Does the Polynomial Hierarchy Collapse if Onto Functions are Invertible?
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.
- Does the Polynomial Hierarchy Collapse if Onto Functions are Invertible?
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Theory of Computing Systems
Volume 46, Issue 1 , pp 143-156
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- Computational complexity
- Polynomial-time hierarchy
- Multi-valued functions
- Kolmogorov complexity
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- 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