If I have an NP-complete graph problem $P1$ whose instance, say, a graph $G$ and a positive integer $k$ and I have another graph problem $P2$ whose instance is a graph $G'$ and a positive integer $k'$. I would like to prove that $P2$ is NP-complete. The reduction is as follows:
- set $G'= G$ and
- set $k'= f(k)$.
- $P1$ is solved iff $P2$ is solved.
My question is: is this reduction correct when $f(k)$ is exponential function?, For simplicity, say, $f(k)=2^k$. I am afraid that this is no longer polynomial-time but then again calculating $2^k$ or $k!$ is polynomial-time problem.
My whole question is general as the title mentions: Can we use exponential function in a reduction to create some parameter in the instance of the concerned problem?
My question is motivated by the validity of using exponentiation in polytime reduction on cstheory.SE where the OP asks about a specific reduction used in some paper. My question is general about using exponentiation in reduction.