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Tradeoffs in the complexity of backdoors to satisfiability: dynamic sub-solvers and learning during search

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Abstract

There has been considerable interest in the identification of structural properties of combinatorial problems that lead to efficient algorithms for solving them. Some forms of structure, such as properties of the underlying constraint graph, are “easily” identifiable. Others, such as backdoor sets, are of interest because they capture key aspects of state-of-the-art constraint solvers as well as of many real-world problem instances. While backdoors were originally introduced to capture propagation based simplification mechanisms of solvers, they have recently been studied also in the context of tractable syntactic classes such as 2CNF and Horn. These syntactic classes, however, do not capture key aspects of solvers such as empty clause (i.e., violated constraint) detection. We show that incorporating inconsequential sounding features such as empty clause detection has a dramatic impact on both the complexity of finding a backdoor of size k (which becomes harder in the worst case) and on the size of the resulting backdoor (which can become arbitrarily smaller). Empirically, we show that commonly employed polynomial-time “dynamic” constraint propagation mechanisms, such as unit propagation, pure literal elimination, and probing, often lead to much smaller backdoor sets in real-world domains than “statically” defined classes such as Horn and RHorn, thereby capturing structure much more succinctly. We also reveal the inherent limits of the simpler concept of deletion backdoors, specifically by looking at renamable Horn sub-formulas. Finally, we extend the notion of backdoors to incorporate learning during search—a key aspect of nearly all state-of-the-art systematic SAT solvers—and show, both theoretically and empirically, that this drastically reduces the size of the resulting backdoor set. Our results suggest that structural notions explored for designing efficient algorithms for combinatorial problems should capture both statically and dynamically identifiable properties of the combinatorial problem being solved.

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References

  1. Aspvall, B.: Recognizing disguised NR(1) instances of the satisfiability problem. J. Algoritm. 1(1), 97–103 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  2. Aspvall, B., Plass, M.F., Tarjan, R.E.: A linear-time algorithm for testing the truth of certain quantified boolean formulas. Inf. Process. Lett. 8(3), 121–123 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  3. Beame, P., Kautz, H., Sabharwal, A.: Understanding and harnessing the potential of clause learning. JAIR: J. Artif. Intell. Res. 22, 319–351 (2004)

    MATH  MathSciNet  Google Scholar 

  4. Biere, A., Cimatti, A., Clarke, E.M., Fujita, M., Zhu, Y.: Symbolic model checking using SAT procedures instead of BDDs. In: DAC-99: 36th Design Automation Conference, pp. 317–320 (1999)

  5. Brockington, M., Culberson, J. C.: Camouflaging independent sets in quasi-random graphs. In: Johnson, D.S., Trick, M.A. (eds.) Cliques, Coloring and Satisfiability: The Second DIMACS Implementation Challenge, volume 26 of DIMACS Series in Discrete Mathematics and Theoretical Computer Science, pp. 75–88. American Mathematical Society (1996)

  6. Burch, J.R., Clarke, E.M., McMillan, K.L., Dill, D.L., Hwang, L.J.: Symbolic model checking: 1020 states and beyond. Inf. Comput. 98, 142–170 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  7. Chandru, V., Hooker, J.N.: Detecting embedded Horn structure in propositional logic. Inf. Process. Lett. 42(2), 109–111 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  8. Chen, H., Dalmau, V.: Beyond hypertree width: decomposition methods without decompositions. In: CP-05: 11th International Conference on Principles and Practice of Constraint Programming, pp. 167–181 (2005)

  9. Chen, H., Gomes, C.P., Selman, B.: Formal models of heavy-tailed behavior in combinatorial search. In: CP-01: 7th International Conference on Principles and Practice of Constraint Programming, volume 2239 of LNCS. (2001)

  10. Davis, M., Putnam, H.: A computing procedure for quantification theory. Commun. ACM 7, 201–215 (1960)

    MATH  MathSciNet  Google Scholar 

  11. Davis, M., Logemann, G., Loveland, D.: A machine program for theorem proving. Commun. ACM 5, 394–397 (1962)

    Article  MATH  MathSciNet  Google Scholar 

  12. Dechter, R.: Constraint Processing. Morgan Kaufmann Publishers Inc. (2003). ISBN 1558608907

  13. Dechter, R., Pearl, J.: Network-based heuristics for constraint-satisfaction problems. Artif. Intell. 34(1), 1–38 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  14. del Val, A.: On 2-SAT and renamable horn. In: AAAI-00: 17th National Conference on Artificial Intelligence, pp. 279–284 (2000)

  15. Deville, Y., Van Hentenryck, P.: An efficient arc consistency algorithm for a class of CSP problems. In: IJCAI-91: 12th International Joint Conference on Artificial Intelligence, pp. 325–330 (1991)

  16. Dilkina, B., Gomes, C.P., Sabharwal, A.: Tradeoffs in the complexity of backdoor detection. In: CP-07: 13th International Conference on Principles and Practice of Constraint Programming, volume 4741 of LNCS, pp. 256–270 (2007)

  17. Dilkina, B., Gomes, C.P., Sabharwal, A.: The impact of network topology on pure Nash equilibria in graphical games. In: AAAI-07: 22nd Conference on Artificial Intelligence, pp. 42–49 (2007)

  18. Dilkina, B., Gomes, C.P., Sabharwal, A.: Tradeoffs in backdoors: inconsistency detection, dynamic simplification, and preprocessing. In: ISAIM-08: 10th International Symposium on Artificial Intelligence and Mathematics (2008)

  19. Dilkina, B., Gomes, C.P., Malitsky, Y., Sabharwal, A., Sellmann, M.: Backdoors to combinatorial optimization: feasibility and optimality. In: CPAIOR-09: 6th International Conference on Integration of AI and OR Techniques in Constraint Programming, pp. 56–70. Pittsburgh (2009)

  20. Dilkina, B., Gomes, C.P., Sabharwal, A.: Backdoors in the context of learning. In: SAT-09: 12th International Conference on Theory and Applications of Satisfiability Testing, volume 5584 of LNCS, pp. 73–79 (2009)

  21. Dowling, W.F., Gallier, J.H.: Linear-time algorithms for testing the satisfiability of propositional horn formulae. J. Log. Program. 1(3), 267–284 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  22. Downey, R.G., Fellows, M.R.: Parameterized Complexity (Monographs in Computer Science). Springer (1998). ISBN 978-0387948836

  23. Eén, N., Sörensson, N.: MiniSat: a SAT solver with conflict-clause minimization. In: SAT-05: 8th International Conference on Theory and Applications of Satisfiability Testing (2005)

  24. Erdos, P., Renyi, A.: On random graphs. Publicationes Mathemticae (Debrecen) 6, 290–297 (1959)

    MathSciNet  Google Scholar 

  25. Fichte, J.K., Szeider, S.: Backdoors to normality for disjunctive logic programs. In: AAAI-13: 27th Conference on Artificial Intelligence (2013)

  26. Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer (2006). ISBN 9783642067570

  27. Freuder, E.C.: A sufficient condition for backtrack-free search. J. ACM 29(1), 24–32 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  28. Freuder, E.C.: A sufficient condition for backtrack-bounded search. J. ACM 32(4), 755–761 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  29. Freuder, E. C.: Complexity of k-tree structured constraint satisfaction problems. In: AAAI-90: 8th National Conference on Artificial Intelligence, pp. 4–9 (1990)

  30. Gomes, C.P., Selman, B., Kautz, H.: Boosting combinatorial search through randomization. In: AAAI-98: 15th National Conference on Artificial Intelligence, pp. 431–437 (1998)

  31. 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)

    Article  MATH  MathSciNet  Google Scholar 

  32. Henschen, L., Wos, L.: Unit refutations and Horn sets. J. ACM 21, 590–605 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  33. Hoffmann, J., Gomes, C., Selman, B.: Structure and problem hardness: Goal asymmetry and DPLL proofs in SAT-based planning. Logical Methods Comput. Sci. 3(1–6), 1–41 (2007)

    Google Scholar 

  34. Hoos, H.H., Stützle, T.: SATLIB: an online resource for research on SAT. In: Gent, I.P., van Maaren, H., Walsh, W (eds.) SAT2000, pp. 283–292. IOS Press (2000). http://www.satlib.org

  35. ILOG, SA. CPLEX 10.1 Reference Manual (2006)

  36. Kautz, H.A., Selman, B.: Planning as satisfiability. In: ECAI-92: 10th European Conference on Artificial Intelligence, pp. 359–363 (1992)

  37. Kautz, H.A., Selman, B.: Pushing the envelope: planning, propositional logic, and stochastic search. In: AAAI-96: 13th Conference on Artificial Intelligence, pp. 1194–1201 (1996)

  38. Kearns, M.J., Littman, M.L., Singh, S.P.: Graphical models for game theory. In: UAI-01: 17th Conference on Uncertainty in Artificial Intelligence, pp. 253–260 (2001)

  39. Kilby, P., Slaney, J.K., Thiébaux, S., Walsh, T.: Backbones and backdoors in satisfiability. In: AAAI-05: 20th National Conference on Artificial Intelligence, pp. 1368–1373 (2005)

  40. Kottler, S., Kaufmann, M., Sinz, C.: Computation of renameable horn backdoors. In: SAT’08: 11th International Conference on Theory and Applications of Satisfiability Testing, pp. 154–160 (2008)

  41. Lewis, H.R.: Renaming a set of clauses as a Horn set. J. ACM 25(1), 134–135 (1978)

    Article  MATH  Google Scholar 

  42. Li, C.M., Anbulagan: Heuristics based on unit propagation for satisfiability problems. In: IJCAI-97: 15th International Joint Conference on Artificial Intelligence, pp. 366–371 (1997)

  43. Lynce, I., Marques-Silva, J.P.: Hidden structure in unsatisfiable random 3-SAT: an empirical study. In: ICTAI-06: 16th IEEE International Conference on Tools with Artificial Intelligence, pp. 246–251 (2004)

  44. Marques-Silva, J.P., Sakallah, K.A.: GRASP–a new search algorithm for satisfiability. In: ICCAD-96: International Conference on Computer Aided Design, pp. 220–227 (1996)

  45. Moskewicz, M.W., Madigan, C.F., Zhao, Y., Zhang, L., Malik, S.: Chaff: engineering an efficient SAT solver. In: Proceedings of DAC-01: 38th Design Automation Conference, pp. 530–535 (2001)

  46. Nash, J.: Noncooperative games. Ann. Math. 54, 289–295 (1951)

    Article  MathSciNet  Google Scholar 

  47. Nishimura, N., Ragde, P., Szeider, S.: Detecting backdoor sets with respect to Horn and binary clauses. In: SAT-04: 7th International Conference on Theory and Applications of Satisfiability Testing (2004)

  48. Nishimura, N., Ragde, P., Szeider, S.: Solving #SAT using vertex covers. In: SAT-06: 9th International Conference on Theory and Applications of Satisfiability Testing, pp. 396–409 (2006)

  49. Paris, L., Ostrowski, R., Siegel, P., Sais, L.: Computing Horn strong backdoor sets thanks to local search. In: ICTAI-06: 17th IEEE International Conference on Tools with Artificial Intelligence, pp. 139–143 (2006)

  50. Pfandler, A., Rümmele, S., Szeider, S.: Backdoors to abduction. In: IJCAI: International Joint Conference on Artificial Intelligence, pp. 1046–1052 (2013)

  51. Pipatsrisawat, K., Darwiche, A.: RS at 2.0: SAT solver description. Technical Report D–153, Automated Reasoning Group, Computer Science Department, UCLA (2007)

  52. Pipatsrisawat, K., Darwiche, A.: On the power of clause-learning SAT solvers with restarts. In CP-09: 15th International Conference on Principles and Practice of Constraint Programming, volume 5732 of LNCS, pp. 654–668 (2009)

  53. Razgon, I., O’Sullivan, B.: Almost 2-sat is fixed-parameter tractable. J. Comput. Syst. Sci. 75(8), 435–450 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  54. Samer, M., Szeider, S.: Constraint satisfaction with bounded treewidth revisited. In: CP-06: 12th International Conference on Principles and Practice of Constraint Programming, pp. 499–513 (2006)

  55. Samer, M., Szeider, S.: Backdoor sets of quantified boolean formulas. J. Autom. Reason. 42(1), 77–97 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  56. Sinz, C., Kaiser, A., Küchlin, W.: Formal methods for the validation of automotive product configuration data. Artif. Intell. Eng. Des, Anal. Manuf. 17(1), 75–97 (2003). Special issue on configuration

    Article  Google Scholar 

  57. Szeider, S.: Backdoor sets for DLL subsolvers. J. Autom. Reason. 35(1–3), 73–88 (2005)

    MATH  MathSciNet  Google Scholar 

  58. van Beek, P., Dechter, R.: On the minimality and global consistency of row-convex constraint networks. J. ACM 42(3), 543–561 (1995)

    Article  MATH  Google Scholar 

  59. Williams, R., Gomes, C., Selman, B.: Backdoors to typical case complexity. In: IJCAI-03: 18th International Joint Conference on Artificial Intelligence, pp. 1173–1178 (2003)

  60. Williams, R., Gomes, C., Selman, B.: On the connections between heavy-tails, backdoors, and restarts in combinatorial search. In SAT-03: 6th International Conference on Theory and Applications of Satisfiability Testing, pp. 222–230 (2003)

  61. Zhang, L., Madigan, C.F., Moskewicz, M.H., Malik, S.: Efficient conflict driven learning in a Boolean satisfiability solver. In: ICCAD-01: International Conference on Computer Aided Design, pp. 279–285 (2001)

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Authors and Affiliations

  1. College of Computing, Georgia Institute of Technology, Atlanta, GA, 30332, USA

    Bistra Dilkina

  2. Department of Computer Science, Cornell University, Ithaca, NY, 14853, USA

    Carla P. Gomes

  3. IBM Watson Research Center, Yorktown Heights, NY, 10598, USA

    Ashish Sabharwal

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  1. Bistra Dilkina
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  2. Carla P. Gomes
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  3. Ashish Sabharwal
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Correspondence to Bistra Dilkina.

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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, 399–431 (2014). https://doi.org/10.1007/s10472-014-9407-9

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