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Reduced Cost Fixing in MaxSAT

Part of the Lecture Notes in Computer Science book series (LNPSE,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.

Work supported in part by Academy of Finland (grants 251170 COIN, 276412, 284591 and 295673), the Research Funds and DoCS Doctoral School in Computer Science of the University of Helsinki, and the Natural Sciences and Engineering Research Council of Canada.

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Notes

  1. 1.

    The variables in the LP solution are either basic or non-basic. All of the non-basic variables will be at their upper or lower bounds in the LP solution [5].

  2. 2.

    In a hitting set problem \(b_i=1\) is always feasible. However, MaxHS can also add other constraints to the hitting set problem via a process of constraint seeding [9]. It is not difficult to show that all of our results continue to hold with seeding.

References

  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

    CrossRef  Google Scholar 

  2. Argelich, J., Li, C.M., Manyà, F., Planes, J.: MaxSAT evaluation (2016). http://maxsat.ia.udl.cat/introduction/. Accessed 27 Apr 2017

  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

    CrossRef  Google Scholar 

  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)

    MATH  Google Scholar 

  5. Chvátal, V.: Linear Programming. Freeman, New York (1983)

    MATH  Google Scholar 

  6. Crowder, H., Johnson, E.L., Padberg, M.: Solving large-scale zero-one linear programming problems. Oper. Res. 31(5), 803–834 (1983)

    CrossRef  MATH  Google Scholar 

  7. Danzig, G., Fulkerson, D., Johnson, S.: Solution of a large-scale traveling-salesman problem. Oper. Res. 2, 393–410 (1954)

    MathSciNet  Google Scholar 

  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

    CrossRef  Google Scholar 

  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

    CrossRef  Google Scholar 

  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. 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

    CrossRef  Google Scholar 

  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. 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

    CrossRef  Google Scholar 

  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

    CrossRef  Google Scholar 

  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

    CrossRef  Google Scholar 

  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. Li, C., Manyà, F.: MaxSAT, hard and soft constraints. In: Handbook of Satisfiability, pp. 613–631. IOS Press, Amsterdam (2009)

    Google Scholar 

  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. 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. 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)

    MathSciNet  CrossRef  MATH  Google Scholar 

  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

    CrossRef  Google Scholar 

  22. Nemhauser, G.L., Wolsey, L.A.: Integer and Combinatorial Optimization. Wiley-Interscience, Hoboken (1999)

    MATH  Google Scholar 

  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. 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. 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. 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)

    MathSciNet  CrossRef  MATH  Google Scholar 

  27. Wolsey, L.A.: Integer Programming. Wiley, Hoboken (1998)

    MATH  Google Scholar 

  28. Yunes, T.H., Aron, I.D., Hooker, J.N.: An integrated solver for optimization problems. Oper. Res. 58(2), 342–356 (2010)

    MathSciNet  CrossRef  MATH  Google Scholar 

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Correspondence to Fahiem Bacchus .

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Bacchus, F., Hyttinen, A., Järvisalo, M., Saikko, P. (2017). Reduced Cost Fixing in MaxSAT. In: Beck, J. (eds) Principles and Practice of Constraint Programming. CP 2017. Lecture Notes in Computer Science(), vol 10416. Springer, Cham. https://doi.org/10.1007/978-3-319-66158-2_41

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  • DOI: https://doi.org/10.1007/978-3-319-66158-2_41

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