Computing Solutions Uniquely Collapses the Polynomial Hierarchy

• Authors: Lane A. Hemaspaandra;Ashish V. Naik;Mitsunori Ogihara;Alan L. Selman
• Venue: SIAM Journal on Computing
• Year: 1996

Is there an NP function that, when given a satisfiable formula as input, outputs one satisfying assignment uniquely? That is, can a nondeterministic function cull just one satisfying assignment from a possibly exponentially large collection of assignments? We show that if there is such a nondeterministic function, then the polynomial hierarchy collapses to $\zppnp$ (and thus, in particular, to $\npnp$). 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. We prove our result via a result of independent interest. We say that a set $A$ is NPSV-selective (NPMV-selective) if there is a 2-ary partial NP function with unique values (a 2-ary partial NP function) that decides which of its inputs (if any) is ``more likely'' to belong to $A$; this is a nondeterministic analog of the recursion-theoretic notion of the semi-recursive sets and the extant complexity-theoretic notion of P-selectivity. Our hierarchy collapse result follows by combining %%foo the easy observation that every set in NP is NPMV-selective with the following result: If $A \in \np$ is NPSV-selective, then $A \in (\np\cap\conp)/\poly$. Relatedly, we prove that if $A \in \np$ is NPSV-selective, then $A$ is Low$_2$. We prove that the polynomial hierarchy collapses even further, namely to NP, if all coNP sets are NPMV-selective. This follows from a more general result we prove: Every self-reducible NPMV-selective set is in NP\@.