The Complexity of Counting Functions with Easy Decision Version

  • Aris Pagourtzis
  • Stathis Zachos
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4162)


We investigate the complexity of counting problems that belong to the complexity class #P and have an easy decision version. These problems constitute the class #PE which has some well-known representatives such as #Perfect Matchings, #DNF-Sat, and NonNegative Permanent. An important property of these problems is that they are all #P-complete, in the Cook sense, while they cannot be #P-complete in the Karp sense unless P = NP.

We study these problems in respect to the complexity class TotP, which contains functions that count the number of all paths of a PNTM. We first compare TotP to #P and #PE and show that FPTotP#PE#P and that the inclusions are proper unless P = NP.

We then show that several natural #PE problems — including the ones mentioned above — belong to TotP. Moreover, we prove that TotP is exactly the Karp closure of self-reducible functions of #PE. Therefore, all these problems share a remarkable structural property: for each of them there exists a polynomial-time nondeterministic Turing machine which has as many computation paths as the output value.


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  1. [AJ90]
    Àlvarez, C., Jenner, B.: A very hard log space counting class. In: Proceedings of Structure in Complexity Theory Conference, pp. 154–168 (1990)Google Scholar
  2. [DHK00]
    Durand, A., Hermann, M., Kolaitis, P.G.: Subtractive reductions and complete problems for counting complexity classes. Theoretical Computer Science 340(3), 496–513 (2005)MATHCrossRefMathSciNetGoogle Scholar
  3. [DGGJ03]
    Dyer, M., Goldberg, L.A., Greenhill, C., Jerrum, M.: On the relative complexity of approximate counting problems. Algorithmica 38(3), 471–500 (2003)CrossRefMathSciNetGoogle Scholar
  4. [HHKW05]
    Hemaspaandra, L.A., Homan, C.M., Kosub, S., Wagner, K.W.: The complexity of computing the size of an interval. Technical Report, ACM Computing Research Repository (2005); Preliminary version: Hemaspaandra, L.A., Kosub, S., Wagner, K.W.: The complexity of computing the size of an interval. In: Orejas, F., Spirakis, P.G., van Leeuwen, J. (eds.) ICALP 2001. LNCS, vol. 2076, pp. 1040–1051. Springer, Heidelberg (2001)Google Scholar
  5. [JS96]
    Jerrum, M., Sinclair, A.: The Markov chain Monte-Carlo method: an approach to approximate counting and integration. In: Hochbaum, D. (ed.) Approximation Algorithms for NP-hard Problems, pp. 482–520, PWS (1996)Google Scholar
  6. [KLM89]
    Karp, R.M., Luby, M., Madras, N.: Monte-Carlo approximation algorithms for enumeration problems. Journal of Algorithms 10, 429–448 (1989)MATHCrossRefMathSciNetGoogle Scholar
  7. [KPSZ98]
    Kiayias, A., Pagourtzis, A., Sharma, K., Zachos, S.: The complexity of determining the order of solutions. In: Proceedings of the First Southern Symposium on Computing, Hattiesburg, Mississippi, December 4-5 (1998)Google Scholar
  8. [KPZ99]
    Kiayias, A., Pagourtzis, A., Zachos, S.: Cook Reductions Blur Structural Differences Between Functional Complexity Classes. In: Proceedings of the 2nd Panhellenic Logic Symposium, Delphi, July 13–17, pp. 132–137 (1999)Google Scholar
  9. [Ko83]
    On Self-Reducibility and Weak P-Selectivity. Journal of Computer and System Sciences 26(2), 209–221 (1983)Google Scholar
  10. [Kre88]
    Krentel, M.W.: The complexity of optimization problems. Journal of Computer and System Sciences 36(3), 490–509 (1988)MATHCrossRefMathSciNetGoogle Scholar
  11. [KST89]
    Köbler, J., Schöning, U., Torán, J.: On counting and approximation. Acta Informatica 26(4), 363–379 (1989)MATHMathSciNetGoogle Scholar
  12. [Pag01]
    Pagourtzis, A.: On the complexity of hard counting problems with easy decision version. In: Proceedings of the 3rd Panhellenic Logic Symposium, Anogia, Crete, July 17–21 (2001)Google Scholar
  13. [SST95]
    Saluja, S., Subrahmanyam, K.V., Thakur, M.N.: Descriptive Complexity of #P Functions. Journal of Computer and System Sciences 50(3), 493–505 (1995)MATHCrossRefMathSciNetGoogle Scholar
  14. [Sim75]
    Simon, J.: On some Central Problems of Computational Complexity. PhD thesis, Cornell University, Ithaca, NY (1975)Google Scholar
  15. [Tod91]
    Toda, S.: PP is as hard as the polynomial-time hierarchy. SIAM Journal on Computing 20(5), 865–877 (1991)MATHCrossRefMathSciNetGoogle Scholar
  16. [TW92]
    Toda, S., Watanabe, O.: Polynomial-time 1-Turing reductions from #PH to #P. Theoretical Computer Science 100(1), 205–221 (1992)MATHCrossRefMathSciNetGoogle Scholar
  17. [Tor88]
    Toran, J.: Structural Properties of the Counting Hierarchies. PhD thesis, Facultat d’Informatica de Barcelona (1988)Google Scholar
  18. [Val79a]
    Valiant, L.G.: The complexity of computing the permanent. Theoretical Computer Science 8(2), 189–201 (1979)MATHCrossRefMathSciNetGoogle Scholar
  19. [Val79b]
    Valiant, L.G.: The complexity of enumeration and reliability problems. SIAM Journal on Computing 8(3), 410–421 (1979)MATHCrossRefMathSciNetGoogle Scholar
  20. [Vol94]
    Vollmer, H.: On different reducibility notions for function classes. In: Enjalbert, P., Mayr, E.W., Wagner, K.W. (eds.) STACS 1994. LNCS, vol. 775, pp. 24–26. Springer, Heidelberg (1994)Google Scholar
  21. [Wag86]
    Wagner, K.W.: Some observations on the connection between counting and recursion. Theoretical Computer Science 47(2), 131–147 (1986)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Aris Pagourtzis
    • 1
  • Stathis Zachos
    • 1
    • 2
  1. 1.Department of Computer Science, School of ECENational Technical University of AthensGreece
  2. 2.CIS DepartmentBrooklyn College, CUNY 

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