On the Complexity of Hard Enumeration Problems

  • Nadia Creignou
  • Markus KröllEmail author
  • Reinhard Pichler
  • Sebastian Skritek
  • Heribert Vollmer
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10168)


Complexity theory provides a wealth of complexity classes for analyzing the complexity of decision and counting problems. Despite the practical relevance of enumeration problems, the tools provided by complexity theory for this important class of problems are very limited. In particular, complexity classes analogous to the polynomial hierarchy and an appropriate notion of problem reduction are missing. In this work, we lay the foundations for a complexity theory of hard enumeration problems by proposing a hierarchy of complexity classes and by investigating notions of reductions for enumeration problems.



This work was supported by the Vienna Science and Technology Fund (WWTF) through project ICT12-015, the Austrian Science Fund (FWF): P25207-N23, P25518-N23, I836-N23, W1255-N23 and the French Agence Nationale de la Recherche, AGGREG project reference ANR-14-CE25-0017-01.


  1. 1.
    Arora, S., Barak, B.: Computational Complexity: A Modern Approach. Cambridge University Press, Cambridge (2009)CrossRefzbMATHGoogle Scholar
  2. 2.
    Bagan, G., Durand, A., Grandjean, E.: On acyclic conjunctive queries and constant delay enumeration. In: Duparc, J., Henzinger, T.A. (eds.) CSL 2007. LNCS, vol. 4646, pp. 208–222. Springer, Heidelberg (2007). doi: 10.1007/978-3-540-74915-8_18 CrossRefGoogle Scholar
  3. 3.
    Brault-Baron, J.: De la pertinence de l’énumération : complexité en logiques propositionnelle et du premier ordre. (The relevance of the list: propositional logic and complexity of the first order). Ph.D. thesis, University of Caen Normandy, France (2013).
  4. 4.
    Creignou, N., Hébrard, J.J.: On generating all solutions of generalized satisfiability problems. Informatique Théorique et Applications 31(6), 499–511 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Creignou, N., Kröll, M., Pichler, R., Skritek, S., Vollmer, H.: On the complexity of hard enumeration problems. CoRR abs/1610.05493 (2016),
  6. 6.
    Creignou, N., Vollmer, H.: Parameterized complexity of weighted satisfiability problems: decision, enumeration, counting. Fundam. Inform. 136(4), 297–316 (2015). MathSciNetzbMATHGoogle Scholar
  7. 7.
    Durand, A., Grandjean, E.: First-order queries on structures of bounded degree are computable with constant delay. ACM Trans. Comput. Log. 8(4), 21/1–21/19 (2007). MathSciNetCrossRefGoogle Scholar
  8. 8.
    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
  9. 9.
    Durand, A., Schweikardt, N., Segoufin, L.: Enumerating answers to first-order queries over databases of low degree. In: Proceedings of PODS 2014, pp. 121–131. ACM (2014)Google Scholar
  10. 10.
    Eiter, T., Gottlob, G.: The complexity of logic-based abduction. J. ACM 42(1), 3–42 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Hemachandra, L.A.: The strong exponential hierarchy collapses. In: Proceedings of STOC 1987, pp. 110–122. ACM (1987)Google Scholar
  12. 12.
    Hemaspaandra, L.A., Vollmer, H.: The satanic notations: counting classes beyond #P and other definitional adventures. SIGACT News 26(1), 2–13 (1995)CrossRefGoogle Scholar
  13. 13.
    Johnson, D.S., Papadimitriou, C.H., Yannakakis, M.: On generating all maximal independent sets. Inf. Process. Lett. 27(3), 119–123 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Kimelfeld, B., Kolaitis, P.G.: The complexity of mining maximal frequent subgraphs. ACM Trans. Database Syst. 39(4), 32:1–32:33 (2014). MathSciNetCrossRefGoogle Scholar
  15. 15.
    Lucchesi, C.L., Osborn, S.L.: Candidate keys for relations. J. Comput. Syst. Sci. 17(2), 270–279 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Schaefer, T.J.: The complexity of satisfiability problems. In: Proceedings of STOC 1978, pp. 216–226. ACM Press (1978)Google Scholar
  17. 17.
    Strozecki, Y.: Enumeration complexity and matroid decomposition. Ph.D. thesis, Universite Paris Diderot - Paris 7, December 2010. ystr/these_strozecki
  18. 18.
    Strozecki, Y.: On enumerating monomials and other combinatorial structures by polynomial interpolation. Theory Comput. Syst. 53(4), 532–568 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Toda, S.: PP is as hard as the polynomial-time hierarchy. SIAM J. Comput. 20(5), 865–877 (1991)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Nadia Creignou
    • 1
  • Markus Kröll
    • 2
    Email author
  • Reinhard Pichler
    • 2
  • Sebastian Skritek
    • 2
  • Heribert Vollmer
    • 3
  1. 1.Aix-Marseille University, CNRSMarseilleFrance
  2. 2.TU WienViennaAustria
  3. 3.Leibniz Universität HannoverHannoverGermany

Personalised recommendations