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Constraints

, Volume 18, Issue 4, pp 478–534 | Cite as

Iterative and core-guided MaxSAT solving: A survey and assessment

  • Antonio Morgado
  • Federico Heras
  • Mark Liffiton
  • Jordi Planes
  • Joao Marques-Silva
Survey

Abstract

Maximum Satisfiability (MaxSAT) is an optimization version of SAT, and many real world applications can be naturally encoded as such. Solving MaxSAT is an important problem from both a theoretical and a practical point of view. In recent years, there has been considerable interest in developing efficient algorithms and several families of algorithms have been proposed. This paper overviews recent approaches to handle MaxSAT and presents a survey of MaxSAT algorithms based on iteratively calling a SAT solver which are particularly effective to solve problems arising in industrial settings. First, classic algorithms based on iteratively calling a SAT solver and updating a bound are overviewed. Such algorithms are referred to as iterative MaxSAT algorithms. Then, more sophisticated algorithms that additionally take advantage of unsatisfiable cores are described, which are referred to as core-guided MaxSAT algorithms. Core-guided MaxSAT algorithms use the information provided by unsatisfiable cores to relax clauses on demand and to create simpler constraints. Finally, a comprehensive empirical study on non-random benchmarks is conducted, including not only the surveyed algorithms, but also other state-of-the-art MaxSAT solvers. The results indicate that (i) core-guided MaxSAT algorithms in general abort in less instances than classic solvers based on iteratively calling a SAT solver and that (ii) core-guided MaxSAT algorithms are fairly competitive compared to other approaches.

Keywords

MaxSAT MaxSMT Boolean optimization Optimization problems 

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References

  1. 1.
    Aksoy, L., daCosta, E.A.C., Flores, P.F., Monteiro, J. (2008). Exact and approximate algorithms for the optimization of area and delay in multiple constant multiplications. IEEE Transactions on CAD of Integrated Circuits and Systems on CAD, 27(6), 1013–1026.CrossRefGoogle Scholar
  2. 2.
    Aloul, F., Ramani, A., Markov, I., Sakallah, K. (2002). PBS: A backtrack search pseudo-Boolean solver. In Symposium on theory and applications of satisfiability testing (pp. 346–353).Google Scholar
  3. 3.
    Aloul, F.A., Ramani, A., Markov, I.L., Sakallah, K.A. (2002). Generic ILP versus specialized 0-1 ILP: An update. In International conference on computer-aided design (pp. 450–457).Google Scholar
  4. 4.
    Andres, B., Kaufmann, B., Matheis, O., Schaub, T. (2012). Unsatisfiability-based optimization in clasp. In International conference on logic programming (Technical communications) (pp. 211–221).Google Scholar
  5. 5.
    Anjos, M.F. (2006). Semidefinite optimization approaches for satisfiability and maximum-satisfiability problems. Journal on Satisfiability, Boolean Modeling and Computation, 1(1), 1–47.MathSciNetzbMATHGoogle Scholar
  6. 6.
    Ansótegui, C., Bonet, M.L., Gabàs, J., Levy, J. (2012). Improving SAT-based weighted MaxSAT solvers. In International conference on principles and practice of constraint programming (pp. 86–101).Google Scholar
  7. 7.
    Ansótegui, C., Bonet, M.L., Levy, J. (2009). On solving MaxSAT through SAT. In International conference of the Catalan Association for artificial intelligence (pp. 284–292).Google Scholar
  8. 8.
    Ansótegui, C., Bonet, M.L., Levy, J. (2009). Solving (weighted) partial MaxSAT through satisfiability testing. In International conference on theory and applications of satisfiability testing (pp. 427–440).Google Scholar
  9. 9.
    Ansótegui, C., Bonet, M.L., Levy, J. (2010). A new algorithm for weighted partial MaxSAT. In AAAI conference on artificial intelligence (pp. 3–8).Google Scholar
  10. 10.
    Ansótegui, C., Bonet, M.L., Levy, J. (2013). SAT-based MaxSAT algorithms. In Artificial inteligence journal (Vol. 196, pp. 77–105).Google Scholar
  11. 11.
    Ansótegui, C., & Manyà, F. (2004). Mapping problems with finite-domain variables into problems with Boolean variables. In International conference on theory and applications of satisfiability testing (pp. 111–119).Google Scholar
  12. 12.
    Ardagna, C.A., diVimercati, S.D.C., Foresti, S., Paraboschi, S., Samarati, P. (2010). Minimizing disclosure of private information in credential-based interactions: A graph-based approach. In International conference on social computing/international conference on privacy, security, risk and trust (pp. 743–750).Google Scholar
  13. 13.
    Argelich, J., Berre, D.L., Lynce, I., Marques-Silva, J., Rapicault, P. (2010). Solving linux upgradeability problems using Boolean optimization. In International workshop on logics for component configuration (pp. 11–22).Google Scholar
  14. 14.
    Argelich, J., Li, C.M., Manya, F., Planes, J. (2011). Experimenting with the instances of the MaxSAT Evaluation. In International conference of the catalan association for artificial intelligence (pp. 360–361).Google Scholar
  15. 15.
    Argelich, J., & Lynce, I. (2008). CNF instances from the software package installation problem. In RCRA international workshop on “Experimental Evaluation of Algorithms for solving problems with combinatorial explosion”.Google Scholar
  16. 16.
    Argelich, J., Lynce, I., Marques-Silva, J. (2009). On solving Boolean multilevel optimization problems. In International joint conference on artificial intelligence (pp. 393–398).Google Scholar
  17. 17.
    Asín, R., & Nieuwenhuis, R. (2010). Curriculum-based course timetabling with SAT and MaxSAT. In International conference on the practice and theory of automated timetabling (pp. 42–56).Google Scholar
  18. 18.
    Asín, R., & Nieuwenhuis, R. (2012). Curriculum-based course timetabling with SAT and MaxSAT. Annals of Operations Research, 1–21.Google Scholar
  19. 19.
    Asín, R., Nieuwenhuis, R., Oliveras, A., Rodríguez-Carbonell, E. (2011). Cardinality networks: a theoretical and empirical study. Constraints, 16(2), 195–221.MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Bailleux, O., & Boufkhad, Y. (2003). Efficient CNF encoding of Boolean cardinality constraints. In International conference on principles and practice of constraint programming (pp. 108–122).Google Scholar
  21. 21.
    Bansal, N., & Raman, V. (1999). Upper bounds for MaxSat: Further improved. In International symposium on algorithms and computation (pp. 247–258).Google Scholar
  22. 22.
    Barrett, C., Sebastiani, R., Seshia, S.A., Tinelli, C. (2009). Satisfiability modulo theories. In Handbook of satisfiability (pp. 825–884). IOS Press.Google Scholar
  23. 23.
    Barth, P. (1995). A Davis-Putnam enumeration algorithm for linear pseudo-Boolean optimization. Technical Report MPI-I-95-2-003, Max Plank Institute for Computer Science.Google Scholar
  24. 24.
    Batcher, K.E. (1968). Sorting networks and their applications. In AFIPS spring joint computing conference (pp. 307–314).Google Scholar
  25. 25.
    Berre, D.L., & Parrain, A. (2010). The Sat4j library, release 2.2. Journal on Satisfiability, Boolean Modeling and Computation, 7, 59–64.Google Scholar
  26. 26.
    Biere, A. (2008). PicoSAT essentials. Journal on Satisfiability, Boolean Modeling and Computation, 2, 75–97.Google Scholar
  27. 27.
    Birnbaum, E., & Lozinskii, E.L. (2003). Consistent subsets of inconsistent systems: structure and behaviour. Journal of Experimental and Theoretical Artificial Intelligence, 15(1), 25–46.zbMATHCrossRefGoogle Scholar
  28. 28.
    Bonet, M.L., Levy, J., Manyà, F. (2007). Resolution for Max-SAT. Artificial Intelligence Journal, 171(8–9), 606–618.zbMATHCrossRefGoogle Scholar
  29. 29.
    Borchers, B., & Furman, J. (1999). A two-phase exact algorithm for MAX-SAT and weighted MAX-SAT problems. Journal of Combinatorial Optimization, 2, 299–306.MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Brihaye, T., Bruyère, V., Doyen, L., Ducobu, M., Raskin, J.-F. (2011). Antichain-based QBF solving. In International symposium on automated technology for verification and analysis (pp. 183–197).Google Scholar
  31. 31.
    Cha, B., Iwama, K., Kambayashi, Y., Miyazaki, S. (1997). Local search algorithms for partial MaxSAT. In AAAI conference on artificial intelligence/IAAI innovative applications of artificial intelligence conference (pp. 263–268).Google Scholar
  32. 32.
    Chai, D., & Kuehlmann, A. (2003). A fast pseudo-Boolean constraint solver. In Design automation conference (pp. 830–835).Google Scholar
  33. 33.
    Chen, Y., Safarpour, S., Marques-Silva, J., Veneris, A.G. (2010). Automated design debugging with maximum satisfiability. IEEE Transactions on CAD of Integrated Circuits and Systems, 29(11), 1804–1817.CrossRefGoogle Scholar
  34. 34.
    Chen, Y., Safarpour, S., Veneris, A., Marques-Silva, J. (2009). Spatial and temporal design debug using partial MaxSAT. In IEEE great lakes symposium on VLSI.Google Scholar
  35. 35.
    Cimatti, A., Franzén, A., Griggio, A., Sebastiani, R., Stenico, C. (2010). Satisfiability modulo the theory of costs: Foundations and applications. In International conference tools and algorithms for the construction and analysis of systems (pp. 99–113).Google Scholar
  36. 36.
    Cimatti, A., Griggio, A., Schaafsma, B.J., Sebastiani, R. (2013). A modular approach to MaxSAT modulo theories. In International conference on theory and applications of satisfiability testing (pp. 150–165).Google Scholar
  37. 37.
    Codish, M., Lagoon, V., Stuckey, P.J. (2008). Logic programming with satisfiability. Journal of Theory and Practice of Logic Programming, 8(1), 121–128.zbMATHGoogle Scholar
  38. 38.
    Cooper, M.C., Cussat-Blanc, S., deRoquemaurel, M., Régnier, P. (2006). Soft arc consistency applied to optimal planning. In International conference on principles and practice of constraint programming (pp. 680–684).Google Scholar
  39. 39.
    Cooper, M.C., deGivry, S., Sanchez, M., Schiex, T., Zytnicki, M., Werner, T. (2010). Soft arc consistency revisited. Artificial Intelligence Journal, 174(7–8), 449–478.MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    Davies, J., & Bacchus, F. (2011). Solving MaxSAT by solving a sequence of simpler SAT instances. In International conference on principles and practice of constraint programming (pp. 225–239).Google Scholar
  41. 41.
    Davies, J., Cho, J., Bacchus, F. (2010). Using learnt clauses in MaxSAT. In International conference on principles and practice of constraint programming (pp. 176–190).Google Scholar
  42. 42.
    deGivry, S., Larrosa, J., Meseguer, P., Schiex, T. (2003). Solving Max-SAT as weighted CSP. In International conference on principles and practice of constraint programming (pp. 363–376).Google Scholar
  43. 43.
    deMoura, L.M., & Bjørner, N. (2008). Z3: An efficient SMT solver. In International conference tools and algorithms for the construction and analysis of systems (pp. 337–340).Google Scholar
  44. 44.
    Eén, N., & Sörensson, N. (2003). An extensible SAT-solver. In International conference on theory and applications of satisfiability testing (pp. 502–518).Google Scholar
  45. 45.
    Een, N., & Sörensson, N. (2006). Translating pseudo-Boolean constraints into SAT. Journal on Satisfiability, Boolean Modeling and Computation, 2, 1–26.zbMATHGoogle Scholar
  46. 46.
    Feldman, A., Provan, G., deKleer, J., Robert, S., van Gemund, A. (2010). Solving model-based diagnosis problems with MaxSAT solvers and vice versa. In International workshop on the principles of diagnosis.Google Scholar
  47. 47.
    Fu, Z., & Malik, S. (2006). On solving the partial MAX-SAT problem. In International conference on theory and applications of satisfiability testing (pp. 252–265).Google Scholar
  48. 48.
    Gebser, M., Kaufmann, B., Schaub, T. (2009). The conflict-driven answer set solver clasp: Progress report. In International conference on logic programming and nonmonotonic reasoning (pp. 509–514).Google Scholar
  49. 49.
    Gent, I.P., & Nightingale, P. (2004). A new encoding of alldifferent into SAT. In International workshop on modelling and reformulating constraint satisfaction problems (pp. 95–110).Google Scholar
  50. 50.
    Giunchiglia, E., Lierler, Y., Maratea, M. (2006). Answer set programming based on propositional satisfiability. Journal of Automated Reasoning, 36(4), 345–377.MathSciNetzbMATHCrossRefGoogle Scholar
  51. 51.
    Giunchiglia, E., & Maratea, M. (2006). Optsat: A tool for solving SAT related optimization problems. In European conference on logics in artificial intelligence (JELIA) (pp. 485–489).Google Scholar
  52. 52.
    Giunchiglia, E., & Maratea, M. (2006). Solving optimization problems with DLL. In European conference on artificial intelligence (pp. 377–381).Google Scholar
  53. 53.
    Giunchiglia, E., & Maratea, M. (2007). Planning as satisfiability with preferences. In AAAI conference on artificial intelligence (pp. 987–992).Google Scholar
  54. 54.
    Gomes, C.P., van Hoeve, W.J., Leahu, L. (2006). The power of semidefinite programming relaxations for Max-SAT. In International conference integration of AI and OR techniques in constraint programming for combinatorial optimization problems (pp. 104–118).Google Scholar
  55. 55.
    Gottlob, G. (1995). NP trees and Carnap’s modal logic. Journal of ACM, 42(2), 421–457.MathSciNetzbMATHCrossRefGoogle Scholar
  56. 56.
    Graca, A., Lynce, I., Marques-Silva, J., Oliveira, A. (2010). Efficient and accurate haplotype inference by combining parsimony and pedigree information. In International conference algebraic and numeric biology (pp. 38–56).Google Scholar
  57. 57.
    Graca, A., Marques-Silva, J., Lynce, I., Oliveira, A. (2011). Haplotype inference with pseudo-Boolean optimization. Annals of Operations Research, 184(1), 137–162.MathSciNetzbMATHCrossRefGoogle Scholar
  58. 58.
    Guerra, J., & Lynce, I. (2012). Reasoning over biological networks using maximum satisfiability. In International conference on principles and practice of constraint programming (pp. 941–956).Google Scholar
  59. 59.
    Hachtel, G.D., & Somenzi, F. (1996). Logic synthesis and verification algorithms. Kluwer.Google Scholar
  60. 60.
    Heras, F., Larrosa, J., deGivry, S., Schiex, T. (2008). 2006 and 2007 Max-SAT evaluations: contributed instances. Journal on Satisfiability, Boolean Modeling and Computation, 4(2–4), 239–250.zbMATHGoogle Scholar
  61. 61.
    Heras, F., Larrosa, J., Oliveras, A. (2008). MiniMaxSat: an efficient weighted Max-SAT solver. Journal of Artificial Intelligence Research, 31, 1–32.MathSciNetzbMATHGoogle Scholar
  62. 62.
    Heras, F., & Marques-Silva, J. (2011). Read-once resolution for unsatisfiability-based Max-SAT algorithms. In International joint conference on artificial intelligence (pp. 572–577).Google Scholar
  63. 63.
    Heras, F., Morgado, A., Marques-Silva, J. (2011). Core-guided binary search for maximum satisfiability. In AAAI conference on artificial intelligence.Google Scholar
  64. 64.
    Hoos, H., & Stützle, T. (2005). Stochastic local search: Foundations and applications. Morgan Kaufmann.Google Scholar
  65. 65.
    Hoos, H.H. (2002). An adaptive noise mechanism for WalkSAT. In AAAI conference on artificial intelligence/IAAI innovative applications of artificial intelligence conference (pp. 655–660).Google Scholar
  66. 66.
    Jose, M., & Majumdar, R. (2011). Bug-assist: Assisting fault localization in ANSI-C programs. In International conference on computer aided verification (pp. 504–509).Google Scholar
  67. 67.
    Jose, M., & Majumdar, R. (2011). Cause clue clauses: Error localization using maximum satisfiability. In ACM SIGPLAN conference on programming language design and implementation (pp. 437–446).Google Scholar
  68. 68.
    Juma, F., Hsu, E.I., McIlraith, S.A. (2012). Preference-based planning via MaxSAT. In Canadian conference on AI (pp. 109–120).Google Scholar
  69. 69.
    Koshimura, M., Zhang, T., Fujita, H., Hasegawa, R. (2012). QMaxSAT: a partial Max-SAT solver. Journal on Satisfiability, Boolean Modeling and Computation, 8, 95–100.MathSciNetGoogle Scholar
  70. 70.
    Kuegel, A. (2010). Improved exact solver for the weighted MAX-SAT problem. In Pragmatics of SAT.Google Scholar
  71. 71.
    Larrosa, J., & Heras, F. (2005). Resolution in Max-SAT and its relation to local consistency in weighted CSPs. In International joint conference on artificial intelligence (pp. 193–198).Google Scholar
  72. 72.
    Larrosa, J., Heras, F., deGivry, S. (2008). A logical approach to efficient Max-SAT solving. Artificial Inteligence Journal, 172(2–3), 204–233.MathSciNetzbMATHCrossRefGoogle Scholar
  73. 73.
    Larrosa, J., & Schiex, T. (2004). Solving weighted CSP by maintaining arc consistency. Artificial Inteligence Journal, 159(1–2), 1–26.MathSciNetzbMATHCrossRefGoogle Scholar
  74. 74.
    Li, C.M., & Manyà, F. (2009). MaxSAT, hard and soft constraints. In Handbook of satisfiability (pp. 613–632). IOS Press.Google Scholar
  75. 75.
    Li, C.M., Manyà, F., Planes, J. (2005). Exploiting unit propagation to compute lower bounds in branch and bound Max-SAT solvers. In International conference on principles and practice of constraint programming (pp. 403–414).Google Scholar
  76. 76.
    Li, C.M., Manyà, F., Planes, J. (2007). New inference rules for Max-SAT. Journal of Artificial Intelligence Research, 30, 321–359.MathSciNetGoogle Scholar
  77. 77.
    Liffiton, M.H., Sakallah, K.A. (2005). On finding all minimally unsatisfiable subformulas. In International conference on theory and applications of satisfiability testing (pp. 173–186).Google Scholar
  78. 78.
    Liffiton, M.H., & Sakallah, K.A. (2008). Algorithms for computing minimal unsatisfiable subsets of constraints. Journal Automated Reasoning, 40(1), 1–33.MathSciNetzbMATHCrossRefGoogle Scholar
  79. 79.
    Lin, H., Su, K., Li, C.M. (2008). Within-problem learning for efficient lower bound computation in Max-SAT solving. In AAAI conference on artificial intelligence (pp. 351–356).Google Scholar
  80. 80.
    Mancinelli, F., Boender, J., diCosmo, R., Vouillon, J., Durak, B., Leroy, X., Treinen, R. (2006). Managing the complexity of large free and open source package-based software distributions. In International conference on automated software engineering (pp. 199–208).Google Scholar
  81. 81.
    Mangassarian, H., Veneris, A.G., Safarpour, S., Najm, F.N., Abadir, M.S. (2007). Maximum circuit activity estimation using pseudo-Boolean satisfiability. In Conference on design, automation and test in Europe (pp. 1538–1543).Google Scholar
  82. 82.
    Manquinho, V., Marques-Silva, J., Planes, J. (2009). Algorithms for weighted Boolean optimization. In International conference on theory and applications of satisfiability testing (pp. 495–508).Google Scholar
  83. 83.
    Manquinho, V., Martins, R., Lynce, I. (2010). Improving unsatisfiability-based algorithms for Boolean optimization. In International conference on theory and applications of satisfiability testing (pp. 181–193).Google Scholar
  84. 84.
    Marques-Silva, J., Argelich, J., Graça, A., Lynce, I. (2011). Boolean lexicographic optimization: Algorithms & applications. Annals of Mathematics and Artificial Intelligence, 62(3–4), 317–343.MathSciNetzbMATHCrossRefGoogle Scholar
  85. 85.
    Marques-Silva, J., & Manquinho, V. (2008). Towards more effective unsatisfiability-based maximum satisfiability algorithms. In International conference on theory and applications of satisfiability testing (pp. 225–230).Google Scholar
  86. 86.
    Marques-Silva, J., & Planes, J. (2007). On using unsatisfiability for solving maximum satisfiability. Computing Research Repository. arXiv: abs/0712.0097.
  87. 87.
    Marques-Silva, J., Planes, J. (2008). Algorithms for maximum satisfiability using unsatisfiable cores. In Conference on design, automation and testing in Europe (pp. 408–413).Google Scholar
  88. 88.
    Marques-Silva, J., & Sakallah, K.A. (1996). GRASP—a new search algorithm for satisfiability. In International conference on computer-aided design (pp. 220–227).Google Scholar
  89. 89.
    Martins, R., Manquinho, V., Lynce, I. (2012). On partitioning for maximum satisfiability. In European conference on artificial intelligence (pp. 913–914).Google Scholar
  90. 90.
    Martins, R., Manquinho, V.M., Lynce, I. (2013). Community-based partitioning for MaxSAT solving. In International conference on theory and applications of satisfiability testing (pp. 182–191).Google Scholar
  91. 91.
    Morgado, A., Heras, F., Marques-Silva, J. (2011). The MSUnCore MaxSAT solver. In Pragmatics of SAT.Google Scholar
  92. 92.
    Morgado, A., Heras, F., Marques-Silva, J. (2012). Improvements to core-guided binary search for MaxSAT. In Theory and applications of satisfiability testing (pp. 284–297).Google Scholar
  93. 93.
    Moskewicz, M.W., Madigan, C.F., Zhao, Y., Zhang, L., Malik, S. (2001). Chaff: Engineering an efficient SAT solver. In Design automation conference (pp. 530–535).Google Scholar
  94. 94.
    Neveu, B., Trombettoni, G., Glover, F. (2004). ID Walk: A candidate list strategy with a simple diversification device. In International conference on principles and practice of constraint programming (pp. 423–437).Google Scholar
  95. 95.
    Niedermeier, R., Rossmanith, P. (2000). New upper bounds for maximum satisfiability. Journal of Algorithms, 36(1), 63–88.MathSciNetzbMATHCrossRefGoogle Scholar
  96. 96.
    Nieuwenhuis, R., & Oliveras, A. (2006). On SAT modulo theories and optimization problems. In International conference on theory and applications of satisfiability testing (pp. 156–169).Google Scholar
  97. 97.
    Nieuwenhuis, R., Oliveras, A., Tinelli, C. (2006). Solving SAT and SAT modulo theories: from an abstract Davis–Putnam–Logemann–Loveland procedure to DPLL(T). Journal of ACM, 53(6), 937–977.MathSciNetCrossRefGoogle Scholar
  98. 98.
    Oikarinen, E., Järvisalo, M. (2009). Max-ASP: Maximum satisfiability of answer set programs. In International conference on logic programming and nonmonotonic reasoning (pp. 236–249).Google Scholar
  99. 99.
    Palubeckis, G. (2009). A new bounding procedure and an improved exact algorithm for the MAX-2-SAT problem. Applied Mathematics and Computation, 215(3), 1106–1117.MathSciNetzbMATHCrossRefGoogle Scholar
  100. 100.
    Papadimitriou, C. (1994). Computational complexity. USA: Addison-Wesley.zbMATHGoogle Scholar
  101. 101.
    Papadimitriou, C., & Zachos, S. (1983). Two remarks on the power of counting. Theoretical Computer Science, 269–276.Google Scholar
  102. 102.
    Park, J.D. (2002). Using weighted MAX-SAT engines to solve MPE. In AAAI conference on artificial intelligence (pp. 682–687).Google Scholar
  103. 103.
    Pipatsrisawat, K., Palyan, A., Chavira, M., Choi, A., Darwiche, A. (2008). Solving weighted Max-SAT problems in a reduced search space: a performance analysis. Journal on Satisfiability Boolean Modeling and Computation, 4, 191–217.zbMATHGoogle Scholar
  104. 104.
    Prestwich, S. (2009). CNF encodings. In Handbook of satisfiability (pp. 75–98). IOS Press.Google Scholar
  105. 105.
    Prestwich, S.D. (2007). Variable dependency in local search: Prevention is better than cure. In International conference on theory and applications of satisfiability testing (pp. 107–120).Google Scholar
  106. 106.
    Ramírez, M., & Geffner, H. (2007). Structural relaxations by variable renaming and their compilation for solving MinCostSAT. In International conference on principles and practice of constraint programming (pp. 605–619).Google Scholar
  107. 107.
    Reiter, R. (1987). A theory of diagnosis from first principles. Artificial Inteligence Journal, 32(1), 57–95.MathSciNetzbMATHCrossRefGoogle Scholar
  108. 108.
    Robinson, N., Gretton, C., Pham, D.N., Sattar, A. (2010). Partial weighted MaxSAT for optimal planning. In Pacific rim international conference on artificial intelligence (pp. 231–243).Google Scholar
  109. 109.
    Rosa, E.D., Giunchiglia, E., Maratea, M. (2010). Solving satisfiability problems with preferences. Constraints, 15(4), 485–515.MathSciNetzbMATHCrossRefGoogle Scholar
  110. 110.
    Roussel, O., & Manquinho, V. (2009). Pseudo-Boolean and cardinality constraints. In Handbook of satisfiability (pp. 695–734). IOS Press.Google Scholar
  111. 111.
    Safarpour, S., Mangassarian, H., Veneris, A., Liffiton, M.H., Sakallah, K.A. (2007). Improved design debugging using maximum satisfiability. In Formal methods in computer-aided design.Google Scholar
  112. 112.
    Sandholm, T. (1999). An algorithm for optimal winner determination in combinatorial auctions. In International joint conference on artificial intelligence (pp. 542–547).Google Scholar
  113. 113.
    Sebastiani, R. (2007). Lazy satisfiability modulo theories. Journal on Satisfiability, Boolean Modeling and Computation, 3, 141–224.MathSciNetzbMATHGoogle Scholar
  114. 114.
    Sebastiani, R., & Tomasi, S. (2012). Optimization in SMT with LA(Q) cost functions. In International joint conference in automated reasoning (pp. 484–498).Google Scholar
  115. 115.
    Selman, B., Kautz, H.A., Cohen, B. (1994). Noise strategies for improving local search. In AAAI conference on artificial intelligence (pp. 337–343).Google Scholar
  116. 116.
    Selman, B., Levesque, H.J., Mitchell, D.G. (1992). A new method for solving hard satisfiability problems. In AAAI conference on artificial intelligence (pp. 440–446).Google Scholar
  117. 117.
    Sheini, H., & Sakallah, K. (2006). Pueblo: a hybrid pseudo-Boolean SAT solver. Journal on Satisfiability, Boolean Modeling and Computation, 2(3–4), 165–189.Google Scholar
  118. 118.
    Shen, H., & Zhang, H. (2005). Improving exact algorithms for MAX-2-SAT. Annals of Mathematics and Artificial Intelligence, 44(4), 419–436.MathSciNetzbMATHCrossRefGoogle Scholar
  119. 119.
    Sinz, C. (2005). Towards an optimal CNF encoding of Boolean cardinality constraints. In International conference on principles and practice of constraint programming (pp. 827–831).Google Scholar
  120. 120.
    Strickland, D., Barnes, E., Sokol, J. (2005). Optimal protein structure alignment using maximum cliques. Operations Research, 53(3), 389–402.MathSciNetzbMATHCrossRefGoogle Scholar
  121. 121.
    Teresa Alsinet, J.P., Manyà, F. (2004). A Max-SAT solver with lazy data structures. In Ibero-American conference on AI (IBERAMIA) (pp. 334–342).Google Scholar
  122. 122.
    Tompkins, D.A.D., & Hoos, H.H. (2004). UBCSAT: An implementation and experimentation environment for SLS algorithms for SAT & Max-SAT. In International conference on theory and applications of satisfiability testing (pp. 37–46).Google Scholar
  123. 123.
    Tucker, C., Shuffelton, D., Jhala, R., Lerner, S. (2007). OPIUM: Optimal package install/uninstall manager. In International conference on software engineering (pp. 178–188).Google Scholar
  124. 124.
    Vasquez, M., & Hao, J. (2001). A logic-constrained knapsack formulation and a tabu algorithm for the daily photograph scheduling of an earth observation satellite. Journal of Computational Optimization and Applications, 20(2), 137–157.MathSciNetzbMATHCrossRefGoogle Scholar
  125. 125.
    Warners, J.P. (1998). A linear-time transformation of linear inequalities into conjunctive normal form. Information Processing Letters, 68(2), 63–69.MathSciNetCrossRefGoogle Scholar
  126. 126.
    Xing, Z., & Zhang, W. (2005). MaxSolver: an efficient exact algorithm for (weighted) maximum satisfiability. Artificial Inteligence Journal, 164(1–2), 47–80.MathSciNetzbMATHCrossRefGoogle Scholar
  127. 127.
    Xu, H., Rutenbar, R., Sakallah, K. (2002). sub-SAT: A formulation for relaxed boolean satisfiability with applications in routing. In International symposium on physical design (pp. 182–187).Google Scholar
  128. 128.
    Zhang, L., & Malik, S. (2003). Validating SAT solvers using an independent resolution-based checker: Practical implementations and other applications. In Conference on design, automation and testing in Europe (pp. 10880–10885).Google Scholar
  129. 129.
    Zhang, L., & Bacchus, F. (2012). MaxSAT heuristics for cost optimal planning. In AAAI conference on artificial intelligence (pp. 1846–1852).Google Scholar
  130. 130.
    Zhu, C.S., Weissenbacher, G., Malik, S. (2011). Post-silicon fault localisation using maximum satisfiability and backbones. In Formal methods in computer-aided design (pp. 63–66).Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Antonio Morgado
    • 1
  • Federico Heras
    • 1
  • Mark Liffiton
    • 2
  • Jordi Planes
    • 3
  • Joao Marques-Silva
    • 1
    • 4
  1. 1.CSI/CASLUniversity College DublinDublinIreland
  2. 2.Illinois Wesleyan UniversityBloomingtonUSA
  3. 3.Universitat de LleidaLleidaSpain
  4. 4.IST/INESC-DUniversidade Tecnica de LisboaLisboaPortugal

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