Backdoors to Satisfaction

  • Serge Gaspers
  • Stefan Szeider
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7370)


A backdoor set is a set of variables of a propositional formula such that fixing the truth values of the variables in the backdoor set moves the formula into some polynomial-time decidable class. If we know a small backdoor set we can reduce the question of whether the given formula is satisfiable to the same question for one or several easy formulas that belong to the tractable class under consideration. In this survey we review parameterized complexity results for problems that arise in the context of backdoor sets, such as the problem of finding a backdoor set of size at most k, parameterized by k. We also discuss recent results on backdoor sets for problems that are beyond NP.


Base Class Vertex Cover Truth Assignment Primal Graph Incidence Graph 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abu-Khzam, F.N.: A kernelization algorithm for d-hitting set. J. of Computer and System Sciences 76(7), 524–531 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Aspvall, B., Plass, M.F., Tarjan, R.E.: A linear-time algorithm for testing the truth of certain quantified Boolean formulas. Information Processing Letters 8(3), 121–123 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bacchus, F., Dalmao, S., Pitassi, T.: Algorithms and complexity results for #SAT and Bayesian inference. In: 44th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2003), pp. 340–351 (2003)Google Scholar
  4. 4.
    Baroni, P., Giacomin, M.: Semantics of abstract argument systems. In: Rahwan, I., Simari, G. (eds.) Argumentation in Artificial Intelligence, pp. 25–44. Springer (2009)Google Scholar
  5. 5.
    Bench-Capon, T.J.M., Dunne, P.E.: Argumentation in artificial intelligence. Artificial Intelligence 171(10-15), 619–641 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Berre, D.L., Parrain, A.: On SAT technologies for dependency management and beyond. In: Thiel, S., Pohl, K. (eds.) Proceedings of 12th International Conference Software Product Lines Workshops, SPLC 2008, Limerick, Ireland, September 8-12, vol. 2, pp. 197–200. Lero Int. Science Centre, University of Limerick, Ireland (2008)Google Scholar
  7. 7.
    Besnard, P., Hunter, A.: Elements of Argumentation. The MIT Press (2008)Google Scholar
  8. 8.
    Bjesse, P., Leonard, T., Mokkedem, A.: Finding Bugs in an Alpha Microprocessor Using Satisfiability Solvers. In: Berry, G., Comon, H., Finkel, A. (eds.) CAV 2001. LNCS, vol. 2102, pp. 454–464. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  9. 9.
    Bodlaender, H.L.: On disjoint cycles. International Journal of Foundations of Computer Science 5(1), 59–68 (1994)CrossRefzbMATHGoogle Scholar
  10. 10.
    Bodlaender, H.L., Downey, R.G., Fellows, M.R., Hermelin, D.: On problems without polynomial kernels. J. of Computer and System Sciences 75(8), 423–434 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Bondarenko, A., Dung, P.M., Kowalski, R.A., Toni, F.: An abstract, argumentation-theoretic approach to default reasoning. Artificial Intelligence 93(1-2), 63–101 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Bonsma, P., Lokshtanov, D.: Feedback Vertex Set in Mixed Graphs. In: Dehne, F., Iacono, J., Sack, J.-R. (eds.) WADS 2011. LNCS, vol. 6844, pp. 122–133. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  13. 13.
    Kleine Büning, H., Karpinski, M., Flögel, A.: Resolution for quantified Boolean formulas. Information and Computation 117(1), 12–18 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Cai, L., Huang, X.: Fixed-parameter approximation: Conceptual framework and approximability results. Algorithmica 57(2), 398–412 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Chavira, M., Darwiche, A.: On probabilistic inference by weighted model counting. Artificial Intelligence 172(6-7), 772–799 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Chen, J., Kanj, I.A.: On approximating minimum vertex cover for graphs with perfect matching. Theoretical Computer Science 337(1-3), 305–318 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Chen, J., Fomin, F.V., Liu, Y., Lu, S., Villanger, Y.: Improved algorithms for feedback vertex set problems. J. of Computer and System Sciences 74(7), 1188–1198 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Chen, J., Kanj, I.A., Jia, W.: Vertex cover: further observations and further improvements. J. Algorithms 41(2), 280–301 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Chen, J., Kanj, I.A., Xia, G.: Improved upper bounds for vertex cover. Theoretical Computer Science 411(40–42), 3736–3756 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Chen, J., Liu, Y., Lu, S., O’Sullivan, B., Razgon, I.: A fixed-parameter algorithm for the directed feedback vertex set problem. J. of the ACM 55(5), Art. 21, 19 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Chen, Y.-J., Grohe, M., Grüber, M.: On Parameterized Approximability. In: Bodlaender, H.L., Langston, M.A. (eds.) IWPEC 2006. LNCS, vol. 4169, pp. 109–120. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  22. 22.
    Cook, S.A.: The complexity of theorem-proving procedures. In: Proc. 3rd Annual Symp. on Theory of Computing, pp. 151–158. Shaker Heights, Ohio (1971)Google Scholar
  23. 23.
    Cook, S.A., Mitchell, D.G.: Finding hard instances of the satisfiability problem: a survey. In: Satisfiability problem: theory and applications, Piscataway, NJ. American Mathematical Society, pp. 1–17 (1997)Google Scholar
  24. 24.
    Coste-Marquis, S., Devred, C., Marquis, P.: Symmetric Argumentation Frameworks. In: Godo, L. (ed.) ECSQARU 2005. LNCS (LNAI), vol. 3571, pp. 317–328. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  25. 25.
    Courcelle, B., Makowsky, J.A., Rotics, U.: On the fixed parameter complexity of graph enumeration problems definable in monadic second-order logic. Discr. Appl. Math. 108(1-2), 23–52 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Courcelle, B.: Graph rewriting: an algebraic and logic approach. In: Handbook of Theoretical Computer Science, vol. B, pp. 193–242. Elsevier Science Publishers, North-Holland (1990)Google Scholar
  27. 27.
    Crama, Y., Ekin, O., Hammer, P.L.: Variable and term removal from Boolean formulae. Discr. Appl. Math. 75(3), 217–230 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Cygan, M., Pilipczuk, M., Pilipczuk, M., Wojtaszczyk, J.O.: Subset Feedback Vertex Set Is Fixed-Parameter Tractable. In: Aceto, L., Henzinger, M., Sgall, J. (eds.) ICALP 2011. LNCS, vol. 6755, pp. 449–461. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  29. 29.
    Davis, M., Putnam, H.: A computing procedure for quantification theory. J. of the ACM 7(3), 201–215 (1960)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Davis, M., Logemann, G., Loveland, D.: A machine program for theorem-proving. Communications of the ACM 5, 394–397 (1962)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Dilkina, B.N., Gomes, C.P., Sabharwal, A.: Tradeoffs in the Complexity of Backdoor Detection. In: Bessière, C. (ed.) CP 2007. LNCS, vol. 4741, pp. 256–270. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  32. 32.
    Dimopoulos, Y., Torres, A.: Graph theoretical structures in logic programs and default theories. Theoretical Computer Science 170(1-2), 209–244 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Monographs in Computer Science. Springer, New York (1999)CrossRefzbMATHGoogle Scholar
  34. 34.
    Downey, R.G., Fellows, M.R., McCartin, C.: Parameterized Approximation Problems. In: Bodlaender, H.L., Langston, M.A. (eds.) IWPEC 2006. LNCS, vol. 4169, pp. 121–129. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  35. 35.
    Dung, P.M.: On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games. Artificial Intelligence 77(2), 321–357 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Dunne, P.E.: Computational properties of argument systems satisfying graph-theoretic constraints. Artificial Intelligence 171(10-15), 701–729 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Dunne, P.E., Bench-Capon, T.J.M.: Coherence in finite argument systems. Artificial Intelligence 141(1-2), 187–203 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Eiter, T., Gottlob, G.: On the computational cost of disjunctive logic programming: propositional case. Ann. Math. Artif. Intell. 15(3-4), 289–323 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Fellows, M.R., Szeider, S., Wrightson, G.: On finding short resolution refutations and small unsatisfiable subsets. Theoretical Computer Science 351(3), 351–359 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Fernau, H.: A top-down approach to search-trees: Improved algorithmics for 3-hitting set. Algorithmica 57(1), 97–118 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Fichte, J.K.: The good, the bad, and the odd: Cycles in answer-set programs. In: ESSLII 2011 (2011)Google Scholar
  42. 42.
    Fichte, J.K., Szeider, S.: Backdoors to tractable answer-set programming. Technical Report 1104.2788, (2012), Extended and updated version of a paper that appeared in the proceedings of IJCAI 2011. The 22nd International Joint Conference on Artificial Intelligence (2012)Google Scholar
  43. 43.
    Fischer, E., Makowsky, J.A., Ravve, E.R.: Counting truth assignments of formulas of bounded tree-width or clique-width. Discr. Appl. Math. 156(4), 511–529 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Fortnow, L., Santhanam, R.: Infeasibility of instance compression and succinct PCPs for NP. In: Dwork, C. (ed.) Proceedings of the 40th Annual ACM Symposium on Theory of Computing, Victoria, British Columbia, Canada, May 17-20, pp. 133–142. ACM (2008)Google Scholar
  45. 45.
    Garey, M.R., Johnson, D.R.: Computers and Intractability. W. H. Freeman and Company, New York (1979)zbMATHGoogle Scholar
  46. 46.
    Gaspers, S., Szeider, S.: Backdoors to acyclic SAT. In: Proceedings of ICALP 2012 (Track A: Algorithms, Complexity and Games), the 39th International Colloquium on Automata, Languages and Programming, University of Warwick, UK, July 9-13. LNCS. Springer (to appear, 2012)Google Scholar
  47. 47.
    Gaspers, S., Szeider, S.: Strong backdoors to nested satisfiabiliy. In: Proceedings of SAT 2012, the 15th International Conference on Theory and Applications of Satisfiability Testing, Trento, Italy, June 17-20, 2012. LNCS. Springer (to appear, 2012)Google Scholar
  48. 48.
    Gaspers, S., Szeider, S.: Strong backdoors to bounded treewidth SAT. Technical report 1204.6233, (2012)Google Scholar
  49. 49.
    Gelfond, M., Lifschitz, V.: Classical negation in logic programs and disjunctive databases. New Generation Comput. 9(3/4), 365–386 (1991)CrossRefzbMATHGoogle Scholar
  50. 50.
    Gomes, C.P., Kautz, H., Sabharwal, A., Selman, B.: Satisfiability solvers. In: Handbook of Knowledge Representation. Foundations of Artificial Intelligence, vol. 3, pp. 89–134. Elsevier (2008)Google Scholar
  51. 51.
    Gottlob, G., Szeider, S.: Fixed-parameter algorithms for artificial intelligence, constraint satisfaction, and database problems. The Computer Journal 51(3), 303–325 (2006); survey paperCrossRefGoogle Scholar
  52. 52.
    Guo, J., Hüffner, F., Niedermeier, R.: A Structural View on Parameterizing Problems: Distance from Triviality. In: Downey, R., Fellows, M., Dehne, F. (eds.) IWPEC 2004. LNCS, vol. 3162, pp. 162–173. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  53. 53.
    Hertli, T.: 3-SAT faster and simpler - unique-SAT bounds for PPSZ hold in general. In: Ostrovsky, R. (ed.) Proceedings of the 52nd Annual IEEE Symposium on Foundations of Computer Science (FOCS 2011). IEEE (2011)Google Scholar
  54. 54.
    Iwama, K.: CNF-satisfiability test by counting and polynomial average time. SIAM J. Comput. 18(2), 385–391 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  55. 55.
    Kakimura, N., Kawarabayashi, K., Kobayashi, Y.: Erdös-Pósa property and its algorithmic applications: parity constraints, subset feedback set, and subset packing. In: Rabani, Y. (ed.) Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012, Kyoto, Japan, January 17-19, pp. 1726–1736. SIAM (2012)Google Scholar
  56. 56.
    Kautz, H.A., Selman, B.: Planning as satisfiability. In: Proceedings ECAI 1992, pp. 359–363 (1992)Google Scholar
  57. 57.
    Kawarabayashi, K., Kobayashi, Y.: Fixed-parameter tractability for the subset feedback set problem and the s-cycle packing problem. Technical report, University of Tokyo, Japan (2010); see also [55]Google Scholar
  58. 58.
    Büning, H.K., Lettman, T.: Propositional logic: deduction and algorithms. Cambridge University Press, Cambridge (1999)Google Scholar
  59. 59.
    Knuth, D.E.: Nested satisfiability. Acta Informatica 28(1), 1–6 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  60. 60.
    Kolaitis, P.G., Vardi, M.Y.: Conjunctive-query containment and constraint satisfaction. J. of Computer and System Sciences 61(2), 302–332 (2000); Special issue on the Seventeenth ACM SIGACT-SIGMOD-SIGART Symposium on Principles of Database Systems, Seattle, WA (1998)Google Scholar
  61. 61.
    Levin, L.: Universal sequential search problems. Problems of Information Transmission 9(3), 265–266 (1973)Google Scholar
  62. 62.
    Lewis, H.R.: Renaming a set of clauses as a Horn set. J. of the ACM 25(1), 134–135 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  63. 63.
    Li, Z., van Beek, P.: Finding Small Backdoors in SAT Instances. In: Butz, C., Lingras, P. (eds.) Canadian AI 2011. LNCS, vol. 6657, pp. 269–280. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  64. 64.
    Gupta, A., Prasad, M., Biere, A.: A survey of recent advances in SAT-based formal verification. Software Tools for Technology Transfer 7(2), 156–173 (2005)CrossRefGoogle Scholar
  65. 65.
    Marek, V.W., Truszczynski, M.: Stable models and an alternative logic programming paradigm. In: The Logic Programming Paradigm: a 25-Year Perspective, pp. 169–181. Springer (1999)Google Scholar
  66. 66.
    Marx, D., Schlotter, I.: Parameterized complexity and local search approaches for the stable marriage problem with ties. Algorithmica 58(1), 170–187 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  67. 67.
    Marx, D., Schlotter, I.: Stable assignment with couples: parameterized complexity and local search. Discrete Optim. 8(1), 25–40 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  68. 68.
    Mishra, S., Raman, V., Saurabh, S., Sikdar, S., Subramanian, C.R.: The Complexity of Finding Subgraphs Whose Matching Number Equals the Vertex Cover Number. In: Tokuyama, T. (ed.) ISAAC 2007. LNCS, vol. 4835, pp. 268–279. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  69. 69.
    Niemelä, I.: Logic programs with stable model semantics as a constraint programming paradigm. Ann. Math. Artif. Intell. 25(3-4), 241–273 (1999); Logic programming with non-monotonic semantics: representing knowledge and its computationMathSciNetCrossRefzbMATHGoogle Scholar
  70. 70.
    Nishimura, N., Ragde, P., Szeider, S.: Detecting backdoor sets with respect to Horn and binary clauses. In: Proceedings of SAT 2004 Seventh International Conference on Theory and Applications of Satisfiability Testing, Vancouver, BC, Canada, May 10-13, pp. 96–103 (2004)Google Scholar
  71. 71.
    Nishimura, N., Ragde, P., Szeider, S.: Solving #SAT using vertex covers. Acta Informatica 44(7-8), 509–523 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  72. 72.
    Ordyniak, S., Paulusma, D., Szeider, S.: Satisfiability of acyclic and almost acyclic CNF formulas. In: Lodaya, K., Mahajan, M. (eds.) IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science, FSTTCS 2010, Chennai, India, December 15-18. LIPIcs, vol. 8, pp. 84–95. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2010)Google Scholar
  73. 73.
    Ordyniak, S., Szeider, S.: Augmenting tractable fragments of abstract argumentation. In: Walsh, T. (ed.) Proceedings of the 22nd International Joint Conference on Artificial Intelligence, IJCAI 2011, pp. 1033–1038. AAAI Press (2011)Google Scholar
  74. 74.
    Otwell, C., Remshagen, A., Truemper, K.: An effective QBF solver for planning problems. In: Proceedings of MSV/AMCS, pp. 311–316. CSREA Press (2004)Google Scholar
  75. 75.
    Pan, G., Vardi, M.Y.: Fixed-parameter hierarchies inside PSPACE. In: Proceedings of 21th IEEE Symposium on Logic in Computer Science (LICS 2006), Seattle, WA, USA, August 12-15, pp. 27–36. IEEE Computer Society Press (2006)Google Scholar
  76. 76.
    Papadimitriou, C.H.: Computational Complexity. Addison-Wesley (1994)Google Scholar
  77. 77.
    Parsons, S., Wooldridge, M., Amgoud, L.: Properties and complexity of some formal inter-agent dialogues. J. Logic Comput. 13(3), 347–376 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  78. 78.
    Pietrzak, K.: On the parameterized complexity of the fixed alphabet shortest common supersequence and longest common subsequence problems. J. of Computer and System Sciences 67(4), 757–771 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  79. 79.
    Pollock, J.L.: How to reason defeasibly. Artificial Intelligence 57(1), 1–42 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  80. 80.
    Rahwan, I., Simari, G.R. (eds.): Argumentation in Artificial Intelligence. Springer (2009)Google Scholar
  81. 81.
    Raman, V., Saurabh, S., Subramanian, C.R.: Faster fixed parameter tractable algorithms for finding feedback vertex sets. ACM Transactions on Algorithms 2(3), 403–415 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  82. 82.
    Razgon, I., O’Sullivan, B.: Almost 2-SAT is fixed parameter tractable. J. of Computer and System Sciences 75(8), 435–450 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  83. 83.
    Reed, B., Smith, K., Vetta, A.: Finding odd cycle transversals. Oper. Res. Lett. 32(4), 299–301 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  84. 84.
    Rintanen, J.: Constructing conditional plans by a theorem-prover. J. Artif. Intell. Res. 10, 323–352 (1999)MathSciNetzbMATHGoogle Scholar
  85. 85.
    Robertson, N., Seymour, P.D., Thomas, R.: Permanents, Pfaffian orientations, and even directed circuits. Ann. of Math (2) 150(3), 929–975 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  86. 86.
    Roth, D.: On the hardness of approximate reasoning. Artificial Intelligence 82(1-2), 273–302 (1996)MathSciNetCrossRefGoogle Scholar
  87. 87.
    Sabharwal, A., Ansotegui, C., Gomes, C.P., Hart, J.W., Selman, B.: QBF Modeling: Exploiting Player Symmetry for Simplicity and Efficiency. In: Biere, A., Gomes, C.P. (eds.) SAT 2006. LNCS, vol. 4121, pp. 382–395. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  88. 88.
    Samer, M., Szeider, S.: Backdoor trees. In: Twenty-Third Conference on Artificial Intelligence, AAAI 2008, Chicago, Illinois, July 13–17, pp. 363–368. AAAI Press (2008)Google Scholar
  89. 89.
    Samer, M., Szeider, S.: Backdoor sets of quantified Boolean formulas. Journal of Automated Reasoning 42(1), 77–97 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  90. 90.
    Samer, M., Szeider, S.: Fixed-parameter tractability. In: Biere, A., Heule, M., van Maaren, H., Walsh, T. (eds.) Handbook of Satisfiability, vol. ch. 13, pp. 425–454. IOS Press (2009)Google Scholar
  91. 91.
    Samer, M., Szeider, S.: Algorithms for propositional model counting. J. Discrete Algorithms 8(1), 50–64 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  92. 92.
    Sang, T., Beame, P., Kautz, H.A.: Performing bayesian inference by weighted model counting. In: Proceedings, The Twentieth National Conference on Artificial Intelligence and the Seventeenth Innovative Applications of Artificial Intelligence Conference, Pittsburgh, Pennsylvania, USA, July 9-13, pp. 475–482. AAAI Press / The MIT Press (2005)Google Scholar
  93. 93.
    Schaefer, T.J.: The complexity of satisfiability problems. In: Conference Record of the Tenth Annual ACM Symposium on Theory of Computing, San Diego, Calif., pp. 216–226. ACM (1978)Google Scholar
  94. 94.
    Stockmeyer, L.J., Meyer, A.R.: Word problems requiring exponential time. In: Proc. Theory of Computing, pp. 1–9. ACM (1973)Google Scholar
  95. 95.
    Szeider, S.: On Fixed-Parameter Tractable Parameterizations of SAT. In: Giunchiglia, E., Tacchella, A. (eds.) SAT 2003. LNCS, vol. 2919, pp. 188–202. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  96. 96.
    Szeider, S.: Backdoor sets for DLL subsolvers. Journal of Automated Reasoning 35(1-3), 73–88 (2005); Reprinted as Giunchiglia, E., Walsh, T.(eds.): SAT 2005 - Satisfiability Research in the Year 2005, ch. 4. Springer Verlag (2006)Google Scholar
  97. 97.
    Szeider, S.: Matched formulas and backdoor sets. J. on Satisfiability, Boolean Modeling and Computation 6, 1–12 (2008)MathSciNetzbMATHGoogle Scholar
  98. 98.
    Szeider, S.: Limits of preprocessing. In: Proceedings of the Twenty-Fifth Conference on Artificial Intelligence, AAAI 2011, pp. 93–98. AAAI Press, Menlo Park (2011)Google Scholar
  99. 99.
    Velev, M.N., Bryant, R.E.: Effective use of Boolean satisfiability procedures in the formal verification of superscalar and VLIW microprocessors. J. Symbolic Comput. 35(2), 73–106 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  100. 100.
    Weld, D.S.: Recent advances in AI planning. AI Magazine 20(2), 93–123 (1999)Google Scholar
  101. 101.
    Williams, R., Gomes, C., Selman, B.: Backdoors to typical case complexity. In: Gottlob, G., Walsh, T. (eds.) Proceedings of the Eighteenth International Joint Conference on Artificial Intelligence, IJCAI 2003, pp. 1173–1178. Morgan Kaufmann (2003)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Serge Gaspers
    • 1
  • Stefan Szeider
    • 1
  1. 1.Institute of Information SystemsVienna University of TechnologyViennaAustria

Personalised recommendations