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Fixed-Parameter Tractable Reductions to SAT

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Theory and Applications of Satisfiability Testing – SAT 2014 (SAT 2014)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8561))

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Abstract

Today’s SAT solvers have an enormous importance and impact in many practical settings. They are used as efficient back-end to solve many NP-complete problems. However, many computational problems are located at the second level of the Polynomial Hierarchy or even higher, and hence polynomial-time transformations to SAT are not possible, unless the hierarchy collapses. In certain cases one can break through these complexity barriers by fixed-parameter tractable (fpt) reductions which exploit structural aspects of problem instances in terms of problem parameters. Recent research established a general theoretical framework that supports the classification of parameterized problems on whether they admit such an fpt-reduction to SAT or not. We use this framework to analyze some problems that are related to Boolean satisfiability. We consider several natural parameterizations of these problems, and we identify for which of these an fpt-reduction to SAT is possible. The problems that we look at are related to minimizing an implicant of a DNF formula, minimizing a DNF formula, and satisfiability of quantified Boolean formulas.

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References

  1. Atserias, A., Oliva, S.: Bounded-width QBF is PSPACE-complete. In: Portier, N., Wilke, T. (eds.) 30th International Symposium on Theoretical Aspects of Computer Science, STACS 2013, Kiel, Germany, February 27 - March 2. LIPIcs, vol. 20, pp. 44–54. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2013)

    Google Scholar 

  2. Ayari, A., Basin, D.: Qubos: Deciding quantified Boolean logic using propositional satisfiability solvers. In: Aagaard, M.D., O’Leary, J.W. (eds.) FMCAD 2002. LNCS, vol. 2517, pp. 187–201. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  3. Belov, A., Lynce, I., Marques-Silva, J.: Towards efficient MUS extraction. AI Commun. 25(2), 97–116 (2012)

    MATH  MathSciNet  Google Scholar 

  4. Benedetti, M., Bernardini, S.: Incremental compilation-to-SAT procedures. In: Hoos, H.H., Mitchell, D.G. (eds.) SAT 2004. LNCS, vol. 3542, pp. 46–58. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  5. Biere, A.: Resolve and expand. In: Hoos, H.H., Mitchell, D.G. (eds.) SAT 2004. LNCS, vol. 3542, pp. 59–70. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  6. Biere, A.: Bounded model checking. In: Biere, A., Heule, M., van Maaren, H., Walsh, T. (eds.) Handbook of Satisfiability, Frontiers in Artificial Intelligence and Applications, vol. 185, pp. 457–481. IOS Press (2009)

    Google Scholar 

  7. Biere, A., Cimatti, A., Clarke, E., Zhu, Y.: Symbolic model checking without BDDs. In: Cleaveland, W.R. (ed.) TACAS/ETAPS 1999. LNCS, vol. 1579, pp. 193–207. Springer, Heidelberg (1999)

    Google Scholar 

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

    Google Scholar 

  9. Bodlaender, H.L.: Dynamic programming on graphs with bounded treewidth. In: Lepistö, T., Salomaa, A. (eds.) ICALP 1988. LNCS, vol. 317, pp. 105–118. Springer, Heidelberg (1988)

    Chapter  Google Scholar 

  10. Bodlaender, H.L.: A partial k-arboretum of graphs with bounded treewidth. Theoretical Computer Science 209(1-2), 1–45 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  11. Bodlaender, H.L.: Fixed-parameter tractability of treewidth and pathwidth. In: Bodlaender, H.L., Downey, R., Fomin, F.V., Marx, D. (eds.) Fellows Festschrift 2012. LNCS, vol. 7370, pp. 196–227. Springer, Heidelberg (2012)

    Chapter  Google Scholar 

  12. Chang, R., Kadin, J.: The Boolean Hierarchy and the Polynomial Hierarchy: A closer connection. SIAM J. Comput. 25(2), 340–354 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  13. Downey, R.G., Fellows, M.R.: Parameterized Complexity. Monographs in Computer Science. Springer, New York (1999)

    Book  Google Scholar 

  14. Dvořák, W., Järvisalo, M., Wallner, J.P., Woltran, S.: Complexity-sensitive decision procedures for abstract argumentation. Artificial Intelligence 206(0), 53–78 (2014)

    MathSciNet  Google Scholar 

  15. Fichte, J.K., Szeider, S.: Backdoors to normality for disjunctive logic programs. In: Proceedings of the Twenty-Seventh AAAI Conference on Artificial Intelligence, AAAI 2013, pp. 320–327. AAAI Press (2013)

    Google Scholar 

  16. Flum, J., Grohe, M.: Describing parameterized complexity classes. Information and Computation 187(2), 291–319 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  17. Flum, J., Grohe, M.: Parameterized Complexity Theory, Texts in Theoretical Computer Science. An EATCS Series, vol. XIV. Springer, Berlin (2006)

    Google Scholar 

  18. Garey, M.R., Johnson, D.R.: Computers and Intractability. W. H. Freeman and Company, New York (1979)

    MATH  Google Scholar 

  19. Goldsmith, J., Hagen, M., Mundhenk, M.: Complexity of DNF minimization and isomorphism testing for monotone formulas. Information and Computation 206(6), 760–775 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  20. 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 

  21. Gottlob, G., Fermüller, C.G.: Removing redundancy from a clause. Artificial Intelligence 61(2), 263–289 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  22. Gottlob, G., Pichler, R., Wei, F.: Bounded treewidth as a key to tractability of knowledge representation and reasoning. Artificial Intelligence 174(1), 105–132 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  23. Grégoire, E., Mazure, B., Piette, C.: On approaches to explaining infeasibility of sets of Boolean clauses. In: 20th IEEE International Conference on Tools with Artificial Intelligence (ICTAI 2008), Daytion, Ohio, USA, November 3-5 , pp. 74–83. IEEE Computer Society (2008)

    Google Scholar 

  24. de Haan, R., Szeider, S.: The parameterized complexity of reasoning problems beyond NP. In: Baral, C., De Giacomo, G., Eiter, T. (eds.) Principles of Knowledge Representation and Reasoning: Proceedings of the Fourteenth International Conference, Vienna, Austria, July 20-24. AAAI Press (2014)

    Google Scholar 

  25. Hartmanis, J.: New developments in structural complexity theory. Theoretical Computer Science 71(1), 79–93 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  26. Hooker, J.N.: Solving the incremental satisfiability problem. J. Logic Programming 15(1&2), 177–186 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  27. Kloks, T.: Treewidth: Computations and Approximations. Springer, Berlin (1994)

    Book  MATH  Google Scholar 

  28. 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 

  29. Krajicek, J.: Bounded arithmetic, propositional logic and complexity theory. Cambridge University Press (1995)

    Google Scholar 

  30. Krentel, M.W.: The complexity of optimization problems. J. of Computer and System Sciences 36(3), 490–509 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  31. Malik, S., Zhang, L.: Boolean satisfiability from theoretical hardness to practical success. Communications of the ACM 52(8), 76–82 (2009)

    Article  Google Scholar 

  32. Marques-Silva, J., Janota, M., Belov, A.: Minimal sets over monotone predicates in Boolean formulae. In: Sharygina, N., Veith, H. (eds.) CAV 2013. LNCS, vol. 8044, pp. 592–607. Springer, Heidelberg (2013)

    Chapter  Google Scholar 

  33. Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford Lecture Series in Mathematics and its Applications. Oxford University Press, Oxford (2006)

    Book  MATH  Google Scholar 

  34. 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 (2006)

    Google Scholar 

  35. Papadimitriou, C.H.: Computational Complexity. Addison-Wesley (1994)

    Google Scholar 

  36. Pfandler, A., Rümmele, S., Szeider, S.: Backdoors to abduction. In: Rossi, F. (ed.) Proceedings of the 23rd International Joint Conference on Artificial Intelligence, IJCAI 2013. AAAI Press/IJCAI (2013)

    Google Scholar 

  37. Prestwich, S.D.: CNF encodings. In: Biere, A., Heule, M., van Maaren, H., Walsh, T. (eds.) Handbook of Satisfiability, pp. 75–97. IOS Press (2009)

    Google Scholar 

  38. Sakallah, K.A., Marques-Silva, J.: Anatomy and empirical evaluation of modern SAT solvers. Bulletin of the European Association for Theoretical Computer Science 103, 96–121 (2011)

    MathSciNet  Google Scholar 

  39. Samer, M., Szeider, S.: Constraint satisfaction with bounded treewidth revisited. J. of Computer and System Sciences 76(2), 103–114 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  40. Stockmeyer, L.J.: The polynomial-time hierarchy. Theoretical Computer Science 3(1), 1–22 (1976)

    Article  MathSciNet  Google Scholar 

  41. Umans, C.: Approximability and Completeness in the Polynomial Hierarchy. Ph.D. thesis. University of California, Berkeley (2000)

    Google Scholar 

  42. Wagner, K.W.: Bounded query classes. SIAM J. Comput. 19(5), 833–846 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  43. Whittemore, J., Kim, J., Sakallah, K.A.: SATIRE: A new incremental satisfiability engine. In: Proceedings of the 38th Design Automation Conference, DAC 2001, Las Vegas, NV, USA, June 18-22, pp. 542–545. ACM (2001)

    Google Scholar 

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Authors and Affiliations

  1. Institute of Information Systems, Vienna University of Technology, Vienna, Austria

    Ronald de Haan & Stefan Szeider

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  1. Ronald de Haan
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  2. Stefan Szeider
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Editors and Affiliations

  1. Karlsruher Institut für Technologie (KIT), Am Fasanengarten 5, 76131, Karlsruhe, Germany

    Carsten Sinz

  2. TU Wien, Favoritenstraße 9-11, 1040, Wien, Austria

    Uwe Egly

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de Haan, R., Szeider, S. (2014). Fixed-Parameter Tractable Reductions to SAT. In: Sinz, C., Egly, U. (eds) Theory and Applications of Satisfiability Testing – SAT 2014. SAT 2014. Lecture Notes in Computer Science, vol 8561. Springer, Cham. https://doi.org/10.1007/978-3-319-09284-3_8

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  • DOI: https://doi.org/10.1007/978-3-319-09284-3_8

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-09283-6

  • Online ISBN: 978-3-319-09284-3

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