Advertisement

Learning-Sensitive Backdoors with Restarts

  • Edward ZulkoskiEmail author
  • Ruben Martins
  • Christoph M. Wintersteiger
  • Robert Robere
  • Jia Hui Liang
  • Krzysztof Czarnecki
  • Vijay Ganesh
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11008)

Abstract

Restarts are a pivotal aspect of conflict-driven clause-learning (CDCL) SAT solvers, yet it remains unclear when they are favorable in practice, and whether they offer additional power in theory. In this paper, we consider the power of restarts through the lens of backdoors. Extending the notion of learning-sensitive (LS) backdoors, we define a new parameter called learning-sensitive with restarts (LSR) backdoors. Broadly speaking, we show that LSR backdoors are a powerful parametric lens through which to understand the impact of restarts on SAT solver performance, and specifically on the kinds of proofs constructed by SAT solvers. First, we prove that when backjumping is disallowed, LSR backdoors can be exponentially smaller than LS backdoors. Second, we demonstrate that the size of LSR backdoors are dependent on the learning scheme used during search. Finally, we present new algorithms to compute upper-bounds on LSR backdoors that intrinsically rely upon restarts, and can be computed with a single run of a CDCL SAT solver. We empirically demonstrate that this can often produce much smaller backdoors than previous approaches to computing LS backdoors. We conclude with empirical results on industrial benchmarks which demonstrate that rapid restart policies tend to produce more “local” proofs than other heuristics, in terms of the number of unique variables found in learned clauses of the proof.

Keywords

SAT solving Backdoors Restarts CDCL 

References

  1. 1.
    Atserias, A., Fichte, J.K., Thurley, M.: Clause-learning algorithms with many restarts and bounded-width resolution. J. Artif. Intell. Res. 40, 353–373 (2011)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Beame, P., Kautz, H., Sabharwal, A.: Towards understanding and harnessing the potential of clause learning. J. Artif. Intell. Res. 22, 319–351 (2004)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Beame, P., Kautz, H., Sabharwal, A.: Understanding and harnessing the potential of clause learning. J. Artif. Intell. Res. 22, 319–351 (2014)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Beame, P., Sabharwal, A.: Non-restarting sat solvers with simple preprocessing can efficiently simulate resolution. In: AAAI Conference on Artificial Intelligence, pp. 2608–2615. AAAI Press (2014)Google Scholar
  5. 5.
    Biere, A., Fröhlich, A.: Evaluating CDCL restart schemes. In: Pragmatics of SAT Workshop (2015)Google Scholar
  6. 6.
    Biere, A., Heule, M., van Maaren, H.: Handbook of satisfiability, vol. 185. IOS Press (2009)Google Scholar
  7. 7.
    Buss, S.R., Hoffmann, J., Johannsen, J.: Resolution trees with lemmas: resolution refinements that characterize DLL algorithms with clause learning. arXiv preprint arXiv:0811.1075 (2008)
  8. 8.
    Dilkina, B., Gomes, C.P., Malitsky, Y., Sabharwal, A., Sellmann, M.: Backdoors to combinatorial optimization: feasibility and optimality. In: van Hoeve, W.-J., Hooker, J.N. (eds.) CPAIOR 2009. LNCS, vol. 5547, pp. 56–70. Springer, Heidelberg (2009).  https://doi.org/10.1007/978-3-642-01929-6_6CrossRefzbMATHGoogle Scholar
  9. 9.
    Dilkina, B., Gomes, C.P., Sabharwal, A.: Backdoors in the context of learning. In: Kullmann, O. (ed.) SAT 2009. LNCS, vol. 5584, pp. 73–79. Springer, Heidelberg (2009).  https://doi.org/10.1007/978-3-642-02777-2_9CrossRefzbMATHGoogle Scholar
  10. 10.
    Dilkina, B., Gomes, C.P., Sabharwal, A.: Tradeoffs in the complexity of backdoors to satisfiability: dynamic sub-solvers and learning during search. Ann. Math. Artif. Intell. 70(4), 399–431 (2014).  https://doi.org/10.1007/s10472-014-9407-9MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Ganian, R., Ramanujan, M.S., Szeider, S.: Backdoor treewidth for SAT. In: Gaspers, S., Walsh, T. (eds.) SAT 2017. LNCS, vol. 10491, pp. 20–37. Springer, Cham (2017).  https://doi.org/10.1007/978-3-319-66263-3_2CrossRefzbMATHGoogle Scholar
  12. 12.
    Godefroid, P., Levin, M.Y., Molnar, D.A., et al.: Automated whitebox fuzz testing. In: Network and Distributed System Security Symposium, pp. 151–166. Internet Society (2008)Google Scholar
  13. 13.
    Haim, S., Heule, M.: Towards ultra rapid restarts. arXiv preprint arXiv:1402.4413 (2014)
  14. 14.
    Hertel, P., Bacchus, F., Pitassi, T., Van Gelder, A.: Clause learning can effectively p-simulate general propositional resolution. In: AAAI Conference on Artificial Intelligence, pp. 283–290. AAAI Press (2008)Google Scholar
  15. 15.
    Huang, J.: The effect of restarts on the efficiency of clause learning. In: International Joint Conference on Artificial Intelligence, pp. 2318–2323. AAAI Press (2007)Google Scholar
  16. 16.
    Kilby, P., Slaney, J., Thiébaux, S., Walsh, T.: Backbones and backdoors in satisfiability. In: AAAI Conference on Artificial Intelligence, pp. 1368–1373. AAAI Press (2005)Google Scholar
  17. 17.
    Li, Z., van Beek, P.: Finding small backdoors in SAT instances. In: Butz, C., Lingras, P. (eds.) AI 2011. LNCS (LNAI), vol. 6657, pp. 269–280. Springer, Heidelberg (2011).  https://doi.org/10.1007/978-3-642-21043-3_33CrossRefGoogle Scholar
  18. 18.
    Liang, J.H., Ganesh, V., Poupart, P., Czarnecki, K.: Learning rate based branching heuristic for SAT solvers. In: Creignou, N., Le Berre, D. (eds.) SAT 2016. LNCS, vol. 9710, pp. 123–140. Springer, Cham (2016).  https://doi.org/10.1007/978-3-319-40970-2_9CrossRefzbMATHGoogle Scholar
  19. 19.
    Luby, M., Sinclair, A., Zuckerman, D.: Optimal speedup of las vegas algorithms. Inf. Process. Lett. 47(4), 173–180 (1993)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Marques-Silva, J.P., Sakallah, K.A.: GRASP: a search algorithm for propositional satisfiability. IEEE Trans. Comput. 48(5), 506–521 (1999)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Moskewicz, M.W., Madigan, C.F., Zhao, Y., Zhang, L., Malik, S.: Chaff: engineering an efficient SAT solver. In: Design Automation Conference, pp. 530–535. ACM (2001)Google Scholar
  22. 22.
    Nadel, A., Ryvchin, V.: Chronological backtracking. In: Beyersdorff, O., Wintersteiger, C.M. (eds.) SAT 2018. LNCS, vol. 10929, pp. 111–121. Springer, Cham (2018).  https://doi.org/10.1007/978-3-319-94144-8_7CrossRefGoogle Scholar
  23. 23.
    Pipatsrisawat, K., Darwiche, A.: A new clause learning scheme for efficient unsatisfiability proofs. In: AAAI Conference on Artificial Intelligence, pp. 1481–1484 (2008)Google Scholar
  24. 24.
    Pipatsrisawat, K., Darwiche, A.: On the power of clause-learning SAT solvers with restarts. In: Gent, I.P. (ed.) CP 2009. LNCS, vol. 5732, pp. 654–668. Springer, Heidelberg (2009).  https://doi.org/10.1007/978-3-642-04244-7_51CrossRefGoogle Scholar
  25. 25.
    Pipatsrisawat, K., Darwiche, A.: On the power of clause-learning SAT solvers as resolution engines. Artif. Intell. 175, 512–525 (2011).  https://doi.org/10.1016/j.artint.2010.002MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Ruan, Y., Kautz, H., Horvitz, E.: The backdoor key: a path to understanding problem hardness. In: AAAI Conference on Artificial Intelligence, pp. 124–130. AAAI Press (2004)Google Scholar
  27. 27.
    Samer, M., Szeider, S.: Backdoor trees. In: AAAI Conference on Artificial Intelligence, pp. 363–368. AAAI Press (2008)Google Scholar
  28. 28.
    Gelder, A.: Pool resolution and its relation to regular resolution and DPLL with clause learning. In: Sutcliffe, G., Voronkov, A. (eds.) LPAR 2005. LNCS (LNAI), vol. 3835, pp. 580–594. Springer, Heidelberg (2005).  https://doi.org/10.1007/11591191_40CrossRefzbMATHGoogle Scholar
  29. 29.
    Williams, R., Gomes, C., Selman, B.: On the connections between backdoors, restarts, and heavy-tailedness in combinatorial search. In: International Conference on Theory and Applications of Satisfiability Testing, pp. 222–230. Springer (2003). https://doi.org/10.1.1.128.5725
  30. 30.
    Zhang, L., Madigan, C.F., Moskewicz, M.H., Malik, S.: Efficient conflict driven learning in a boolean satisfiability solver. In: Proceedings of the 2001 IEEE/ACM International Conference on Computer-Aided Design, pp. 279–285. IEEE Press (2001)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Edward Zulkoski
    • 1
    Email author
  • Ruben Martins
    • 2
  • Christoph M. Wintersteiger
    • 3
  • Robert Robere
    • 4
  • Jia Hui Liang
    • 1
  • Krzysztof Czarnecki
    • 1
  • Vijay Ganesh
    • 1
  1. 1.University of WaterlooWaterlooCanada
  2. 2.Carnegie Mellon UniversityPittsburghUSA
  3. 3.Microsoft ResearchCambridgeUK
  4. 4.University of TorontoTorontoCanada

Personalised recommendations