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Does the Polynomial Hierarchy Collapse if Onto Functions are Invertible?

Theory of Computing Systems volume 46, Article number: 143 (2010) Cite this article

Abstract

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.

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Authors and Affiliations

  1. CWI and University of Amsterdam, Amsterdam, The Netherlands

    Harry Buhrman

  2. University of Chicago, Chicago, USA

    Lance Fortnow

  3. Institute of Mathematics, Academy of Sciences of the Czech Republic, Prague, Czech Republic

    Michal Koucký

  4. DePaul University, Chicago, USA

    John D. Rogers

  5. Lomonosov Moscow State University, Moscow, Russia

    Nikolay Vereshchagin

Authors
  1. Harry Buhrman
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  2. Lance Fortnow
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  3. Michal Koucký
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  4. John D. Rogers
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  5. Nikolay Vereshchagin
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Corresponding author

Correspondence to John D. Rogers.

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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

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Buhrman, H., Fortnow, L., Koucký, M. et al. Does the Polynomial Hierarchy Collapse if Onto Functions are Invertible?. Theory Comput Syst 46, 143 (2010). https://doi.org/10.1007/s00224-008-9160-8

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  • DOI: https://doi.org/10.1007/s00224-008-9160-8

Keywords

  • Computational complexity
  • Polynomial-time hierarchy
  • Multi-valued functions
  • Kolmogorov complexity