Advertisement

On Fixed-Parameter Tractable Parameterizations of SAT

  • Stefan Szeider
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2919)

Abstract

We survey and compare parameterizations of the propositional satisfiability problem (SAT) in the framework of Parameterized Complexity (Downey and Fellows, 1999). In particular, we consider (a) parameters based on structural graph decompositions (tree-width, branch-width, and clique-width), (b) a parameter emerging from matching theory (maximum deficiency), and (c) a parameter defined by translating clause-sets into certain implicational formulas (falsum number).

Keywords

Conjunctive Normal Form Tree Decomposition Primal Graph Satisfying Assignment Graph Minor 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aharoni, R., Linial, N.: Minimal non-two-colorable hypergraphs and minimal unsatisfiable formulas. J. Combin. Theory Ser. A 43, 196–204 (1986)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Alekhnovich, M., Razborov, A.A.: Satisfiability, branch-width and Tseitin tautologies. In: 43rd Annual IEEE Symposium on Foundations of Computer Science (FOCS 2002), pp. 593–603 (2002)Google Scholar
  3. 3.
    Arnborg, S., Corneil, D.G., Proskurowski, A.: Complexity of finding embeddings in a k-tree. SIAM J. Algebraic Discrete Methods 8(2), 277–284 (1987)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    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 (2003) (to appear)Google Scholar
  5. 5.
    Bodlaender, H.L.: A linear-time algorithm for finding tree-decompositions of small treewidth. SIAM J. Comput. 25(6), 1305–1317 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Bodlaender, H.L.: A partial k-arboretum of graphs with bounded treewidth. Theoret. Comput. Sci. 209(1-2), 1–45 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Cesati, M.: Compendium of parameterized problems (2001), http://bravo.ce.uniroma2.it/home/cesati/research/
  8. 8.
    Cook, S.A.: An exponential example for analytic tableaux (1972) (manuscript)Google Scholar
  9. 9.
    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)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Courcelle, B., Olariu, S.: Upper bounds to the clique width of graphs. Discr. Appl. Math. 101(1-3), 77–114 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Davis, M., Logemann, G., Loveland, D.: A machine program for theoremproving. Comm. ACM 5, 394–397 (1962)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, Heidelberg (1999)Google Scholar
  13. 13.
    Fleischner, H., Földes, S., Szeider, S.: Remarks on the concept of robust algorithm. Technical Report RRR 26-2001, Rutgers Center for Operations Research (RUTCOR) (April 2001)Google Scholar
  14. 14.
    Fleischner, H., Kullmann, O., Szeider, S.: Polynomial-time recognition of minimal unsatisfiable formulas with fixed clause-variable difference. Theoret. Comput. Sci. 289(1), 503–516 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Franco, J., Goldsmith, J., Schlipf, J., Speckenmeyer, E., Swaminathan, R.P.: An algorithm for the class of pure implicational formulas. Discr. Appl. Math. 97, 89–106 (1999)CrossRefMathSciNetGoogle Scholar
  16. 16.
    Golumbic, M.C., Rotics, U.: On the clique-width of some perfect graph classes. In: Widmayer, P., Neyer, G., Eidenbenz, S. (eds.) WG 1999. LNCS, vol. 1665, pp. 423–443. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  17. 17.
    Gottlob, G., Pichler, R.: Hypergraphs in model checking: Acyclicity and hypertree-width versus clique-width. In: Orejas, F., Spirakis, P.G., van Leeuwen, J. (eds.) ICALP 2001. LNCS, vol. 2076, pp. 708–719. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  18. 18.
    Gottlob, G., Scarcello, F., Sideri, M.: Fixed-parameter complexity in AI and nonmonotonic reasoning. Artificial Intelligence 138(1-2), 55–86 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Heusch, P.: The complexity of the falsifiability problem for pure implicational formulas. Discr. Appl. Math. 96/97, 127–138 (1999)CrossRefMathSciNetGoogle Scholar
  20. 20.
    Heusch, P., Porschen, S., Speckenmeyer, E.: Improving a fixed parameter tractability time bound for the shadow problem. Technical Report 2001-425, Universität zu Köln (2001)Google Scholar
  21. 21.
    Kullmann, O.: Investigating a general hierarchy of polynomially decidable classes of CNF’s based on short tree-like resolution proofs. Technical Report TR99–041, Electronic Colloquium on Computational Complexity, ECCC (1999)Google Scholar
  22. 22.
    Kullmann, O.: An application of matroid theory to the SAT problem. In: Fifteenth Annual IEEE Conference on Computational Complexity, pp. 116–124 (2000)Google Scholar
  23. 23.
    Pretolani, D.: Hierarchies of polynomially solvable satisfiability problems. Ann. Math. Artif. Intell. 17(3-4), 339–357 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Robertson, N., Seymour, P.D.: Graph minors, II. Algorithmic aspects of treewidth. J. Algorithms 7(3), 309–322 (1986)zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Robertson, N., Seymour, P.D.: Graph minors, X. Obstructions to treedecomposition. J. Combin. Theory Ser. B 52(2), 153–190 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Robertson, N., Seymour, P.D.: Graph minors, XIII. The disjoint paths problem. J. Combin. Theory Ser. B 63(1), 65–110 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Spinrad, J.P.: Representations of graphs. Book manuscript, Vanderbilt University (1997)Google Scholar
  28. 28.
    Szeider, S.: Generalizations of matched CNF formulas. Ann. Math. Artif. Intell., Special issue with selected papers from the 5th Int. Symp. on the Theory and Applications of Satisfiability Testing, SAT 2002 (2002) (to appear)Google Scholar
  29. 29.
    Szeider, S.: Minimal unsatisfiable formulas with bounded clause-variable difference are fixed-parameter tractable. In: Warnow, T.J., Zhu, B. (eds.) COCOON 2003. LNCS, vol. 2697, pp. 548–558. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  30. 30.
    Urquhart, A.: The complexity of propositional proofs. Bull. of Symbolic Logic 1(4), 425–467 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Yannakakis, M.: Algorithms for acyclic database schemes. In: Zaniolo, C., Delobel, C. (eds.) Very Large Data Bases, 7th International Conference, Cannes, France, September 9-11, pp. 81–94. IEEE Computer Society, Los Alamitos (1981)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Stefan Szeider
    • 1
  1. 1.Department of Computer ScienceUniversity of TorontoTorontoCanada

Personalised recommendations