Completeness Results for Counting Problems with Easy Decision

  • Eleni Bakali
  • Aggeliki Chalki
  • Aris Pagourtzis
  • Petros Pantavos
  • Stathis Zachos
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10236)

Abstract

Counting problems with easy decision are the only ones among problems in complexity class \(\#\textsc {P}\) that are likely to be (randomly) approximable, under the assumption \(\textsc {RP}\ne \textsc {NP}\). \(\text {TotP} \) is a subclass of \(\#\textsc {P}\) that contains many of these problems. \(\text {TotP} \) and \(\#\textsc {P}\) share some complete problems under Cook reductions, the approximability of which does not extend to all problems in these classes (if \(\textsc {RP}\ne \textsc {NP}\)); the reason is that such reductions do not preserve the function value. Therefore Cook reductions do not seem useful in obtaining (in)approximability results for counting problems in \(\text {TotP} \) and \(\#\textsc {P}\).

On the other hand, the existence of \(\text {TotP} \)-complete problems (apart from the generic one) under stronger reductions that preserve the function value has remained an open question thus far. In this paper we present the first such problems, the definitions of which are related to satisfiability of Boolean circuits and formulas. We also discuss implications of our results to the complexity and approximability of counting problems in general.

References

  1. 1.
    Achlioptas, D.: Random Satisfiability. In: Biere, A., et al. (eds.) Handbook of Satisfiability, pp. 245–270. IOS Press, Amsterdam (2009)Google Scholar
  2. 2.
    Achlioptas, D., Coja-Oghlan, A., Ricci-Tersenghi, F.: On the solution-space geometry of random constraint satisfaction problems. Random Struct. Algorithms 38(3), 251–268 (2011)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Achlioptas, D., Ricci-Tersenghi, F.: Random formulas have frozen variables. SIAM J. Comput. 39(1), 260–280 (2009)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Àlvarez, C., Jenner, B.: A very hard log-space counting class. Theoret. Comput. Sci. 107(1), 3–30 (1993)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Arora, S., Barak, B.: Computational Complexity: A Modern Approach. Cambridge University Press, New York (2009)CrossRefMATHGoogle Scholar
  6. 6.
    Bakali, E.: Self-reducible with easy decision version counting problems admit additive error approximation. Connections to counting complexity, exponential time complexity, and circuit lower bounds. CoRR abs/1611.01706 (2016)Google Scholar
  7. 7.
    Bampas, E., Gobel, A., Pagourtzis, A., Tentes, A.: On the connection between interval size functions and path counting. Comput. Complex., 1–47 (2016). doi:10.1007/s00037-016-0137-8. Springer
  8. 8.
    Dyer, M.: Approximate counting by dynamic programming. In: Proceedings of 35th Annual ACM Symposium on Theory of Computing (STOC), pp. 693–699 (2003)Google Scholar
  9. 9.
    Dyer, M.E., Goldberg, L.A., Greenhill, C.S., Jerrum, M.: The relative complexity of approximate counting problems. Algorithmica 38(3), 471–500 (2003)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Galanis, A., Goldberg, L.A., Jerrum, M.: Approximately counting H-colourings is \(\#\)BIS-hard. SIAM J. Comput. 45(3), 680–711 (2016)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Goldberg, L.A., Jerrum, M.: The complexity of ferromagnetic ising with local fields. Comb. Probab. Comput. 16(1), 43–61 (2007)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Gopalan, P., Klivans, A., Meka, R., Štefankovič, D., Vempala, S., Vigoda, E.: An FPTAS for \(\#\)knapsack and related counting problems. In: Proceedings of 52nd Annual Symposium on Foundations of Computer Science (FOCS), pp. 817–826 (2011)Google Scholar
  13. 13.
    Hemaspaandra, L.A., Homan, C.M., Kosub, S., Wagner, K.W.: The complexity of computing the size of an interval. SIAM J. Comput. 36(5), 1264–1300 (2007)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    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, Boston (1996)Google Scholar
  15. 15.
    Karp, R.M., Luby, M., Madras, N.: Monte-Carlo approximation algorithms for enumeration problems. J. Algorithms 10(3), 429–448 (1989)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Kiayias, A., Pagourtzis, A., Sharma, K., Zachos, S.: Acceptor-definable counting classes. In: Manolopoulos, Y., Evripidou, S., Kakas, A.C. (eds.) PCI 2001. LNCS, vol. 2563, pp. 453–463. Springer, Heidelberg (2003). doi:10.1007/3-540-38076-0_29 CrossRefGoogle Scholar
  17. 17.
    Kiayias, A., Pagourtzis, A., Zachos, S.: Cook reductions blur structural differences between functional complexity classes. In: Proceedings of 2nd Panhellenic Logic Symposium, pp. 132–137 (1999)Google Scholar
  18. 18.
    Knuth, D.E.: Estimating the efficiency of backtrack programs. Math. Comput. 29(129), 121–136 (1975)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Köbler, J., Schöning, U., Toran, J.: On counting and approximation. Acta Inform. 26, 363–379 (1989)MathSciNetMATHGoogle Scholar
  20. 20.
    Pagourtzis, A.: On the complexity of hard counting problems with easy decision version. In: Proceedings of 3rd Panhellenic Logic Symposium, Anogia, Crete (2001)Google Scholar
  21. 21.
    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). doi:10.1007/11821069_64 CrossRefGoogle Scholar
  22. 22.
    Papadimitriou, C.H.: Computational Complexity. Addison-Wesley, New York (1994)MATHGoogle Scholar
  23. 23.
    Saluja, S., Subrahmanyam, K.V., Thakur, M.: Descriptive complexity of #P functions. J. Comput. Syst. Sci. 50(3), 169–184 (1992)MathSciNetMATHGoogle Scholar
  24. 24.
    Sinclair, A.J., Jerrum, M.R.: Approximate counting, uniform generation and rapidly mixing Markov chains. Inf. Comput. 82, 93–133 (1989)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Valiant, L.G.: The complexity of computing the permanent. Theoret. Comput. Sci. 8(2), 189–201 (1979)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Valiant, L.G.: The complexity of enumeration and reliability problems. SIAM J. Comput. 8(3), 410–421 (1979)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.School of Electrical and Computer EngineeringNational Technical University of AthensAthensGreece

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