Tradeoffs in the Complexity of Backdoor Detection

  • Bistra Dilkina
  • Carla P. Gomes
  • Ashish Sabharwal
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4741)

Abstract

There has been considerable interest in the identification of structural properties of combinatorial problems that lead to efficient algorithms for solving them. Some of these properties are “easily” identifiable, while others are of interest because they capture key aspects of state-of-the-art constraint solvers. In particular, it was recently shown that the problem of identifying a strong Horn- or 2CNF-backdoor can be solved by exploiting equivalence with deletion backdoors, and is NP-complete. We prove that strong backdoor identification becomes harder than NP (unless NP=coNP) as soon as the inconsequential sounding feature of empty clause detection (present in all modern SAT solvers) is added. More interestingly, in practice such a feature as well as polynomial time constraint propagation mechanisms often lead to much smaller backdoor sets. In fact, despite the worst-case complexity results for strong backdoor detection, we show that Satz-Rand is remarkably good at finding small strong backdoors on a range of experimental domains. Our results suggest that structural notions explored for designing efficient algorithms for combinatorial problems should capture both statically and dynamically identifiable properties.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Brockington, M., Culberson, J.C.: Camouflaging independent sets in quasi-random graphs. In: Johnson, D.S., Trick, M.A. (eds.) Cliques, Coloring, and Satisfiability: Second DIMACS Implementation Challenge, American Mathematical Society, vol. 26, pp. 75–88 (1996)Google Scholar
  2. 2.
    Chandru, V., Hooker, J.N.: Detecting embedded Horn structure in propositional logic. Information Processing Letters 42(2), 109–111 (1992)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Chen, H., Dalmau, V.: Beyond hypertree width: Decomposition methods without decompositions. In: van Beek, P. (ed.) CP 2005. LNCS, vol. 3709, pp. 167–181. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  4. 4.
    Chen, H., Gomes, C., Selman, B.: Formal models of heavy-tailed behavior in combinatorial search. In: Walsh, T. (ed.) CP 2001. LNCS, vol. 2239, Springer, Heidelberg (2001)CrossRefGoogle Scholar
  5. 5.
    Dechter, R.: Constraint Processing. Morgan Kaufmann Publishers Inc., San Francisco (2003)Google Scholar
  6. 6.
    Dechter, R., Pearl, J.: Network-based heuristics for constraint-satisfaction problems. Artif. Intell. 34(1), 1–38 (1987)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Deville, Y., Van Hentenryck, P.: An efficient arc consistency algorithm for a class of csp problems. In: IJCAI 1991, pp. 325–330 (1991)Google Scholar
  8. 8.
    Freuder, E.C.: A sufficient condition for backtrack-free search. J. ACM 29(1), 24–32 (1982)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Freuder, E.C.: A sufficient condition for backtrack-bounded search. J. ACM 32(4), 755–761 (1985)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Freuder, E.C.: Complexity of k-tree structured constraint satisfaction problems. In: AAAI 1990, Boston, MA, pp. 4–9 (1990)Google Scholar
  11. 11.
    Gomes, C., Selman, B., Kautz, H.: Boosting Combinatorial Search Through Randomization. In: AAAI 1998, New Providence, RI, pp. 431–438 (1998)Google Scholar
  12. 12.
    Gomes, C.P., Selman, B., Crato, N., Kautz, H.: Heavy-tailed phenomena in satisfiability and constraint satisfaction problems. J. Autom. Reason. 24(1-2), 67–100 (2000)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Hoffmann, J., Gomes, C., Selman, B.: Structure and problem hardness: Goal asymmetry and DPLL proofs in SAT-based planning. Logical Methods in Computer Science 3(1:6) (2007)Google Scholar
  14. 14.
    ILOG, SA: CPLEX 10.1 Reference Manual (2006)Google Scholar
  15. 15.
    Kilby, P., Slaney, J.K., Thiébaux, S., Walsh, T.: Backbones and backdoors in satisfiability. In: AAAI 2005, pp. 1368–1373 (2005)Google Scholar
  16. 16.
    Li, C.M., Anbulagan: Heuristics based on unit propagation for satisfiability problems. In: IJCAI 1997, pp. 366–371 (1997)Google Scholar
  17. 17.
    Lynce, I., Marques-Silva, J.: Hidden structure in unsatisfiable random 3-SAT: An empirical study. In: ICTAI 2004 (2004)Google Scholar
  18. 18.
    Moskewicz, M.W., Madigan, C.F., Zhao, Y., Zhang, L., Malik, S.: Chaff: engineering an efficient SAT solver. In: DAC 2001, pp. 530–535 (2001) ISBN 1-58113-297-2. Google Scholar
  19. 19.
    Nishimura, N., Ragde, P., Szeider, S.: Detecting backdoor sets with respect to Horn and binary clauses. In: Hoos, H.H., Mitchell, D.G. (eds.) SAT 2004. LNCS, vol. 3542. Springer, Heidelberg (2005)Google Scholar
  20. 20.
    Paris, L., Ostrowski, R., Siegel, P., Sais, L.: Computing Horn strong backdoor sets thanks to local search. In: ICTAI 2006, pp. 139–143 (2006), http://doi.ieeecomputersociety.org/10.1109/ICTAI.2006.43 ISSN 1082-3409
  21. 21.
    Samer, M., Szeider, S.: Constraint satisfaction with bounded treewidth revisited. In: Benhamou, F. (ed.) CP 2006. LNCS, vol. 4204, pp. 499–513. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  22. 22.
    Sinz, C., Kaiser, A., Küchlin, W.: Formal methods for the validation of automotive product configuration data. Artificial Intelligence for Engr. Design, Analysis and Manufacturing 17(1), 75–97 (2003)Google Scholar
  23. 23.
    Szeider, S.: Backdoor sets for DLL subsolvers. J. of Automated Reasoning (2005)Google Scholar
  24. 24.
    van Beek, P., Dechter, R.: On the minimality and global consistency of row-convex constraint networks. J. ACM 42(3), 543–561 (1995)MATHCrossRefGoogle Scholar
  25. 25.
    Williams, R., Gomes, C., Selman, B.: Backdoors to typical case complexity. In: IJCAI 2003, pp. 1173–1178 (2003)Google Scholar
  26. 26.
    Williams, R., Gomes, C., Selman, B.: On the connections between heavy-tails, backdoors, and restarts in combinatorial search. In: Giunchiglia, E., Tacchella, A. (eds.) SAT 2003. LNCS, vol. 2919, pp. 222–230. Springer, Heidelberg (2004)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Bistra Dilkina
    • 1
  • Carla P. Gomes
    • 1
  • Ashish Sabharwal
    • 1
  1. 1.Cornell University, Department of Computer Science, Ithaca, NY 14850USA

Personalised recommendations