Backdoor Treewidth for SAT

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10491)

Abstract

A strong backdoor in a CNF formula is a set of variables such that each possible instantiation of these variables moves the formula into a tractable class. The algorithmic problem of finding a strong backdoor has been the subject of intensive study, mostly within the parameterized complexity framework. Results to date focused primarily on backdoors of small size. In this paper we propose a new approach for algorithmically exploiting strong backdoors for SAT: instead of focusing on small backdoors, we focus on backdoors with certain structural properties. In particular, we consider backdoors that have a certain tree-like structure, formally captured by the notion of backdoor treewidth.

First, we provide a fixed-parameter algorithm for SAT parameterized by the backdoor treewidth w.r.t. the fundamental tractable classes Horn, Anti-Horn, and 2CNF. Second, we consider the more general setting where the backdoor decomposes the instance into components belonging to different tractable classes, albeit focusing on backdoors of treewidth 1 (i.e., acyclic backdoors). We give polynomial-time algorithms for SAT and #SAT for instances that admit such an acyclic backdoor.

References

  1. 1.
    Arnborg, S., Courcelle, B., Proskurowski, A., Seese, D.: An algebraic theory of graph reduction. In: Ehrig, H., Kreowski, H.-J., Rozenberg, G. (eds.) Graph Grammars 1990. LNCS, vol. 532, pp. 70–83. Springer, Heidelberg (1991). doi:10.1007/BFb0017382 CrossRefGoogle Scholar
  2. 2.
    Biere, A., Heule, M., van Maaren, H., Walsh, T. (eds.): Handbook of Satisfiability. Frontiers in Artificial Intelligence and Applications, vol. 185. IOS Press, Amsterdam (2009)MATHGoogle Scholar
  3. 3.
    Bodlaender, H.L., Fluiter, B.: Reduction algorithms for constructing solutions in graphs with small treewidth. In: Cai, J.-Y., Wong, C.K. (eds.) COCOON 1996. LNCS, vol. 1090, pp. 199–208. Springer, Heidelberg (1996). doi:10.1007/3-540-61332-3_153 CrossRefGoogle Scholar
  4. 4.
    Bodlaender, H.L., van Antwerpen-de Fluiter, B.: Reduction algorithms for graphs of small treewidth. Inf. Comput. 167(2), 86–119 (2001)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bodlaender, H.L., Hagerup, T.: Parallel algorithms with optimal speedup for bounded treewidth. In: Fülöp, Z., Gécseg, F. (eds.) ICALP 1995. LNCS, vol. 944, pp. 268–279. Springer, Heidelberg (1995). doi:10.1007/3-540-60084-1_80 CrossRefGoogle Scholar
  6. 6.
    Boros, E., Hammer, P.L., Sun, X.: Recognition of \(q\)-Horn formulae in linear time. Discr. Appl. Math. 55(1), 1–13 (1994)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms, 3rd edn. MIT Press, Cambridge (2009). http://mitpress.mit.edu/books/introduction-algorithms MATHGoogle Scholar
  8. 8.
    Cygan, M., Fomin, F.V., Kowalik, L., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, Cham (2015). doi:10.1007/978-3-319-21275-3 CrossRefMATHGoogle Scholar
  9. 9.
    Diestel, R.: Graph Theory. Graduate Texts in Mathematics, vol. 173, 4th edn. Springer, Heidelberg (2012). doi:10.1007/978-3-662-53622-3 MATHGoogle Scholar
  10. 10.
    Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity. Texts in Computer Science. Springer, London (2013). doi:10.1007/978-1-4471-5559-1 CrossRefMATHGoogle Scholar
  11. 11.
    Fellows, M.R., Langston, M.A.: An analogue of the Myhill-Nerode theorem and its use in computing finite-basis characterizations (extended abstract). In: FOCS, pp. 520–525 (1989)Google Scholar
  12. 12.
    de Fluiter, B.: Algorithms for graphs of small treewidth. Ph.D. thesis, Utrecht University (1997)Google Scholar
  13. 13.
    Fomin, F.V., Lokshtanov, D., Misra, N., Ramanujan, M.S., Saurabh, S.: Solving d-SAT via backdoors to small treewidth. In: Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015, San Diego, CA, USA, pp. 630–641, 4–6 January 2015 (2015)Google Scholar
  14. 14.
    Ganian, R., Ramanujan, M.S., Szeider, S.: Combining treewidth and backdoors for CSP. In: 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), vol. 66, pp. 36:1–36:17. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik (2017)Google Scholar
  15. 15.
    Ganian, R., Ramanujan, M.S., Szeider, S.: Discovering archipelagos of tractability for constraint satisfaction and counting. ACM Trans. Algorithms 13(2), 29:1–29:32 (2017). http://doi.acm.org/10.1145/3014587 MathSciNetCrossRefGoogle Scholar
  16. 16.
    Gaspers, S., Szeider, S.: Backdoors to satisfaction. In: Bodlaender, H.L., Downey, R., Fomin, F.V., Marx, D. (eds.) The Multivariate Algorithmic Revolution and Beyond. LNCS, vol. 7370, pp. 287–317. Springer, Heidelberg (2012). doi:10.1007/978-3-642-30891-8_15 CrossRefGoogle Scholar
  17. 17.
    Grohe, M., Kawarabayashi, K., Marx, D., Wollan, P.: Finding topological subgraphs is fixed-parameter tractable. In: Proceedings of the 43rd ACM Symposium on Theory of Computing, STOC 2011, San Jose, CA, USA, pp. 479–488, 6–8 June 2011Google Scholar
  18. 18.
    Hopcroft, J.E., Tarjan, R.E.: Efficient algorithms for graph manipulation [H] (algorithm 447). Commun. ACM 16(6), 372–378 (1973)CrossRefGoogle Scholar
  19. 19.
    Kleine Büning, H., Kullmann, O.: Minimal unsatisfiability and autarkies, Chap. 11. In: Biere, A., Heule, M.J.H., van Maaren, H., Walsh, T. (eds.) Handbook of Satisfiability. Frontiers in Artificial Intelligence and Applications, vol. 185, pp. 339–401. IOS Press, Amsterdam (2009)Google Scholar
  20. 20.
    Kleine Büning, H., Zhao, X.: Satisfiable formulas closed under replacement. In: Kautz, H., Selman, B. (eds.) Proceedings for the Workshop on Theory and Applications of Satisfiability. Electronic Notes in Discrete Mathematics, vol. 9. Elsevier Science Publishers, North-Holland (2001)Google Scholar
  21. 21.
    Kloks, T. (ed.): Treewidth: Computations and Approximations. LNCS, vol. 842. Springer, Heidelberg (1994). doi:10.1007/BFb0045375 MATHGoogle Scholar
  22. 22.
    Nishimura, N., Ragde, P., Szeider, S.: Detecting backdoor sets with respect to Horn and binary clauses. In: Proceedings of Seventh International Conference on Theory and Applications of Satisfiability Testing (SAT 2004), Vancouver, BC, Canada, pp. 96–103, 10–13 May 2004Google Scholar
  23. 23.
    Ordyniak, S., Paulusma, D., Szeider, S.: Satisfiability of acyclic and almost acyclic CNF formulas. Theor. Comput. Sci. 481, 85–99 (2013)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Robertson, N., Seymour, P.D.: Graph minors. II. Algorithmic aspects of tree-width. J. Algorithms 7(3), 309–322 (1986)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Samer, M., Szeider, S.: Fixed-parameter tractability, Chap. 13. In: Biere, A., Heule, M., van Maaren, H., Walsh, T. (eds.) Handbook of Satisfiability, pp. 425–454. IOS Press, Amsterdam (2009)Google Scholar
  26. 26.
    Samer, M., Szeider, S.: Algorithms for propositional model counting. J. Discrete Algorithms 8(1), 50–64 (2010)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Samer, M., Szeider, S.: Constraint satisfaction with bounded treewidth revisited. J. Comput. Syst. Sci. 76(2), 103–114 (2010)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Williams, R., Gomes, C., Selman, B.: On the connections between backdoors, restarts, and heavy-tailedness in combinatorial search. In: Informal Proceedings of the Sixth International Conference on Theory and Applications of Satisfiability Testing (SAT 2003), S. Margherita Ligure - Portofino, Italy, pp. 222–230, 5–8 May 2003Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Robert Ganian
    • 1
  • M. S. Ramanujan
    • 1
  • Stefan Szeider
    • 1
  1. 1.Algorithms and Complexity GroupTU WienViennaAustria

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