The Isomorphism Conjecture of Berman and Hartmanis states that all $NP$-complete sets are polynomial time isomorphic to each other. This means that $NP$-complete problems are efficiently reducible to each other via polynomial time computable and invertible bijections. The conjecture implies $P\neq NP$.

The isomorphism conjecture implies an exponential lower bound on the density of $NP$-complete sets since Satisfiability problem is dense. I am wondering if it also implies an exponential lower bound on the density of witnesses for $NP$-complete set.

Does the isomorphism conjecture imply exponential lower bounds on witnesses density? Does it imply that $NP$-complete problems can not be in $FewP$?

The best result I am aware of is the following:

If $P=UP$ and $NP=EXP$ then the isomorphism conjecture holds. The converse implication would answer my questions.

Density $D$ of a set $S$ refers to the number of strings of length less than $n$ in the language. A set $S$ is exponentially dense if its density is $D=\Omega(2^{n^\epsilon})$ for some $\epsilon \gt 0$ and for infinitely many $n$ and sparse if $D$= $O(poly(n))$.

The Isomorphism Conjecture of Berman and Hartmanis states that all $NP$-complete sets are polynomial time isomorphic to each other. This means that $NP$-complete problems are efficiently reducible to each other via polynomial time computable and invertible bijections. The conjecture implies $P\neq NP$.

The isomorphism conjecture implies an exponential lower bound on the density of $NP$-complete sets since Satisfiability problem is dense. I am wondering if it also implies an exponential lower bound on the density of witnesses for $NP$-complete set.

Does the isomorphism conjecture imply exponential lower bounds on witnesses density? Does it imply that $NP$-complete problems can not be in $FewP$?

The best result I am aware of is the following:

If $P=UP$ and $NP=EXP$ then the isomorphism conjecture holds.

Density $D$ of a set $S$ refers to the number of strings of length less than $n$ in the language. A set $S$ is exponentially dense if its density is $D=\Omega(2^{n^\epsilon})$ for some $\epsilon \gt 0$ and for infinitely many $n$ and sparse if $D$= $O(poly(n))$.

The Isomorphism Conjecture of Berman and Hartmanis states that all $NP$-complete sets are polynomial time isomorphic to each other. This means that $NP$-complete problems are efficiently reducible to each other via polynomial time computable and invertible bijections. The conjecture implies $P\neq NP$.

The isomorphism conjecture implies an exponential lower bound on the density of $NP$-complete sets since Satisfiability problem is dense. I am wondering if it also implies an exponential lower bound on the density of witnesses for $NP$-complete set.

Does the isomorphism conjecture imply exponential lower bounds on witnesses density? Does it imply that $NP$-complete problems can not be in $FewP$?

Density $D$ of a set $S$ refers to the number of strings of length less than $n$ in the language. A set $S$ is exponentially dense if its density is $D=\Omega(2^{n^\epsilon})$ for some $\epsilon \gt 0$ and for infinitely many $n$ and sparse if $D$= $O(poly(n))$.

The Isomorphism Conjecture of Berman and Hartmanis states that all $NP$-complete sets are polynomial time isomorphic to each other. This means that $NP$-complete problems are efficiently reducible to each other via polynomial time computable and invertible bijections. The conjecture implies $P\neq NP$.

The isomorphism conjecture implies an exponential lower bound on the density of $NP$-complete sets since Satisfiability problem is dense. I am wondering if it also implies an exponential lower bound on the density of witnesses for $NP$-complete set.

Does the isomorphism conjecture imply exponential lower bounds on witnesses density? Does it imply that $NP$-complete problems can not be in $FewP$?

The best result I am aware of is the following:

If $P=UP$ and $NP=EXP$ then the isomorphism conjecture holds. The converse implication would answer my questions.

Density $D$ of a set $S$ refers to the number of strings of length less than $n$ in the language. A set $S$ is exponentially dense if its density is $D=\Omega(2^{n^\epsilon})$ for some $\epsilon \gt 0$ and for infinitely many $n$ and sparse if $D$= $O(poly(n))$.