Constraints

, Volume 18, Issue 2, pp 202–235 | Cite as

Reformulation based MaxSAT robustness

  • Miquel Bofill
  • Dídac Busquets
  • Víctor Muñoz
  • Mateu Villaret
Article
  • 179 Downloads

Abstract

The presence of uncertainty in the real world makes robustness a desirable property of solutions to constraint satisfaction problems (CSP). A solution is said to be robust if it can be easily repaired when unexpected events happen. This issue has already been addressed in the frameworks of Boolean satisfiability (SAT) and Constraint Programming (CP). Most existing works on robustness implement search algorithms to look for robust solutions instead of taking the declarative approach of reformulation, since reformulation tends to generate prohibitively large formulas, especially in the CP setting. In this paper we consider the unaddressed problem of robustness in weighted MaxSAT, by showing how robust solutions to weighted MaxSAT instances can be effectively obtained via reformulation into pseudo-Boolean formulae. Our encoding provides a reasonable balance between increase in size and performance, as shown by our experiments in the robust resource allocation framework. We also address the problem of flexible robustness, where some of the breakages may be left unrepaired if a totally robust solution does not exist. In a sense, since the use of SAT and MaxSAT encodings for solving CSP has been gaining wide acceptance in recent years, we provide an easy-to-implement new method for achieving robustness in combinatorial optimization problems.

Keywords

Robustness Reformulation MaxSAT Pseudo-Boolean Combinatorial optimization Resource allocation Auctions 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Abío, I., Deters, M., Nieuwenhuis, R., Stuckey, P.J. (2011). Reducing chaos in SAT-like search: Finding solutions close to a given one. In 14th international conference on theory and applications of satisfiability testing, SAT 2011, LNCS (Vol. 6695, pp. 273–286). Springer.Google Scholar
  2. 2.
    Ansótegui, C., Bonet, M.L., Levy, J. (2009). Solving (weighted) partial MaxSAT through satisfiability testing. In 12th international conference on theory and applications of satisfiability testing, SAT 2009, LNCS (Vol. 5584, pp. 427–440). Springer.Google Scholar
  3. 3.
    Argelich, J., Cabiscol, A., Lynce, I., Manyà, F. (2008). Modelling Max-CSP as partial Max-SAT. In 11th international conference on theory and applications of satisfiability testing, SAT 2008, LNCS (Vol. 4996, pp. 1–14). Springer.Google Scholar
  4. 4.
    Argelich, J., Li, C.M., Manyà, F., Planes, J. (2008). The first and second Max-SAT evaluations. Journal on Satisfiability, Boolean Modeling and Computation, 4(2–4), 251–278.MATHGoogle Scholar
  5. 5.
    Argelich, J., Lynce, I., Marques-Silva, J.P. (2009). On solving boolean multilevel optimization problems. In: 21st international joint conference on artificial intelligence, IJCAI 2009 (pp. 393–398).Google Scholar
  6. 6.
    Bertsimas, D., & Sim, M. (2004). The price of robustness. Operations Research, 52(1), 35–53.MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Bofill, M., Busquets, D., Villaret, M. (2010). A declarative approach to robust weighted Max-SAT. In: 12th international ACM SIGPLAN conference on principles and practice of declarative programming, PPDP 2010 (pp. 67–76). ACM.Google Scholar
  8. 8.
    Cramton, P., Shoham, Y., Steinberg, R. (Eds.) (2006). Combinatorial auctions. MIT Press.Google Scholar
  9. 9.
    Eén, N., & Sörensson, N. (2006). Translating pseudo-boolean constraints into SAT. Journal on Satisfiability, Boolean Modeling and Computation, 2(1–4), 1–26.MATHGoogle Scholar
  10. 10.
    Gebser, M., Kaminski, R., Kaufmann, B., Ostrowski, M., Schaub, T., Schneider, M. (2011). Potassco: the Potsdam answer set solving collection. AI Communications, 24(2), 105–124.MathSciNetGoogle Scholar
  11. 11.
    Ginsberg, M.L., Parkes, A.J., Roy, A. (1998). Supermodels and robustness. In: 15th national conference on artificial intelligence and 10th innovative applications of artificial intelligence conference, AAAI/IAAI 1998 (pp. 334–339). AAAI Press/The MIT Press.Google Scholar
  12. 12.
    Gu, Z., Rothberg, E., Bixby, R. (2010). Gurobi 4.0.2. http://www.gurobi.com. Accessed 10 Apr 2011.
  13. 13.
    Hebrard, E. (2006). Robust solutions for constraint satisfaction and optimisation under uncertainty. Ph.D. thesis, University of New South Wales.Google Scholar
  14. 14.
    Hebrard, E., Hnich, B., O’Sullivan, B., Walsh, T. (2005). Finding diverse and similar solutions in constraint programming. In: 20th national conference on artificial intelligence and 17th innovative applications of artificial intelligence conference, AAAI/IAAI 2005 (pp. 372–377). AAAI Press/The MIT Press.Google Scholar
  15. 15.
    Hebrard, E., Hnich, B., Walsh, T. (2004). Robust solutions for constraint satisfaction and optimization. In: 16th Eureopean conference on artificial intelligence, ECAI 2004 (pp. 186–190). IOS Press.Google Scholar
  16. 16.
    Hebrard, E., Hnich, B., Walsh, T. (2004). Super solutions in constraint programming. In: 8th international conference on integration of AI and OR techniques in constraint programming for combinatorial optimization problems, CPAIOR 2004, LNCS (Vol. 3011, pp. 157–172). Springer.Google Scholar
  17. 17.
    Hebrard, E., Hnich, B., Walsh, T. (2005). Improved algorithm for finding (a,b)-super solutions. In: Workshop on constraint programming for planning and scheduling (pp. 236–248).Google Scholar
  18. 18.
    Heras, F., Larrosa, J., de Givry, S., Schiex, T. (2008). 2006 and 2007 Max-SAT evaluations: contributed instances. Journal on Satisfiability, Boolean Modeling and Computation, 4(2–4), 239–250.MATHGoogle Scholar
  19. 19.
    Holland, A. (2005). Risk management for combinatorial auctions. Ph.D. thesis, Department of Computer Science, National University of Ireland, Cork.Google Scholar
  20. 20.
    Holland, A., & O’Sullivan, B. (2005). Robust solutions for combinatorial auctions. In: 6th ACM conference on electronic commerce, EC 2005 (pp. 183–192). ACM.Google Scholar
  21. 21.
    Holland, A., & O’Sullivan, B. (2005). Weighted super solutions for constraint programs. In: 20th national conference on artificial intelligence and 17th innovative applications of artificial intelligence conference, AAAI/IAAI 2005 (pp. 378–383). AAAI Press/The MIT Press.Google Scholar
  22. 22.
    Hoos, H.H., & Boutilier, C. (2000). Solving combinatorial auctions using stochastic local search. In: 17th national conference on artificial intelligence and 12th innovative applications of artificial intelligence conference, AAAI/IAAI 2000 (pp. 22–29). AAAI Press/The MIT Press.Google Scholar
  23. 23.
    IBM ILOG CPLEX Optimizer (2011). www-01.ibm.com/software/integration/optimization/cplex-optimizer/. Accessed 15 Nov 2011.
  24. 24.
    Le Berre, D., & Parrain, A. (2010). The Sat4j library, release 2.2. Journal on Satisfiability, Boolean Modeling and Computation, 7(2–3), 59–6.Google Scholar
  25. 25.
    Leyton-Brown, K., Pearson, M., Shoham, Y. (2000). Towards a universal test suite for combinatorial auction algorithms. In: ACM conference on electronic commerce, EC 2000 (pp. 66–76). ACM.Google Scholar
  26. 26.
    Li, C.M., & Manyà, F. (2009). Handbook of satisfiability (Chap. MaxSAT, Hard and Soft Constraints, pp. 613–631). IOS Press.Google Scholar
  27. 27.
    Liu, L., & Truszczynski, M. (2007). Satisfiability testing of boolean combinations of pseudo-boolean constraints using local-search techniques. Constraints, 12(3), 345–369.MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    Manquinho, V.M., Marques-Silva, J.P., Planes, J. (2009). Algorithms for weighted boolean optimization. In: 12th international conference on theory and applications of satisfiability testing, SAT 2009, LNCS (Vol. 5584, pp. 495–508). Springer.Google Scholar
  29. 29.
    Muñoz, V. (2011). Robustness in resource allocation problems. Ph.D. thesis, Department d’Enginyeria Elèctrica, Electrònica i Automàtica, Universitat de Girona, Spain.Google Scholar
  30. 30.
    Rossi, F., van Beek, P., Walsh, T. (Eds.) (2006) Handbook of constraint programming (Foundations of Artificial Intelligence). Elsevier Science.Google Scholar
  31. 31.
    Roussel, O., & Manquinho, V. (2009) Handbook of satisfiability (Chap. Pseudo-Boolean and Cardinality Constraints, pp. 695–734). IOS Press.Google Scholar
  32. 32.
    Roy, A. (2006). Fault tolerant boolean satisfiability. Journal of Artificial Intelligence Research, 25, 503–527.MathSciNetMATHGoogle Scholar
  33. 33.
    Soos, M. (2009). CryptoMiniSat—A SAT solver for cryptographic problems. http://planete.inrialpes.fr/~soos/CryptoMiniSat2/index.php. Accessed 18 Jan 2012.
  34. 34.
    Weigel, R., & Bliek, C. (1998). On reformulation of constraint satisfaction problems. In: 13th European conference on artificial intelligence, ECAI 1998 (pp. 254–258).Google Scholar
  35. 35.
    Williams, H.P. (1999). Model building in mathematical programming, 4th Edition. Wiley.Google Scholar

Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  • Miquel Bofill
    • 1
  • Dídac Busquets
    • 2
  • Víctor Muñoz
    • 3
  • Mateu Villaret
    • 1
  1. 1.Departament d’Informàtica i Matemàtica AplicadaUniversitat de GironaGironaSpain
  2. 2.Department of Electrical and Electronic EngineeringImperial College LondonLondonUK
  3. 3.Newronia SL, Parc Científic i TecnològicGironaSpain

Personalised recommendations