Does the Polynomial Hierarchy Collapse if Onto Functions are Invertible?
 Harry Buhrman,
 Lance Fortnow,
 Michal Koucký,
 John D. Rogers,
 Nikolay Vereshchagin
 … show all 5 hide
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 polynomialtime does this imply the polynomialtime hierarchy collapses? By computing a multivalued function in deterministic polynomialtime 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 polynomialtime hierarchy is infinite. We also show that relative to this same oracle, P≠UP and TFNP^{NP} functions are not computable in polynomialtime with an NP oracle.
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 Title
 Does the Polynomial Hierarchy Collapse if Onto Functions are Invertible?
 Open Access
 Available under Open Access This content is freely available online to anyone, anywhere at any time.
 Journal

Theory of Computing Systems
Volume 46, Issue 1 , pp 143156
 Cover Date
 20100101
 DOI
 10.1007/s0022400891608
 Print ISSN
 14324350
 Online ISSN
 14330490
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Keywords

 Computational complexity
 Polynomialtime hierarchy
 Multivalued functions
 Kolmogorov complexity
 Industry Sectors
 Authors

 Harry Buhrman ^{(1)}
 Lance Fortnow ^{(2)}
 Michal Koucký ^{(3)}
 John D. Rogers ^{(4)}
 Nikolay Vereshchagin ^{(5)}
 Author Affiliations

 1. CWI and University of Amsterdam, Amsterdam, The Netherlands
 2. University of Chicago, Chicago, USA
 3. Institute of Mathematics, Academy of Sciences of the Czech Republic, Prague, Czech Republic
 4. DePaul University, Chicago, USA
 5. Lomonosov Moscow State University, Moscow, Russia