Theory of Computing Systems

, 46:143

Does the Polynomial Hierarchy Collapse if Onto Functions are Invertible?

  • Harry Buhrman
  • Lance Fortnow
  • Michal Koucký
  • John D. Rogers
  • Nikolay Vereshchagin
Open AccessArticle

DOI: 10.1007/s00224-008-9160-8

Cite this article as:
Buhrman, H., Fortnow, L., Koucký, M. et al. Theory Comput Syst (2010) 46: 143. doi:10.1007/s00224-008-9160-8


The class TFNP, defined by Megiddo and Papadimitriou, consists of multivalued functions with values that are polynomially verifiable and guaranteed to exist. Do we have evidence that such functions are hard, for example, if TFNP is computable in polynomial-time does this imply the polynomial-time hierarchy collapses? By computing a multivalued function in deterministic polynomial-time we mean on every input producing one of the possible values of the function on that input.

We give a relativized negative answer to this question by exhibiting an oracle under which TFNP functions are easy to compute but the polynomial-time hierarchy is infinite. We also show that relative to this same oracle, P≠UP and TFNPNP functions are not computable in polynomial-time with an NP oracle.


Computational complexityPolynomial-time hierarchyMulti-valued functionsKolmogorov complexity
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© The Author(s) 2008

Authors and Affiliations

  • Harry Buhrman
    • 1
  • Lance Fortnow
    • 2
  • Michal Koucký
    • 3
  • John D. Rogers
    • 4
  • Nikolay Vereshchagin
    • 5
  1. 1.CWI and University of AmsterdamAmsterdamThe Netherlands
  2. 2.University of ChicagoChicagoUSA
  3. 3.Institute of Mathematics, Academy of Sciences of the Czech RepublicPragueCzech Republic
  4. 4.DePaul UniversityChicagoUSA
  5. 5.Lomonosov Moscow State UniversityMoscowRussia