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Abstract

Partial Maximum Boolean Satisfiability (Partial Max-SAT or PMSAT) is an optimization variant of Boolean satisfiability (SAT) problem, in which a variable assignment is required to satisfy all hard clauses and a maximum number of soft clauses in a Boolean formula. PMSAT is considered as an interesting encoding domain to many real-life problems for which a solution is acceptable even if some constraints are violated. Amongst the problems that can be formulated as such are planning and scheduling. New insights into the study of PMSAT problem have been gained since the introduction of the Max-SAT evaluations in 2006. Indeed, several PMSAT exact solvers have been developed based mainly on the Davis-Putnam-Logemann-Loveland (DPLL) procedure and Branch and Bound (B&B) algorithms. In this paper, we investigate and analyze a number of exact methods for PMSAT. We propose a taxonomy of the main exact methods within a general framework that integrates their various techniques into a unified perspective. We show its effectiveness by using it to classify PMSAT exact solvers which participated in the 2007 ~ 2011 Max-SAT evaluations, emphasizing on the most promising research directions.

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References

  1. Cook S A. The complexity of theorem-proving procedures. In Proc. the 3rd STOC, May 1971, pp.151-158.

  2. Kautz H, Selman B. Pushing the envelope: Planning, propositional logic, and stochastic search. In Proc. the 13th National Conf. Artificial Intelligence, Aug. 1996, pp.1194-1201.

  3. Gomes C P, Selman B, McAloon K, Tretkoff C. Randomization in backtrack search: Exploiting heavy-tailed profiles for solving hard scheduling problems. In Proc. the 4th AIPS, June 1998, pp.208-213.

  4. Biere A, Cimatti A, Clarke E M, Fujita M, Zhu Y. Symbolic model checking using SAT procedures instead of BDDs. In Proc. the 36th DAC, June 1999, pp.317-320.

  5. Abdulla P A, Bjesse P, Eén N. Symbolic reachability analysis based on SAT solvers. In Proc. the 6th TACAS, March 25-April 2, 2000, pp.411-425.

  6. Bjesse P, Leonard T, Mokkedem A. Finding bugs in an Alpha microprocessor using satisfiability solvers. In Proc. the 13th CAV, July 2001, pp.454-464.

  7. Jackson D, Schechter I, Shlyakhter I. Alcoa: The alloy constraint analyzer. In Proc. the 22nd ICSE, June 2000, pp.730-733.

  8. Clarke E M, Kroening D, Lerda F. A tool for checking ANSIC programs. In Proc. the 10th TACAS, Mar. 29-Apr. 2 2004, pp.168-176.

  9. Khurshid S, Marinov D. Testera: Specification-based testing of Java programs using SAT. Automated Software Engineering, 2004, 11(4): 403–434.

    Article  Google Scholar 

  10. Buro M, Büning H K. Report on a SAT competition. Bulletin of the European Association for Theoretical Computer Science, 1992, 49: 143–151.

    Google Scholar 

  11. Simon L, Le Berre D, Hirsch E A. The SAT2002 competition. Annals of Mathematics and Artificial Intelligence, 2005, 43(1): 307–342.

    Article  Google Scholar 

  12. Purdom P W, Le Berre D, Simon L. A parsimony tree for the SAT2002 competition. Annals of Mathematics and Artificial Intelligence, 2005, 43(1): 343–365.

    Article  Google Scholar 

  13. Marques-Silva J P, Sakallah K A. GRASP-A new search algorithm for satisfiability. In Proc. ICCAD 1996, Nov. 1996, pp.220-227.

  14. Moskewicz M W, Madigan C F, Zhao Y, Zhang L, Malik S. Chaff: Engineering an efficient SAT solver. In Proc. the 38th DAC, June 2001, pp.530-535.

  15. Goldberg E, Novikov Y. Berkmin: A fast and robust SAT-solver. In Proc. DATE 2002, Mar. 2002, pp.142-149.

  16. Nadel A. The Jerusat SAT solver [Master’s Thesis]. Hebrew University of Jerusalem, 2002.

  17. Eén N, Sörensson N. MiniSat: A SAT solver with conflict-clause minimization. In Proc. SAT 2005, June 2005, pp.502-518.

  18. Gu J. Efficient local search for very large-scale satisfiability problems. SIGART Bulletin, 1992, 3(1): 8–12.

    Article  Google Scholar 

  19. Gent I P, Walsh T. Towards an understanding of hill-climbing procedures for SAT. In Proc. the 11th AAAI, July 1993, pp.28-33.

  20. Ginsberg M L, McAllester D A. GSAT and dynamic back-tracking. In Proc. the 2nd PPCP 1994, May 1994, pp.243-265.

  21. Konolige K. Easy to be hard: Difficult problems for greedy algorithms. In Proc. the 4th KR, May 1994, pp.374-378.

  22. Cha B, Iwama K. Adding new clauses for faster local search. In Proc. the 13th AAAI, Aug. 1996, Vol.1, pp.332-337.

  23. Frank J, Cheeseman P, Stutz J. When gravity fails: Local search topology. Journal of Artificial Intelligence Research, 1997, 7: 249–281.

    MathSciNet  MATH  Google Scholar 

  24. Gu J, Purdom P W, Franco J, Wah B W. Algorithms for the satisfiability (SAT) problem: A survey. In Satisfiability Problem: Theory and Applications, Du D, Gu J, Pardalos R M (eds.), 1997, Vol.35, pp.19-152.

  25. Hoos H H. On the run-time behaviour of stochastic local search algorithms for SAT. In Proc. the 16th National Conf. Artificial Intelligence and 11th Innovation Applications of Artificial Intelligence, July 1999, pp.661-666.

  26. Hoos H H. An adaptive noise mechanism for walkSAT. In Proc. the 18th AAAI, July 28-Aug. 1, 2002, pp.655-660.

  27. Li X Y, Stallmann M F, Brglez F. A local search SAT solver using an effective switching strategy and an efficient unit propagation. In Proc. the 6th SAT, May 2003, pp.53-68.

  28. Hoos H, Stützle T. Stochastic Local Search: Foundations & Applications. San Francisco, CA, USA: Morgan Kaufmann, 2004.

    Google Scholar 

  29. Hirsch E A, Kojevnikov A. UnitWalk: A new SAT solver that uses local search guided by unit clause elimination. Annals of Mathematics and Artificial Intelligence, 2005, 43(1/4): 91–111.

    MathSciNet  MATH  Google Scholar 

  30. Mazure B, Saïs L, Grégoire E. Boosting complete techniques thanks to local search methods. Annals of Mathematics and Artificial Intelligence, 1998, 22(3/4): 319–331.

    Article  MathSciNet  MATH  Google Scholar 

  31. Rish I, Dechter R. To guess or to think? Hybrid algorithms for SAT. In Proc. the 2nd CP 1996, Aug. 1996, pp.555-556.

  32. Habet D, Li C M, Devendeville L, Vasquez M. A hybrid approach for SAT. In Proc. the 8th CP, Sept. 2002, pp.172-184.

  33. Anbulagan A, Pham D N, Slaney J, Sattar A. Old resolution meets modern SLS. In Proc. the 20th AAAI, July 2005, pp.354-359.

  34. Cook S A, Mitchell D G. Finding hard instances of the satisfiability problem: A survey. In Satisfiability Problem: Theory and Applications, Du D, Gu J, Pardalos P M (eds.), 1997, Vol.35, pp.1-17.

  35. Dixon H E, Ginsberg M L, Parkes A J. Generalizing Boolean satisfiability I: Background and survey of existing work. Artificial Intelligence Research, 2004, 21: 193–243.

    Article  MathSciNet  MATH  Google Scholar 

  36. Kautz H, Selman B. The state of SAT. Discrete Applied Mathematics, 2007, 155(12): 1514–1524.

    Article  MathSciNet  MATH  Google Scholar 

  37. Marques-Silva J. Practical applications of Boolean satisfiability. In Proc. the 9th WODES, May 2008, pp.74-80.

  38. Gomes C P, Kautz H, Sabharwal A, Selman B. Satisfiability solvers. In Foundations of Artificial Intelligence, van Harmelen F, Lifschit Z, Porter B (eds.), 2008, Vol.3, pp.89-134.

  39. Biere A, Heule M J, van Maaren H, Walsh T (eds.). Handbook of Satisfiability. Frontiers in Artificial Intelligence and Applications, Vol.185, IOS Press, 2009.

  40. Xu K, Boussemart F, Hemery F, Lecoutre C. Random constraint satisfaction: Easy generation of hard (satisfiable) instances. Artificial Intelligence, 2007, 171(8/9): 514–534.

    Article  MathSciNet  MATH  Google Scholar 

  41. Shostak R E. An algorithm for reasoning about equality. Communications of the ACM, 1978, 21(7): 583–585.

    Article  MathSciNet  MATH  Google Scholar 

  42. Shostak R E. A practical decision procedure for arithmetic with function symbols. Journal of the ACM, 1979, 26(2): 351–360.

    Article  MathSciNet  MATH  Google Scholar 

  43. Nelson G, Oppen D C. Simplification by cooperating decision procedures. ACM Transactions on Programming Languages and Systems, 1979, 1(2): 245–257.

    Article  MATH  Google Scholar 

  44. Nelson G, Oppen D C. Fast decision procedures based on congruence closure. Journal of the ACM, 1980, 27(2): 356–364.

    Article  MathSciNet  MATH  Google Scholar 

  45. Shostak R E. Deciding combinations of theories. Journal of the ACM, 1984, 31(1): 1–12.

    Article  MathSciNet  MATH  Google Scholar 

  46. Boyer R S, Moore J S. Integrating decision procedures into heuristic theorem provers: A case study of linear arithmetic. Machine Intelligence, Hayes J E, Michie D, Richards J (eds.), New York, NY, USA: Oxford University Press, Inc., 1988, pp.83-124.

  47. Boyer R S, Moore J S. A theorem prover for a computational logic. In Proc. the 10th CADE, July 1990, pp.1-15.

  48. Johnson D S. Approximation algorithms for combinatorial problems. Journal of Computer and System Sciences, 1974, 9(3): 256–278.

    Article  MathSciNet  MATH  Google Scholar 

  49. Kautz H A, Selman B. Planning as satisfiability. In Proc. the 10th ECAI, Aug. 1992, pp.359-363.

  50. Rintanen J, Heljanko K, Niemelä I. Planning as satisfiability: Parallel plans and algorithms for plan search. Artificial Intelligence, 2006, 170(12/13): 1031–1080.

    Article  MathSciNet  MATH  Google Scholar 

  51. Vasquez M, Hao J K. A “logic-constrained” knapsack formulation and a tabu algorithm for the daily photograph scheduling of an earth observation satellite. Computational Optimization and Applications, 2001, 20(2): 137–157.

    Article  MathSciNet  MATH  Google Scholar 

  52. Strickland D M, Barnes E R, Sokol J S. Optimal protein structure alignment using maximum cliques. Operations Research, 2005, 53(3): 389–402.

    Article  MathSciNet  MATH  Google Scholar 

  53. Miyazaki S, Iwama K, Kambayashi Y. Database queries as combinatorial optimization problems. In Proc. CODAS 1996, Dec. 1996, pp.477-483.

  54. Sandholm T. Algorithm for optimal winner determination in combinatorial auctions. Artificial Intelligence, 2002, 135(1/2): 1–54.

    Article  MathSciNet  MATH  Google Scholar 

  55. Nguyen T A, Perkins W A, Laffey T J, Pecora D. Checking an expert systems knowledge base for consistency and completeness. In Proc. the 9th IJCAI, Aug. 1985, pp.375-378.

  56. Li X Y, Stallmann M F, Berglez F. Optimization algorithms for the minimum-cost satisfiability problem [PhD Thesis]. North Carolina State University, 2004.

  57. Argelich J, Li C M, Manyà F, Planes J. First evaluation of Max-SAT solvers. http://www.iiia.csic.es/conferences/max-sat06/, 2006.

  58. Li C M, Manyà F. Max-SAT, hard and soft constraints. In Handbook of Satisfiability, Biere A, Heule M, van Maaren H, Walsh T (eds.), IOS Press, 2009, pp.613-631.

  59. Cha B, Iwama K, Kambayashi Y, Miyazaki S. Local search algorithms for partial MaxSAT. In Proc. the 14th AAAI and 9th IAAI, July 1997, pp.263-268.

  60. Kautz H, Selman B, Jiang Y. A general stochastic approach to solving problems with hard and soft constraints. In The Satisfiability Problem: Theory and Applications, Gu D, Du J, Pardalos P (eds.), American Mathematical Society, 1997, Vol.35, pp.573-586.

  61. Menai M E B. Solution reuse in Partial Max-SAT problem. In Proc. the 2004 IEEE-IRI, Nov. 2004, pp.481-486.

  62. Menai M E B. A two-phase backbone-based search heuristic for Partial Max-SAT: An initial investigation. In Proc. the 18th IEA/AIE, June 2005, pp.681-684.

  63. Menai M E B, Batouche M. A backbone-based co-evolutionary heuristic for Partial Max-SAT. In Proc. the 7th EA, Oct. 2005, pp.155-166.

  64. Mouhoub M. Systematic versus local search and GA techniques for incremental SAT. International Journal of Computational Intelligence and Applications, 2008, 7(1): 77–96.

    Article  MATH  Google Scholar 

  65. Argelich J, Li C M, Manyà F, Planes J. The first and second Max-SAT evaluations. Journal on Satisfiability, Boolean Modeling and Computation, 2008, 4(2/4): 251–278.

    MATH  Google Scholar 

  66. Argelich J, Li C M, Manyà F, Planes J. Third evaluation of Max-SAT solvers. http://maxsat.ia.udl.cat:81/08/, Feb. 2013.

  67. Argelich J, Li C M, Manyà F, Planes J. Fourth evaluation of Max-SAT solvers. http://maxsat.ia.udl.cat:81/09/, Feb. 2013.

  68. Argelich J, Li C M, Manyà F, Planes J. Fifth evaluation of Max-SAT solvers. http://maxsat.ia.udl.cat:81/10/, Feb. 2013.

  69. Argelich J, Li C M, Manyà F, Planes J. Sixth evaluation of Max-SAT solvers. http://maxsat.ia.udl.cat:81/11/, Feb. 2013.

  70. Robinson J A. A machine-oriented logic based on the resolution principle. Journal of the ACM, 1965, 12(1): 23–41.

    Article  MATH  Google Scholar 

  71. Davis M, Putnam H. A computing procedure for quantification theory. Journal of the ACM, 1960, 7(3): 201–215.

    Article  MathSciNet  MATH  Google Scholar 

  72. Davis M, Logemann G, Loveland D. A machine program for theorem-proving. Communications of the ACM, 1962, 5(7): 394–397.

    Article  MathSciNet  MATH  Google Scholar 

  73. Urquhart A. Hard examples for resolution. Journal of the ACM, 1987, 34(1): 209–219.

    Article  MathSciNet  MATH  Google Scholar 

  74. Chatalic P, Simon L. ZRES: The old Davis-Putman procedure meets ZBDD. In Proc. the 17th CADE, June 2000, pp.449-454.

  75. Land A H, Doig A G. An automatic method for solving discrete programming problems. Econometrica, 1960, 28(3): 497–520.

    Article  MathSciNet  MATH  Google Scholar 

  76. Kuegel A. Improved exact solver for the weighted Max-SAT problem. In Pragmatics of SAT Workshop of the SAT Conference, July 2010.

  77. Pipatsrisawat K, Darwiche A. Clone: Solving weighted Max-SAT in a reduced search space. In Proc. the 20th AUS-AI 2007, Dec. 2007, pp.223-233.

  78. Pipatsrisawat K, Palyan A, Chavira M, Choi A, Darwiche A. Solving weighted Max-SAT problems in a reduced search space: A performance analysis. Journal on Satisfiability, Boolean Modeling and Computation, 2008, 4(2/4): 191–217.

    MATH  Google Scholar 

  79. Lin H, Su K. Exploiting inference rules to compute lower bounds for Max-SAT solving. In Proc. the 20th IJCAI, Jan. 2007, pp.2334-2339.

  80. Lin H, Su K, Li C M. Within-problem learning for efficient lower bound computation in Max-SAT solving. In Proc. the 23rd AAAI, July 2008, pp.351-356.

  81. Heras F, Larrosa J, Oliveras A. MiniMaxSAT: A new weighted Max-SAT solver. In Proc. the 10th SAT, May 2007, pp.41-55.

  82. Heras F, Larrosa J, Oliveras A. MiniMaxSAT: An efficient weighted Max-SAT solver. Journal of Artificial Intelligence Research, 2008, 31: 1–32.

    MathSciNet  MATH  Google Scholar 

  83. Argelich J, Manyà F. Partial Max-SAT solvers with clause learning. In Proc. the 10th SAT, May 2007, pp.28-40.

  84. Ramírez M, Geffner H. Structural relaxations by variable renaming and their compilation for solving MinCostSAT. In Proc. the 13th CP, Sept. 2007, pp.605-619.

  85. de Givry S, Heras F, Zytnicki M, Larrosa J. Existential arc consistency: Getting closer to full arc consistency in weighted CSPs. In Proc. the 19th IJCAI, July 30-Aug. 5, 2005, pp.84-89.

  86. Heras F, Larrosa J. New inference rules for efficient Max-SAT solving. In Proc. the 21st AAAI, July 2006, pp.68-73.

  87. Larrosa J, Heras F, de Givry S. A logical approach to efficient Max-SAT solving. Artificial Intelligence, 2008, 172(2/3): 204–233.

    Article  MathSciNet  MATH  Google Scholar 

  88. Li C M, Manyà F, Mohamedou N, Planes J. Exploiting cycle structures in Max-SAT. In Proc. the 12th SAT, June 30-July 3, 2009, pp.467-480.

  89. Darras S, Dequen G, Devendeville L, Li C M. On inconsistent clause-subsets for Max-SAT solving. In Proc. the 13th CP, Sept. 2007, pp.225-240.

  90. Li C M, Manyà F, Planes J. Exploiting unit propagation to compute lower bounds in branch and bound Max-SAT solvers. In Proc. the 11th CP, Oct. 2005, pp.403-414.

  91. Li C M, Manyà M, Planes J. Detecting disjoint inconsistent subformulas for computing lower bounds for Max-SAT. In Proc. the 21st AAAI, July 2006, pp.86-91.

  92. Darwiche A, Marquis P. A knowledge compilation map. Artificial Intelligence Research, 2002, 17: 229–264.

    MathSciNet  MATH  Google Scholar 

  93. Darwiche A. New advances in compiling CNF into decomposable negational normal form. In Proc. the 16th ECAI, Aug. 2004, pp.328-332.

  94. Larrosa J, Heras F. Resolution in Max-SAT and its relation to local consistency in weighted CSPs. In Proc. the 19th IJCAI, July 30-Aug. 5, 2005, pp.193-198.

  95. de Givry S, Larrosa J, Meseguer P, Schiex T. Solving Max-SAT as weighted CSP. In Proc. the 9th CP, Sept. 29-Oct. 3, 2003, pp.363-376.

  96. Fu Z, Malik S. On solving the Partial Max-SAT problem. In Proc. the 9th SAT, Aug. 2006, pp.252-265.

  97. Marques-Silva J, Planes J. On using unsatisfiability for solving maximum satisfiability. Computing Research Repository, arXiv: 0712.1097, 2007.

  98. Marques-Silva J, Manquinho V. Towards more effective unsatisfiability-based maximum satisfiability algorithms. In Proc. the 11th SAT, May 2008, pp.225-230.

  99. Marques-Silva J, Planes J. Algorithms for maximum satisfiability using unsatisfiable cores. In Proc. DATE, Mar. 2008, pp.408-413.

  100. Manquinho V, Marques-Silva J, Planes J. Algorithms for weighted Boolean optimization. In Proc. the 12th SAT, June 30-July 3, 2009, pp.495-508.

  101. Ansótegui C, Bonet M L, Levy J. Solving (weighted) partial Max-SAT through satisfiability testing. In Proc. the 12th SAT, June 30-July 3, 2009, pp.427-440.

  102. Le Berre D, Parrain A. The Sat4j library (release 2.2). Journal on Satisfiability, Boolean Modeling and Computation, 2010, 7(2/3): 59–64.

    Google Scholar 

  103. Barth P. A Davis-Putnam based enumeration algorithm for linear pseudo-Boolean optimization. Technical Report MPI-I-95-2-003, Max-Planck-Institut fÄur Informatik, Im Stadtwald, D-66123 Saarbrücken, Germany, Jan. 1995.

  104. Xu L, Hutter F, Hoos H H, Leyton-Brown K. SATzilla: Portfolio-based algorithm selection for SAT. Journal of Artificial Intelligence Research, 2008, 32: 565–606.

    MATH  Google Scholar 

  105. Mouhoub M, Sadaoui S. Solving incremental satisfiability. International Journal on Artificial Intelligence Tools, 2007, 16(1): 139–147.

    Article  Google Scholar 

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Correspondence to Mohamed El Bachir Menai.

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This work was supported by the Research Center of College of Computer and Information Sciences at King Saud University, Saudi Arabia.

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El Bachir Menai, M., Al-Yahya, T.N. A Taxonomy of Exact Methods for Partial Max-SAT. J. Comput. Sci. Technol. 28, 232–246 (2013). https://doi.org/10.1007/s11390-013-1325-5

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