UnambiguousSAT reductions Let $\Pi$ be an $\mathsf{NP}$-complete problem. It is standard that $3SAT$ and $\Pi$ are reducible from each other.

Let UnambiguousSAT, or USAT for short, denote the promise problem which is 3SAT but with the promise that there is $\leq 1$ solution (that is, it is in $\mathsf{PromiseUP}$). Valiant-Vazirani gives a randomized reduction from SAT to USAT.

For some natural problem $\Pi$, let U$\Pi$ is the corresponding promise problem in $\mathsf{PromiseUP}$, and suppose that $\Pi$ randomly reduces to U$\Pi$.

Is it reasonable to expect reduction from U$\Pi$ to USAT and vice versa? Would that mean $\mathsf{PromiseUP}$ has complete problem?

In general if USAT is in $\mathsf{RP}$ then $\mathsf{NP}=\mathsf{RP}$. Can we get something similar for U$\Pi$ as above?
"Is it reasonable to expect reduction from ..."

unambiguous$\Pi$ to unambiguousSAT ​ ?

Yes, since SAT is NP-hard under reductions that preserve the number of solutions.

"and vice versa" ?
$\Pi$ is NP-complete and you've assumed that U$\Pi$ "has a randomized reduction from $\Pi$", so there is

a randomized reduction from USAT to U$\Pi$, since there's one from SAT to U$\Pi$. ​ For most natural problems, if U$\Pi$ is non-trivial then one can find a simple reduction from SAT to $\Pi$ that preserves number of solutions. ​ In particular, it's reasonable to expect a reduction from USAT to U$\Pi$. ​ However, I have no clue regarding whether-or-not there's necessarily

a better reduction from USAT to U$\Pi$ (than the one from $\Pi$ to U$\Pi$).



"Would that mean promiseNP has a complete problem?" ​ Yes, since it follows

from NP having complete languages that promiseNP unconditionally does too.



"In general if" unambiguousSAT "is in RP then NP=RP."

"Can we get something similar for" unambiguous$\Pi$?

Yes, as follows:

By NP-hardness of U$\Pi$ under randomized reductions, ​ NP $\subseteq$ BPP . ​ ​ ​ By search-to-decision, probabilistic polynomial-time algorithms can solve all of FNP. ​ By the definition of FNP,

one can efficiently deterministically check whether-or-not an alleged witness is valid.

Finally, if there are no valid witnesses, then one cannot incorrectly find such a witness.