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.
Keywords Computational complexity - Polynomial-time hierarchy - Multi-valued functions - Kolmogorov complexity