfiability. Frontiers in Artificial Intelligence and Applications, vol. 185. IOS Press, Amsterdam (2009)MATHGoogle Scholar
  5. 5.
    Chvátal, V.: Linear Programming. Freeman, New York (1983)MATHGoogle Scholar
  6. 6.
    Crowder, H., Johnson, E.L., Padberg, M.: Solving large-scale zero-one linear programming problems. Oper. Res. 31(5), 803–834 (1983)CrossRefMATHGoogle Scholar
  7. 7.
    Danzig, G., Fulkerson, D., Johnson, S.: Solution of a large-scale traveling-salesman problem. Oper. Res. 2, 393–410 (1954)MathSciNetGoogle Scholar
  8. 8.
    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). doi: 10.1007/978-3-642-23786-7_19 CrossRefGoogle Scholar
  9. 9.
    Davies, J., Bacchus, F.: Exploiting the power of mip solvers in maxsat. In: Järvisalo, M., Van Gelder, A. (eds.) SAT 2013. LNCS, vol. 7962, pp. 166–181. Springer, Heidelberg (2013). doi: 10.1007/978-3-642-39071-5_13 CrossRefGoogle Scholar
  10. 10.
    Davies, J.: Solving MAXSAT by decoupling optimization and satisfaction. Ph.D. thesis, University of Toronto (2013). http://www.cs.toronto.edu/~jdavies/Davies_Jessica_E_201311_PhD_thesis.pdf
  11. 11.
    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). doi: 10.1007/978-3-642-40627-0_21 CrossRefGoogle Scholar
  12. 12.
    Focacci, F., Lodi, A., Milano, M.: Cost-based domain filtering. In: Jaffar, J. (ed.) CP 1999. LNCS, vol. 1713, pp. 189–203. Springer, Heidelberg (1999). doi: 10.1007/978-3-540-48085-3_14 Google Scholar
  13. 13.
    Heras, F., Morgado, A., Marques-Silva, J.: Lower bounds and upper bounds for MaxSAT. In: Hamadi, Y., Schoenauer, M. (eds.) LION 2012. LNCS, pp. 402–407. Springer, Heidelberg (2012). doi: 10.1007/978-3-642-34413-8_35 CrossRefGoogle Scholar
  14. 14.
    Karp, R.M.: Implicit hitting set problems and multi-genome alignment. In: Amir, A., Parida, L. (eds.) CPM 2010. LNCS, vol. 6129, p. 151. Springer, Heidelberg (2010). doi: 10.1007/978-3-642-13509-5_14 CrossRefGoogle Scholar
  15. 15.
    Li, C.M., Manyà, F., Mohamedou, N.O., Planes, J.: Transforming inconsistent subformulas in MaxSAT lower bound computation. In: Stuckey, P.J. (ed.) CP 2008. LNCS, vol. 5202, pp. 582–587. Springer, Heidelberg (2008). doi: 10.1007/978-3-540-85958-1_46 CrossRefGoogle Scholar
  16. 16.
    Li, C.M., Manyà, F., Planes, J.: Detecting disjoint inconsistent subformulas for computing lower bounds for Max-SAT. In: Proceedings of AAAI, pp. 86–91. AAAI Press (2006)Google Scholar
  17. 17.
    Li, C., Manyà, F.: MaxSAT, hard and soft constraints. In: Handbook of Satisfiability, pp. 613–631. IOS Press, Amsterdam (2009)Google Scholar
  18. 18.
    Lin, H., Su, K.: Exploiting inference rules to compute lower bounds for MAX-SAT solving. In: Proceedings of IJCAI, pp. 2334–2339 (2007)Google Scholar
  19. 19.
    Lin, H., Su, K., Li, C.M.: Within-problem learning for efficient lower bound computation in Max-SAT solving. In: Proceedings of AAAI, pp. 351–356. AAAI Press (2008)Google Scholar
  20. 20.
    Moreno-Centeno, E., Karp, R.M.: The implicit hitting set approach to solve combinatorial optimization problems with an application to multigenome alignment. Oper. Res. 61(2), 453–468 (2013)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    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). doi: 10.1007/978-3-642-31612-8_22 CrossRefGoogle Scholar
  22. 22.
    Nemhauser, G.L., Wolsey, L.A.: Integer and Combinatorial Optimization. Wiley-Interscience, Hoboken (1999)MATHGoogle Scholar
  23. 23.
    Saikko, P., Berg, J., Järvisalo, M.: LMHS: a SAT-IP hybrid MaxSAT solver. In: Creignou, N., Le Berre, D. (eds.) SAT 2016. LNCS, vol. 9710, pp. 539–546. Springer, Cham (2016). doi: 10.1007/978-3-319-40970-2_34 Google Scholar
  24. 24.
    Saikko, P.: Re-implementing and extending a hybrid SAT-IP approach to maximum satisfiability. Master’s thesis, University of Helsinki (2015). http://hdl.handle.net/10138/159186
  25. 25.
    Saikko, P., Wallner, J.P., Järvisalo, M.: Implicit hitting set algorithms for reasoning beyond NP. In: Proceedings of KR, pp. 104–113. AAAI Press (2016)Google Scholar
  26. 26.
    Thorsteinsson, E.S., Ottosson, G.: Linear relaxations and reduced-cost based propagation of continuous variable subscripts. Ann. Oper. Res. 115(1–4), 15–29 (2002)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Wolsey, L.A.: Integer Programming. Wiley, Hoboken (1998)MATHGoogle Scholar
  28. 28.
    Yunes, T.H., Aron, I.D., Hooker, J.N.: An integrated solver for optimization problems. Oper. Res. 58(2), 342–356 (2010)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Fahiem Bacchus
    • 1
  • Antti Hyttinen
    • 2
  • Matti Järvisalo
    • 2
  • Paul Saikko
    • 2
  1. 1.Department of Computer ScienceUniversity of TorontoTorontoCanada
  2. 2.HIIT, Department of Computer ScienceUniversity of HelsinkiHelsinkiFinland

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