Complexity of Counting the Optimal Solutions

  • Miki Hermann
  • Reinhard Pichler
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5092)


Following the approach of Hemaspaandra and Vollmer, we can define counting complexity classes #\(\cdot\mathcal{C}\) for any complexity class Open image in new window of decision problems. In particular, the classes Open image in new window with k ≥ 1 corresponding to all levels of the polynomial hierarchy have thus been studied. However, for a large variety of counting problems arising from optimization problems, a precise complexity classification turns out to be impossible with these classes. In order to remedy this unsatisfactory situation, we introduce a hierarchy of new counting complexity classes #·Opt k P and #·Opt k P[log n] with k ≥ 1. We prove several important properties of these new classes, like closure properties and the relationship with the Open image in new window -classes. Moreover, we establish the completeness of several natural counting complexity problems for these new classes.


Turing Machine Complexity Class Vertex Cover Conjunctive Normal Form Propositional Formula 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Cadoli, M., Lenzerini, M.: The Complexity of Propositional Closed World Reasoning and Circumscription. Journal of Computer and System Sciences 48(2), 255–310 (1994)CrossRefMathSciNetzbMATHGoogle Scholar
  2. 2.
    Creignou, N., Hermann, M.: Complexity of Generalized Satisfiability Counting Problems. Information and Computation 125(1), 1–12 (1996)CrossRefMathSciNetzbMATHGoogle Scholar
  3. 3.
    Durand, A., Hermann, M., Kolaitis, P.G.: Subtractive Reductions and Complete Problems for Counting Complexity Classes. Theoretical Computer Science 340(3), 496–513 (2005)CrossRefMathSciNetzbMATHGoogle Scholar
  4. 4.
    Fortnow, L.: Counting complexity. In: Hemaspaandra, L.A., Selman, A.L. (eds.) Complexity Theory Retrospective II, pp. 81–107. Springer, Heidelberg (1997)Google Scholar
  5. 5.
    Galil, Z.: On Some Direct Encodings of Nondeterministic Turing Machines Operating in Polynomial Time into P-complete Problems. SIGACT News 6(1), 19–24 (1974)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Gasarch, W.I., Krentel, M.W., Rappoport, K.J.: OptP as the Normal Behavior of NP-complete Problems. Mathematical Systems Theory 28(6), 487–514 (1995)CrossRefMathSciNetzbMATHGoogle Scholar
  7. 7.
    Hemaspaandra, L.A., Vollmer, H.: The Satanic Notations: Counting Classes beyond #P and other Definitional Adventures. SIGACT News, Complexity Theory Column 8 26(1), 2–13 (1995)CrossRefGoogle Scholar
  8. 8.
    Jenner, B., Torán, J.: Computing Functions with Parallel Queries to NP. Theoretical Computer Science 141(1-2), 175–193 (1995)CrossRefMathSciNetzbMATHGoogle Scholar
  9. 9.
    Köbler, J., Schöning, U., Torán, J.: On Counting and Approximation. Acta Informatica 26(4), 363–379 (1989)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Krentel, M.W.: The Complexity of Optimization Problems. Journal of Computer and System Sciences 36(3), 490–509 (1988)CrossRefMathSciNetzbMATHGoogle Scholar
  11. 11.
    Krentel, M.W.: Generalizations of OptP to the Polynomial Hierarchy. Theoretical Computer Science 97(2), 183–198 (1992)CrossRefMathSciNetzbMATHGoogle Scholar
  12. 12.
    Pagourtzis, A., Zachos, S.: The Complexity of Counting Functions with Easy Decision Version. In: Královič, R., Urzyczyn, P. (eds.) MFCS 2006. LNCS, vol. 4162, pp. 741–752. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  13. 13.
    Papadimitriou, C.H.: Computational Complexity. Addison-Wesley, Reading (1994)zbMATHGoogle Scholar
  14. 14.
    Selman, A., Mei-Rui, X., Book, R.: Positive Relativizations of Complexity Classes. SIAM Journal on Computing 12(3), 565–579 (1983)CrossRefMathSciNetzbMATHGoogle Scholar
  15. 15.
    Toda, S., Ogiwara, M.: Counting Classes are at least as Hard as the Polynomial-Time Hierarchy. SIAM Journal on Computing 21(2), 316–328 (1992)CrossRefMathSciNetzbMATHGoogle Scholar
  16. 16.
    Toda, S., Watanabe, O.: Polynomial-Time 1-Turing Reductions from #PH to #P. Theoretical Computer Science 100(1), 205–221 (1992)CrossRefMathSciNetzbMATHGoogle Scholar
  17. 17.
    Valiant, L.G.: The Complexity of Computing the Permanent. Theoretical Computer Science 8(2), 189–201 (1979)CrossRefMathSciNetzbMATHGoogle Scholar
  18. 18.
    Valiant, L.G.: The Complexity of Enumeration and Reliability Problems. SIAM Journal on Computing 8(3), 410–421 (1979)CrossRefMathSciNetzbMATHGoogle Scholar
  19. 19.
    Wagner, K.: Bounded Query Classes. SIAM Journal on Computing 19(5), 833–846 (1990)CrossRefMathSciNetzbMATHGoogle Scholar
  20. 20.
    Wrathall, C.: Complete Sets and the Polynomial-Time Hierarchy. Theoretical Computer Science 3(1), 23–33 (1976)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Miki Hermann
    • 1
  • Reinhard Pichler
    • 2
  1. 1.LIX (CNRS, UMR 7161)École PolytechniquePalaiseauFrance
  2. 2.Institut für InformationssystemeTechnische Universität WienWienAustria

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