Advertisement

The Weight in Enumeration

  • Johannes SchmidtEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10168)

Abstract

In our setting enumeration amounts to generate all solutions of a problem instance without duplicates. We address the problem of enumerating the models of B-formulæ. A B-formula is a propositional formula whose connectives are taken from a fixed set B of Boolean connectives. Without imposing any specific order to output the solutions, this task is solved. We completely classify the complexity of this enumeration task for all possible sets of connectives B imposing the orders of (1) non-decreasing weight, (2) non-increasing weight; the weight of a model being the number of variables assigned to 1. We consider also the weighted variants where a non-negative integer weight is assigned to each variable and show that this add-on leads to more sophisticated enumeration algorithms and even renders previously tractable cases intractable, contrarily to the constraint setting. As a by-product we obtain also complexity classifications for the optimization problems known as \(\textsc {Min}\hbox {-}\textsc {Ones}\) and \(\textsc {Max}\hbox {-}\textsc {Ones}\) which are in the B-formula setting two different tasks.

Keywords

Computational complexity Enumeration Non-decreasing weight Polynomial delay Post’s lattice MaxOnes 

Notes

Acknowledgements

The author would like to thank Johan Thapper for combinatorial support.

References

  1. 1.
    Beyersdorff, O., Meier, A., Thomas, M., Vollmer, H.: The complexity of propositional implication. Inf. Process. Lett. 109(18), 1071–1077 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Böhler, E., Creignou, N., Galota, M., Reith, S., Schnoor, H., Vollmer, H.: Complexity classifications for different equivalence and audit problems for Boolean circuits. Logical Methods Comput. Sci. 8(3) (2012). https://lmcs.episciences.org/1172
  3. 3.
    Bulatov, A.A., Dyer, M.E., Goldberg, L.A., Jalsenius, M., Jerrum, M., Richerby, D.: The complexity of weighted and unweighted #csp. J. Comput. Syst. Sci. 78(2), 681–688 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Creignou, N., Hébrard, J.-J.: On generating all solutions of generalized satisfiability problems. ITA 31(6), 499–511 (1997)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Creignou, N., Olive, F., Schmidt, J.: Enumerating all solutions of a Boolean CSP by non-decreasing weight. In: Sakallah, K.A., Simon, L. (eds.) SAT 2011. LNCS, vol. 6695, pp. 120–133. Springer, Heidelberg (2011). doi: 10.1007/978-3-642-21581-0_11 CrossRefGoogle Scholar
  6. 6.
    Creignou, N., Schmidt, J., Thomas, M.: Complexity classifications for propositional abduction in post’s framework. J. Log. Comput. 22(5), 1145–1170 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Durand, A., Hermann, M., Kolaitis, P.G.: Subtractive reductions and complete problems for counting complexity classes. Theor. Comput. Sci. 340(3), 496–513 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Erdős, P., Ko, C., Rado, R.: Intersection theorem for system of finite sets. Quart. J. Math. Oxford Ser. 12, 313–318 (1961)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Johnson, D.S., Papadimitriou, C.H., Yannakakis, M.: On generating all maximal independent sets. Inf. Process. Lett. 27(3), 119–123 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Khanna, S., Sudan, M., Williamson, D.P.: A complete classification of the approximability of maximization problems derived from Boolean constraint satisfaction. In: STOC, pp. 11–20 (1997)Google Scholar
  11. 11.
    Kimelfeld, B., Sagiv, Y.: Incrementally computing ordered answers of acyclic conjunctive queries. In: Etzion, O., Kuflik, T., Motro, A. (eds.) NGITS 2006. LNCS, vol. 4032, pp. 141–152. Springer, Heidelberg (2006). doi: 10.1007/11780991_13 CrossRefGoogle Scholar
  12. 12.
    Lewis, H.: Satisfiability problems for propositional calculi. Math. Syst. Theory 13, 45–53 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Post, E.: The two-valued iterative systems of mathematical logic. Ann. Math. Stud. 5, 1–122 (1941)MathSciNetGoogle Scholar
  14. 14.
    Reith, S.: On the complexity of some equivalence problems for propositional calculi. In: Rovan, B., Vojtáš, P. (eds.) MFCS 2003. LNCS, vol. 2747, pp. 632–641. Springer, Heidelberg (2003). doi: 10.1007/978-3-540-45138-9_57 CrossRefGoogle Scholar
  15. 15.
    Schaefer, T.J.: The complexity of satisfiability problems. In: Proceedings 10th Symposium on Theory of Computing, pp. 216–226. ACM Press (1978)Google Scholar
  16. 16.
    Schmidt, J.: Enumeration: Algorithms and complexity. Preprint (2009). http://www.thi.uni-hannover.de/fileadmin/forschung/arbeiten/schmidt-da.pdf
  17. 17.
    Schnoor, H., Schnoor, I.: Enumerating all solutions for constraint satisfaction problems. In: Thomas, W., Weil, P. (eds.) STACS 2007. LNCS, vol. 4393, pp. 694–705. Springer, Heidelberg (2007). doi: 10.1007/978-3-540-70918-3_59 CrossRefGoogle Scholar
  18. 18.
    Schnorr, C.: Optimal algorithms for self-reducible problems. In: ICALP, pp. 322–337 (1976)Google Scholar
  19. 19.
    Strozecki, Y.: Enumeration complexity and matroid decomposition. Ph.D. thesis (2010)Google Scholar
  20. 20.
    Thomas, M.: The complexity of circumscriptive inference in post’s lattice. In: Erdem, E., Lin, F., Schaub, T. (eds.) LPNMR 2009. LNCS (LNAI), vol. 5753, pp. 290–302. Springer, Heidelberg (2009). doi: 10.1007/978-3-642-04238-6_25 CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Jönköping International Business SchoolJönköping UniversityJönköpingSweden

Personalised recommendations