Skip to main content
SpringerLink
Log in

Algorithms for the Satisfiability Problem

  • Reference work entry
  • First Online:
Handbook of Combinatorial Optimization

  • 7689 Accesses

Abstract

This chapter discusses advances in SAT algorithm design, including the use of SAT algorithms as theory drivers, classic implementations of SAT solvers, and some theoretical aspects of SAT. Some applications to which SAT solvers have been successfully applied are also presented. The intention is to assist someone interested in applying SAT technology in solving some stubborn class of combinatorial problems.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 3,400.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 549.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

Notes

  1. 1.

    Each satisfiable set of Horn clauses has a unique minimal model with respect to 1, which can be computed in linear time by a well-known algorithm [53, 78] which is discussed in Sect. 6.2. Implications of the minimal model semantics may be found in Sect. 6.5, among others.

  2. 2.

    Horn formulas are solved efficiently (see Sect. 6.2).

  3. 3.

    Actual circuit voltage levels are abstracted to the values 0 and 1.

  4. 4.

    The meaning of ⇋ is given on Page 318.

  5. 5.

    See [39] for other results along these lines.

  6. 6.

    Random 3-SAT formulas are defined in Sect. 7, Page 440.

  7. 7.

    Finding inferences is referred to in the BDD literature as finding essential values to variables, and a set of inferences (a conjunction of literals) is referred to as a cube.

  8. 8.

    Random k-SAT formulas are defined in Sect. 7, Page 440.

  9. 9.

    Random k-SAT formulas are defined in Sect. 7, Page 440

References

  1. D. Achlioptas, C. Moore, The asymptotic order of the random k-SAT thresholds, in 43rd Annual Symposium on Foundations of Computer Science (IEEE Computer Society, Los Alamitos, 2002)

    Google Scholar 

  2. D. Achlioptas, Y. Peres, The threshold for random k-SAT is 2klog2 − O(k). J. Am. Math. Soc. 17, 947–973 (2004)

    MathSciNet  MATH  Google Scholar 

  3. D. Achlioptas, G. Sorkin, Optimal myopic algorithms for random 3-SAT, in 41st Annual Symposium on Foundations of Computer Science, Redondo Beach, California (IEEE Computer Society, Los Alamitos, 2000), pp. 590–600

    Google Scholar 

  4. A. Agrawal, P. Klein, R. Ravi, When trees collide: an approximation algorithm for the generalized Steiner problem on networks. SIAM J. Comput. 24, 440–456 (1995)

    MathSciNet  MATH  Google Scholar 

  5. R. Aharoni, N. Linial, Minimal non-two-colorable hypergraphs and minimal unsatisfiable formulas. J. Comb. Theory A 43, 196–204 (1986)

    MathSciNet  MATH  Google Scholar 

  6. S.B. Akers, Binary decision diagrams. IEEE Trans. Comput. C-27, 509–516 (1978)

    Google Scholar 

  7. M. Alekhnovich, J. Johannsen, T. Pitassi, A. Urquhart, An exponential separation between regular and general resolution. Theory Comput. 3, 81–102 (2007). http://theoryof computing.org

    Google Scholar 

  8. R. Amara, A.J. Lipinski, Business Planning for an Uncertain Future: Scenarios & Strategies (Elsevier, Amsterdam, 1983)

    Google Scholar 

  9. H. Andersen, An introduction to binary decision diagrams. Technical report, Department of Information Technology, Technical University of Denmark, 1998

    Google Scholar 

  10. B. Aspvall, Recognizing disguised NR(1) instances of the satisfiability problem. J. Algorithms 1, 97–103 (1980)

    MathSciNet  MATH  Google Scholar 

  11. G. Audemard, L. Simon, Predicting learnt clauses quality in modern SAT solvers, in Proceedings of the 21st International Joint Conference on Artificial Intelligence, Pasadena, CA, 2009, pp. 399–404

    Google Scholar 

  12. C. Barrett, A. Stump, C. Tinelli, The SMT-LIB Standard: Version 1.2. Linked from http://combination.cs.uiowa.edu/smtlib/

  13. C. Barrett, R. Sebastiani, S.A. Sanjit, C. Tinelli, Satisfiability Modulo Theories, in Handbook of Satisfiability, ed. by A. Biere, M. Heule, H. van Maaren, T. Walsh (IOS Press, Amsterdam, 2009), pp. 825–886

    Google Scholar 

  14. P. Beame, R.M. Karp, T. Pitassi, M. Saks, On the complexity of unsatisfiability proofs for random k-CNF formulas, in 30th Annual Symposium on the Theory of Computing (Association for Computing Machinery, New York, 1998), pp. 561–571

    Google Scholar 

  15. P. Beame, H. Kautz, A. Sabharwal, Towards understanding and harnessing the potential of clause learning. J. Artif. Intell. Res. 22, 319–351 (2004)

    MathSciNet  MATH  Google Scholar 

  16. P. Beame, H. Kautz, A. Sabharwal, Understanding the power of clause learning, in Proceedings of the 20th International Joint Conference on Artificial Intelligence, Hyderabad, India, 2007, pp. 1194–1201

    Google Scholar 

  17. E. Ben-Sasson, A. Wigderson, Short proofs are narrow – resolution made simple. J. Assoc. Comput. Mach. 48, 149–169 (2001)

    MathSciNet  MATH  Google Scholar 

  18. A. Biere, A. Cimatti, E. Clarke, Y. Zhu, Symbolic model checking without BDDs. Lect. Notes Comput. Sci. 1579, 193–207 (1999)

    Google Scholar 

  19. A. Biere, M. Heule, H. van Maaren, T. Walsh (eds.) Handbook of Satisfiability (IOS Press, Amsterdam, 2009)

    MATH  Google Scholar 

  20. A. Blake, Canonical expressions in Boolean algebra. Ph.D. Dissertation, Department of Mathematics, University of Chicago, 1937

    Google Scholar 

  21. E. Boros, Y. Crama, P.L. Hammer, Polynomial-time inference of all valid implications for Horn and related formulae. Ann. Math. Artif. Intell. 1, 21–32 (1990)

    MATH  Google Scholar 

  22. E. Boros, P.L. Hammer, X. Sun, Recognition of q-Horn formulae in linear time. Discret. Appl. Math. 55, 1–13 (1994)

    MathSciNet  MATH  Google Scholar 

  23. E. Boros, Y. Crama, P.L. Hammer, M. Saks, A complexity index for satisfiability problems. SIAM J. Comput. 23, 45–49 (1994)

    MathSciNet  MATH  Google Scholar 

  24. E. Boros, P.L. Hammer, T. Ibaraki, A. Kogan, E. Mayoraz, I. Muchnik, An implementation of logical analysis of data. RUTCOR Research Report RRR 04-97, RUTCOR, Rutgers University, 1996

    Google Scholar 

  25. E. Boros, P.L. Hammer, T. Ibaraki, A. Kogan, Logical analysis of numerical data. Math. Program. 79, 163–190 (1997)

    MathSciNet  MATH  Google Scholar 

  26. D. Brand, Verification of large synthesized designs, in Proceeding of the International Conference on Computer Aided Design (IEEE Computer Society Press, Los Alamitos, 1993), pp. 534–537

    Google Scholar 

  27. R.E. Bryant, Graph-based algorithms for Boolean function manipulation. IEEE Trans. Comput. C-35, 677–691 (1986)

    Google Scholar 

  28. R.E. Bryant, Symbolic Boolean manipulation with ordered binary decision diagrams. ACM Comput. Surv. 24, 293–318 (1992)

    Google Scholar 

  29. B. Buchberger, An algorithm for finding a basis for the residue of a class ring of a zero-dimensional polynomial ideal. Ph.D. Thesis, Universität Innsbruck, Institut für Mathematik, 1965

    MATH  Google Scholar 

  30. R. Chandrasekaran, Integer programming problems for which a simple rounding type of algorithm works, in Progress in Combinatorial Optimization, ed. by W. Pulleyblank (Academic, Toronto, 1984), pp. 101–106

    Google Scholar 

  31. V. Chandru, J.N. Hooker, Extended Horn sets in propositional logic. J. Assoc. Comput. Mach. 38, 205–221 (1991)

    MathSciNet  MATH  Google Scholar 

  32. M.-T. Chao, J. Franco, Probabilistic analysis of two heuristics for the 3-Satisfiability problem. SIAM J. Comput. 15, 1106–1118 (1986)

    MathSciNet  MATH  Google Scholar 

  33. M.-T. Chao, J. Franco, Probabilistic analysis of a generalization of the unit-clause literal selection heuristics for the k-satisfiability problem. Inf. Sci. 51, 289–314 (1990)

    MathSciNet  MATH  Google Scholar 

  34. V. Chvátal, B. Reed, Mick gets some (the odds are on his side), in 33th Annual Symposium on Foundations of Computer Science, Pittsburgh, Pennsylvania (IEEE Computer Society, Los Alamitos, 1992), pp. 620–627

    Google Scholar 

  35. V. Chvátal, E. Szemerédi, Many hard examples for resolution. J. Assoc. Comput. Mach. 35, 759–768 (1988)

    MathSciNet  MATH  Google Scholar 

  36. E. Clarke, A. Biere, R. Raimi, Y. Zhu, Bounded model checking using satisfiability solving. Form. Methods Syst. Des. 19(1), 7–34 (2001)

    MATH  Google Scholar 

  37. M. Clegg, J. Edmonds, R. Impagliazzo, Using the Groebner basis algorithm to find proofs of unsatisfiability, in 28th ACM Symposium on the Theory of Computing (ACM, Philadelphia, 1996), pp. 174–183

    Google Scholar 

  38. M. Conforti, G. Cornuéjols, A. Kapoor, K. VuŠković, M.R. Rao, Balanced matrices, in Mathematical Programming: State of the Art, ed. by J.R. Birge, K.G. Murty, Braun-Brumfield, United States. Produced in association with the 15th International Symposium on Mathematical Programming (University of Michigan, Ann Arbor, 1994)

    Google Scholar 

  39. S.A. Cook, R.A. Reckhow, Corrections for “On the Lengths of Proofs in the Propositional Calculus.” SIGACT News (ACM Special Interest Group on Automata and Computability Theory) 6 (1974)

    Google Scholar 

  40. T.H. Cormen, C.E. Leiserson, R.L. Rivest, C. Stein, Introduction to Algorithms, 2nd edn. (MIT/McGraw-Hill, 2001)

    MATH  Google Scholar 

  41. O. Coudert, On solving covering problems, in Proceedings of the 33rd Design Automation Conference (IEEE Computer Society, Los Alimitos, 1996), pp. 197–202

    Google Scholar 

  42. O. Coudert, J.C. Madre, A unified framework for the formal verification of sequential circuits, in Proceedings of the 1990 International Conference on Computer-Aided Design (ICCAD ’90) (IEEE Computer Society, Los Alamitos, 1990), pp. 126–129

    Google Scholar 

  43. O. Coudert, C. Berthet, J.C. Madre, Verification of synchronous sequential machines based on symbolic execution. Lect. Notes Comput. Sci. 407, 365–373 (1990). Springer

    Google Scholar 

  44. Y. Crama, P.L. Hammer, T. Ibaraki, Cause-effect relationships and partially defined Boolean functions. Ann. Oper. Res. 16, 299–326 (1998)

    MathSciNet  Google Scholar 

  45. W. Craig, Linear reasoning: a new form of the Herbrand-Gentzen theorem. J. Symb. Logic 22(3), 250–268 (1957)

    MathSciNet  MATH  Google Scholar 

  46. J.M. Crawford, A.B. Baker, Experimental results on the application of satisfiability algorithms to scheduling problems, in Proceedings of the National Conference on Artificial Intelligence (AAAI/MIT, Menlo Park, 1994), pp. 1092–1097

    Google Scholar 

  47. E. Dantsin, A. Wolpert, Derandomization of Schuler’s algorithm for SAT, in 6th International Conference on the Theory and Applications of Satisfiability Testing. Lecture Notes in Computer Science, vol. 2919 (Springer, New York, 2004), pp. 69–75

    Google Scholar 

  48. E. Dantsin, A. Wolpert, An improved upper bound for SAT, in 8th International Conference on the Theory and Applications of Satisfiability Testing. Lecture Notes in Computer Science, vol. 3569 (Springer, New York, 2005), pp. 400–407

    Google Scholar 

  49. E. Dantsin, A. Goerdt, E.A. Hirsch, R. Kannan, J. Kleinberg, C. Papadimitriou, P. Raghavan, U. Schöning, A deterministic (2 − 2∕(k+1))n algorithm for k-SAT based on local search. Theor. Comput. Sci. 289, 69–83 (2002)

    MATH  Google Scholar 

  50. A. Darwiche, K. Pipatsrisawat, Complete algorithms, in Handbook of Satisfiability, ed. by A. Biere, M. Heule, H. van Maaren, T. Walsh (IOS Press, Amsterdam, 2009), pp. 99–130

    Google Scholar 

  51. M. Davis, H. Putnam, A computing procedure for quantification theory. J. ACM 7, 201–215 (1960)

    MathSciNet  MATH  Google Scholar 

  52. M. Davis, G. Logemann, D. Loveland, A machine program for theorem proving. Commun. ACM 5, 394–397 (1962)

    MathSciNet  MATH  Google Scholar 

  53. W.F. Dowling, J.H. Gallier, Linear-time algorithms for testing the satisfiability of propositional Horn formulae. J. Logic Program. 1, 267–284 (1984)

    MathSciNet  MATH  Google Scholar 

  54. C. Ducot, G.J. Lubben, A typology for scenarios. Future 12, 51–57 (1980)

    Google Scholar 

  55. B. Dutertre, L. de Moura, A Fast Linear-Arithmetic Solver for DPLL(T)*. Lecture Notes in Computer Science, vol. 4144 (Springer, Berlin/Heidelberg, 2006), pp. 81–94

    Google Scholar 

  56. S. Even, A. Itai, A. Shamir, On the complexity of timetable and multi-commodity flow problems. SIAM J. Comput. 5, 691–703 (1976)

    MathSciNet  MATH  Google Scholar 

  57. R. Feldmann, N. Sensen, Efficient algorithms for the consistency analysis in scenario projects. Technical Report, Universität Paderborn, Germany, 1997

    Google Scholar 

  58. H. Fleischner, O. Kullmann, S. Szeider, Polynomial-time recognition of minimal unsatisfiable formulas with fixed clause-variable difference. Theor. Comput. Sci. 289(1), 503–516 (2002)

    MathSciNet  MATH  Google Scholar 

  59. J. Franco, A. Van Gelder, A perspective on certain polynomial time solvable classes of satisfiability. Discret. Appl. Math. 125(2–3), 177–214 (2003)

    MATH  Google Scholar 

  60. J.W. Freeman, Improvements to propositional satisfiability search algorithms. Ph.D. Thesis, University of Pennsylvania, Computer and Information Science, 1995

    Google Scholar 

  61. Z. Fu, S. Malik, Solving the minimum-cost Satisfiability problem using SAT based branch-and-bound search, in Proceedings of the International Conference on Computer Aided Design (Association for Computing Machinery, New York, 2006), pp. 852–859

    Google Scholar 

  62. J. Gausemeier, A. Fink, O. Schlake, Szenario-Management (Carl Hanser Verlag, München-Wien, 1995)

    Google Scholar 

  63. M. Godet, Scenarios and Strategic Management (Butterworths Publishing (Pty) Ltd., South Africa, 1987)

    Google Scholar 

  64. A. Goerdt, M. Krivelevich, Efficient Recognition of Random Unsatisfiable k-SAT Instances by Spectral Methods. Lecture Notes in Computer Science, vol. 2010 (Springer, Berlin, 2001), pp. 294–304

    Google Scholar 

  65. E. Goldberg, Y. Novikov, How Good Can a Resolution Based SAT-Solver Be? Lecture Notes in Computer Science, vol. 2919 (Springer, Berlin, 2003), pp. 35–52

    Google Scholar 

  66. E. Grégoire, B. Mazure, L. Saïs, Logically-complete local search for propositional non-monotonic knowledge bases, in Proceedings of the 7th International Workshop on Non-monotonic Reasoning, Trente, 1998, pp. 37–45

    Google Scholar 

  67. J.F. Groote, H. Zantema, Resolution and Binary Decision Diagrams cannot simulate each other polynomially, in Perspectives of System Informatics: 4th International Andrei Ershov Memorial Conference, ed. by D. Bjorner, M. Broy, A.V. Zamulin. Lecture Notes in Computer Science, vol. 2244 (Springer, New York, 2001), pp. 33–38

    Google Scholar 

  68. S. Haim, T. Walsh, Restart strategy selection using machine learning techniques, in Proceedings of the 12th International Conference on Theory and Applications of Satisfiability Testing, Swansea, UK, 2009, pp. 312–325

    Google Scholar 

  69. M.T. Hajiaghayi, G.B. Sorkin, The satisfiability threshold of random 3-SAT is at least 3.52. Available from http://arxiv.org/pdf/math.CO/0310193

  70. P. Hall, On representatives of subsets. J. Lond. Math. Soc. 10, 26–30 (1935)

    Google Scholar 

  71. P.L. Hammer, Partially defined Boolean functions and cause-effect relationships, in Proceedings of the International Conference on Multi-Attribute Decision Making Via OR-Based Expert Systems, University of Passau, Passau, Germany, 1986

    Google Scholar 

  72. P. Hansen, B. Jaumard, G. Plateau, An extension of nested satisfiability. Les Cahiers du GERAD, G-93-27, 1993

    Google Scholar 

  73. H.H. Hoos, On the run-time behaviour of stochastic local search algorithms for SAT, in Proceedings of the 16th National Conference on Artificial Intelligence (AAAI/MIT, Menlo Park, 1999), pp. 661–666

    Google Scholar 

  74. H.H. Hoos, An adaptive noise mechanism for WalkSAT, in Proceedings of the National Conference on Artificial Intelligence (AAAI/MIT, Menlo Park, 2002), pp. 655–660

    Google Scholar 

  75. H.H. Hoos, D.A.D. Tompkins, Adaptive Novelty+. Technical Report, Computer Science Department, University of British Columbia, 2007. Available from: http://www.satcompetition.org/2007/adaptnovelty.pdf

  76. J. Huang, The effect of restarts on the efficiency of clause learning, in Proceedings of the Eighteenth International Joint Conference on Artificial Intelligence, Acapulco, Mexico, 2003, pp. 2318–2323

    Google Scholar 

  77. W. Hunt, S. Swords, Centaur technology media unit verification, in Proceedings of the International Conference on Computer Aided Design (Association for Computing Machinery, Grenoble, 2009), pp. 353–367

    Google Scholar 

  78. A. Itai, J. Makowsky, On the complexity of Herbrand’s theorem. Working paper 243, Department of Computer Science, Israel Institute of Technology, 1982

    Google Scholar 

  79. K. Iwama, S. Tamaki, Improved upper bounds for 3-SAT, in 15th ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, Philadelphia, 2003), pp. 328–328

    Google Scholar 

  80. R.E. Jeroslow, J. Wang, Solving propositional satisfiability problems. Ann. Math. AI 1, 167–187 (1990)

    MATH  Google Scholar 

  81. D.S. Johnson, Approximation algorithms for combinatorial problems. J. Comput. Syst. Sci. 9, 256–278 (1974)

    MATH  Google Scholar 

  82. E.L. Johnson, G.L. Nemhauser, M.W.P. Savelsbergh, Progress in linear programming-based algorithms for integer programming: an exposition. INFORMS J. Comput. 12(1), 2–23 (2000)

    MathSciNet  MATH  Google Scholar 

  83. A.C. Kaporis, L.M. Kirousis, E.G. Lalas, The probabilistic analysis of a greedy satisfiability algorithm, in 10th Annual European Symposium on Algorithms, Rome, Italy, 2002

    Google Scholar 

  84. A.C. Kaporis, L.M. Kirousis, E.G. Lalas, Selecting complementary pairs of literals. Electron. Notes Discret. Math. 16(1), 1–24 (2004)

    Google Scholar 

  85. R.M. Karp, Reducibility among combinatorial problems, in Complexity of Computer Computations, ed. by R.E. Miller, J.W. Thatcher (Plenum, New York, 1972), pp. 85–103

    Google Scholar 

  86. H. Kautz, B. Selman, MAXWalkSat. Available from: http://www.cs.rochester.edu/~kautz/walksat/

  87. H. Kautz, B. Selman, Y. Jiang, Stochastic Search for Solving Weighted MAX-SAT Encodings of Constraint Satisfaction Problems (AT&T Laboratories, Murray Hill, 1998, Preprint)

    Google Scholar 

  88. H. Kautz, E. Horvitz, Y. Ruan, C. Gomes, B. Selman, Dynamic restart policies, in Proceedings of the National Conference on Artificial Intelligence (AAAI/MIT, Menlo Park, 2002), pp. 674–681

    Google Scholar 

  89. H. Kautz, A. Sabharwal, B. Selman, Incomplete algorithms, in Handbook of Satisfiability, ed. by A. Biere, M. Heule, H. van Maaren, T. Walsh (IOS Press, Amsterdam, 2009), pp. 185–204

    Google Scholar 

  90. F. Klaedtke, Decision Procedures for Logical Theories, Lecture 12: Combining Decision Procedures: Nelson-Oppen, Department of Computer Science, ETH Zurich, 2006

    Google Scholar 

  91. D. Knuth, Nested satisfiability. Acta Inform. 28, 1–6 (1990)

    MathSciNet  MATH  Google Scholar 

  92. F. Krohm, A. Kuehlmann, Equivalence checking using cuts and heaps, in Proceedings of the 34 th Design Automation Conference (IEEE Computer Society, Los Alimitos, 1997), pp. 263–268

    Google Scholar 

  93. F. Krohm, A. Kuehlmann, A. Mets, The use of random simulation in formal verification, in Proceedings of the IEEE International Conference on Computer Design (IEEE Computer Society, Los Alimitos, 1997), pp. 371–376

    Google Scholar 

  94. A. Kuehlmann, Dynamic transition relation simplification for bounded property checking, in Proceeding of the IEEE International Conference on Computer Aided Design (IEEE Computer Society, Los Alamitos, 2004), pp. 50–57

    Google Scholar 

  95. A. Kuehlmann, V. Paruthi, F. Krohm, M.K. Ganai, Robust Boolean reasoning for equivalence checking and functional property verification. IEEE Trans. Comput. Aided Des. 21(12), 1377–1394 (2002)

    Google Scholar 

  96. O. Kullmann, A first summary on minimal unsatisfiable clause-sets. Technical report, University of Toronto, June, 1998

    Google Scholar 

  97. O. Kullmann, Investigations on autark assignments. Discret. Appl. Math. 107, 99–137 (2000)

    MathSciNet  MATH  Google Scholar 

  98. O. Kullmann, An application of Matroid theory to the SAT problem, in Proceedings of the 15 th Annual IEEE Conference on Computational Complexity (IEEE, 2000), pp. 116–124

    Google Scholar 

  99. H. Kleine Büning, On the minimal unsatisfiability problem for some subclasses of CNF, in Abstracts of 16th Int’l Symposium on Mathematical Programming, Lausanne, 1997

    Google Scholar 

  100. M. Kouril, J. Franco, Resolution tunnels for improved SAT solver performance, in Proceedings of the 8th International Conference on Theory and Applications of Satisfiability Testing, St. Andrews, UK, 2005, pp. 136–141

    Google Scholar 

  101. M. Kouril, J. Paul, The van der Waerden Number W(2, 6) Is 1132. Exp. Math. 17(1), 53–61 (2008)

    MathSciNet  MATH  Google Scholar 

  102. R.A. Kowalski, A proof procedure using connection graphs. J. ACM 22, 572–595 (1974)

    Google Scholar 

  103. C.Y. Lee, Representation of switching circuits by binary decision programs. Bell Syst. Tech. J. 38, 985–999 (1959)

    Google Scholar 

  104. H.R. Lewis, Renaming a set of clauses as a Horn set. J. Assoc. Comput. Mach. 25, 134–135 (1978)

    MathSciNet  MATH  Google Scholar 

  105. D. Lichtenstein, Planar formulae and their uses. SIAM J. Comput. 11, 329–343 (1982)

    MathSciNet  MATH  Google Scholar 

  106. I. Lynce, J.P.M. Silva, Efficient data structures for backtrack search SAT solvers. Ann. Math. Artif. Intell. 43(1), 137–152 (2005)

    MathSciNet  MATH  Google Scholar 

  107. I. Lynce, J.P.M. Silva, SAT in Bioinformatics: making the Case with Haplotype Inference, in Proceedings of the 9th International Conference on Theory and Applications of Satisfiability Testing, Seattle, WA, 2006, pp. 136–141

    Google Scholar 

  108. V.M. Manquinho, J.P.M. Silva, Search pruning techniques in SAT-based branch-and-bound algorithms for the binate covering problem. IEEE Trans. CAD Integr. Circuits Syst. 21(5), 505–516 (2002)

    Google Scholar 

  109. D. McAllester, B. Selman, H. Kautz, Evidence for invariants in local search, in Proceedings of the 14th National Conference on Artificial Intelligence (AAAI/MIT, Menlo Park, 1997), pp. 321–326

    Google Scholar 

  110. I. Mironov, L. Zhang, Applications of SAT solvers to cryptanalysis of hash functions, in Proceedings of the 9th International Conference on Theory and Applications of Satisfiability Testing, Seattle, WA, 2006, pp. 102–115

    Google Scholar 

  111. B. Monien, E. Speckenmeyer, Solving satisfiability in less than 2n steps. Discret. Appl. Math. 10, 287–295 (1985)

    MathSciNet  MATH  Google Scholar 

  112. M.W. Moskewicz, C.F. Madigan, Y. Zhao, L. Zhang, S. Malik, Chaff: engineering an efficient SAT solver, in Proceedings of the 38th Design Automation Conference, Las Vegas, NV, USA, 2001, pp. 530–535

    Google Scholar 

  113. G. Nelson, D.C. Oppen, Simplification by cooperating decision procedures. ACM Trans. Program. Lang. Syst. 1(2), 245–257 (1979)

    MATH  Google Scholar 

  114. D.C. Oppen, Complexity, convexity and combinations of theories. Theor. Comput. Sci. 12, 291–302, 1980

    MathSciNet  MATH  Google Scholar 

  115. R. Paturi, P. Pudlák, M.E. Saks, F. Zane, An improved exponential-time algorithm for k-SAT, in 39th Annual Symposium on Foundations of Computer Science (IEEE Computer Society, Los Alamitos, 1998)

    Google Scholar 

  116. W.V.O. Quine, A way to simplify truth functions. Am. Math. Mon. 62, 627–631 (1955)

    MathSciNet  MATH  Google Scholar 

  117. R. Reiter, A theory of diagnosis from first principles. Artif. Intell. 32, 57–95 (1987)

    MathSciNet  MATH  Google Scholar 

  118. J.A. Robinson, A machine-oriented logic based on the resolution principle. J. ACM 12, 23–41 (1965)

    MATH  Google Scholar 

  119. L. Ryan, Efficient algorithms for clause-learning SAT solvers. Masters Thesis, Simon Fraser University, 2004

    Google Scholar 

  120. E.W. Samson, B.E. Mills, Circuit minimization: algebra and algorithms for new Boolean canonical expressions. Air Force Cambridge Research Center Technical Report 54–21, 1954

    Google Scholar 

  121. T.S. Schaefer, The complexity of satisfiability problems, in Proceedings of 10 th Annual ACM Symposium on Theory of Computing (ACM, New York, 1978), pp. 216–226

    Google Scholar 

  122. J.S. Schlipf, F. Annexstein, J. Franco, R. Swaminathan, On finding solutions for extended Horn formulas. Inf. Process. Lett. 54, 133–137 (1995)

    MathSciNet  MATH  Google Scholar 

  123. U. Schöning, A probabilistic algorithm for k-SAT and constraint satisfaction problems, in 40th Annual Symposium on Foundations of Computer Science (IEEE Computer Society, Los Alamitos, 1999), pp. 410–414

    Google Scholar 

  124. R. Schuler, An algorithm for the satisfiability problem of formulas in conjunctive normal form. J. Algorithms 54(1), 40–44 (2005)

    MathSciNet  MATH  Google Scholar 

  125. M.G. Scutella, A note on Dowling and Gallier’s top-down algorithm for propositional Horn satisfiability. J. Logic Program. 8, 265–273 (1990)

    MathSciNet  MATH  Google Scholar 

  126. R. Sebastiani, Lazy Satisfiability Modulo Theories. J. Satisfiability Boolean Model. Comput. 3(3–4), 141–224 (2007)

    MathSciNet  MATH  Google Scholar 

  127. B. Selman, H. Kautz, B. Cohen, Local search strategies for satisfiability testing, in Cliques, Coloring, and Satisfiability: The Second DIMACS Implementation Challenge, ed. by D.S. Johnson, M.A. Trick (American Mathematical Society, Providence, 1996), pp. 521–532

    Google Scholar 

  128. Y. Shang, B.W. Wah, A discrete Lagrangian-based global search method for solving satisfiability problems. J. Glob. Optim. 12(1), 61–100 (1998)

    MathSciNet  MATH  Google Scholar 

  129. C.E. Shannon, A symbolic analysis of relay and switching circuits. Masters Thesis, Massachusetts Institute of Technology, 1940. Available from http://hdl.handle.net/1721.1/11173

  130. J.P.M. Silva, K.A. Sakallah, GRASP – a new search algorithm for satisfiability, in Proceedings of the 1996 International Conference on Computer-Aided Design (ICCAD ’96) (IEEE Computer Society, Los Alamitos, 1996), pp. 220–227

    Google Scholar 

  131. J.P.M. Silva, I. Lynce, S. Malik, Conflict-driven clause learning SAT solvers, in Handbook of Satisfiability, ed. by A. Biere, M. Heule, H. van Maaren, T. Walsh (IOS Press, Amsterdam, 2009), pp. 131–153

    Google Scholar 

  132. E.W. Smith, D.L. Dill, Automatic formal verification of block Cipher implementations, in Proceedings of the Formal Methods in Computer-Aided Design Conference, Portland, Oregon, 2008, pp. 1–7

    Google Scholar 

  133. R.P. Swaminathan, D.K. Wagner, The arborescence–realization problem. Discret. Appl. Math. Comb. Oper. Res. Comput. Sci. 59(3), 267–283 (1995)

    MathSciNet  MATH  Google Scholar 

  134. S. Szeider, Minimal unsatisfiable formulas with bounded clause-variable difference are fixed-parameter tractable. J. Comput. Syst. Sci. 69, 656–674 (2004)

    MathSciNet  MATH  Google Scholar 

  135. S. Szeider, Generalizations of matched CNF formulas. Ann. Math. Artif. Intell. 43(1), 223–238 (2005)

    MathSciNet  MATH  Google Scholar 

  136. C.A. Tovey, A simplified NP-complete satisfiability problem. Discret. Appl. Math. 8, 85–89 (1984)

    MathSciNet  MATH  Google Scholar 

  137. K. Truemper, Monotone decomposition of matrices. Technical Report UTDCS-1-94, University of Texas at Dallas, 1994

    Google Scholar 

  138. K. Truemper, Effective Logic Computation (Wiley, New York, 1998)

    MATH  Google Scholar 

  139. G.S. Tseitin, On the complexity of derivations in propositional calculus, in Structures in Constructive Mathematics and Mathematical Logic, Part II, ed. by A.O. Slisenko (Nauka, Moscow, 1968), pp. 115–125 (translated from Russian)

    Google Scholar 

  140. L.G. Valiant, A theory of the learnable. Commun. ACM 27, 1134–1142 (1984)

    MATH  Google Scholar 

  141. A. Van Gelder, Generalizations of watched literals for backtracking search, in Seventh International Symposium on Artificial Intelligence and Mathematics, 2002

    Google Scholar 

  142. A. Van Gelder, Verifying propositional unsatisfiability: Pitfalls to avoid, in Proceedings of the 10th International Conference on Theory and Applications of Satisfiability Testing, Lisbon, Portugal, 2007, pp. 328–333

    Google Scholar 

  143. A. Van Gelder, Improved conflict-clause minimization leads to improved propositional proof traces, in Proceedings of the 12th International Conference on Theory and Applications of Satisfiability Testing, Swansea, UK, 2009, pp. 141–146

    Google Scholar 

  144. A. Van Gelder, Y.K. Tsuji, Satisfiability testing with more reasoning and less guessing, in Cliques, Coloring, and Satisfiability: Second DIMACS Implementation Challenge, ed. by D.S. Johnson, M. Trick (American Mathematical Society, Providence, 1996)

    Google Scholar 

  145. H. van Maaren, A short note on some tractable classes of the Satisfiability problem. Inf. Comput. 158(2), 125–130 (2000)

    MATH  Google Scholar 

  146. S. Weaver, J. Franco, J. Schlipf, Extending existential quantification in conjunctions of BDDs. J. Satisfiability Boolean Model. Comput. 1, 89–110 (2006)

    MATH  Google Scholar 

  147. Z. Wu, B.W. Wah, Trap escaping strategies in discrete Lagrangian methods for solving hard satisfiability and maximum satisfiability problems, in Proceedings of the 16th National Conference on Artificial Intelligence (AAAI/MIT, Menlo Park, 1999), pp. 673–678

    Google Scholar 

  148. Z. Wu, B.W. Wah, An efficient global-search strategy in discrete Lagrangian methods for solving hard satisfiability problems, in Proceedings of the 17th National Conference on Artificial Intelligence (AAAI/MIT, Menlo Park, 2000), pp. 310–315

    Google Scholar 

  149. M. Yoeli, Formal Verification of Hardware Design (IEEE Computer Society, Los Alamitos, 1988)

    Google Scholar 

  150. L. Zhang, S. Malik, Validating SAT solvers using an independent resolution-based checker: practical implementations and other applications, in Proceedings of the Conference on Design, Automation and Test in Europe, Washington, DC, USA, 2003, pp. 880–885

    Google Scholar 

  151. H. Zhang, D. Li, H. Shen, A SAT based scheduler for tournament schedules, in Proceedings of the 7th International Conference on Theory and Applications of Satisfiability Testing, Vancouver, BC, 2004

    Google Scholar 

Download references

Acknowledgements

We thank John S. Schlipf for his help in writing some of the text, particularly the sections on the diagnosis of circuit faults and Binary Decision Diagrams. We especially appreciate John’s attention to detail which was quite valuable to us.

Author information

Authors and Affiliations

  1. School of Electronic and Computing Systems, University of Cincinnati, Cincinnati, OH, USA

    John Franco

  2. U.S. Department of Defense, Ft. George G. Meade, MD, USA

    Sean Weaver

Authors
  1. John Franco
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sean Weaver
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to John Franco or Sean Weaver .

Editor information

Editors and Affiliations

  1. Department of Industrial and Systems Eng, University of Florida, Gainesville, Florida, USA

    Panos M. Pardalos

  2. Department of Computer Science, University of Texas, Dallas, Richardson, Texas, USA

    Ding-Zhu Du

  3. Dept. Comp. Sci. & Engineering, University of California, San Diego, La Jolla, California, USA

    Ronald L. Graham

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer Science+Business Media New York

About this entry

Cite this entry

Franco, J., Weaver, S. (2013). Algorithms for the Satisfiability Problem. In: Pardalos, P., Du, DZ., Graham, R. (eds) Handbook of Combinatorial Optimization. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-7997-1_31

Download citation

Publish with us

Policies and ethics

Search

Navigation