4
$\begingroup$

Suppose $A$ is any #P-complete problem. Now, $A$ is modified to obtain a decision problem $A'$ not by asking whether there is a solution but whether at least half of the potential solutions are actually true solutions. Question: Is $A'$ PP-complete?

This works if $A$ is #Sat, since MajSat is PP-complete. My approach so far: If $A$ is #P-complete, then there is a reduction from #Sat. So it should be possible to adapt this reduction to obtain a reduction from MajSat to $A'$. However, I ran into the problem that the reduction from #Sat to $A$ often requires a Turing (or Cook) reduction and it seemed far from clear how to obtain a many-one reduction from it.

$\endgroup$
8
$\begingroup$

Not necessary. Imagine the following Fake-#SAT problem: possible solutions are extended by one bit, and all vectors with this bit set are solutions. That is, the number of satisfying assignments for the new problem is $2^n+f$, where $f$ is the number of satisfying assignments for the original #SAT problem ($0\le f\le 2^n$).

The problem remains #P-complete; however, the corresponding "majority" problem (formulates as $\ge$ vs $<$) has always the answer "yes".

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.