Computing Solutions Uniquely collapses the Polynomial Hierarchy

Lane A. Hemaspaandra, Ashish V. Naik, Mitsunori Ogihara, Alan L. Selman

Abstract: We show that if there is an NP function that, when given a satisfiable formula as input, outputs one satisfying assignment uniquely, then the polynomial hierarchy collapses to its second level. As the existence of such a function is known to be equivalent to the statement “every NP function has an NP refinement with unique outputs,” our result provides the strongest evidence yet that NP functions cannot be refined.