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? 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. To create the oracle, we introduce Kolmogorov-generic oracles where the strings placed in the oracle are derived from an exponentially long Kolmogorov-random string. We also show that relative to this same oracle, P is not equal to UP and TFNP functions with a SAT oracle are not computable in polynomial-time with a SAT oracle.