Assume we have a cryptographic protocol for secure multi-party computation that supports generic computation.
Let assume the "protocol 1" uses fully homomorphic encryption (FHE) and the overall computation complexity of the protocol is $O(c)$, where $c$ is input size (i.e. number of inputs). So it involves $O(c)$ homomorphic operations.
On the other, we have " protocol 2" designed for specific operation e.g. private set intersection (PSI). Its computation complexity is $O(c)$, but it does not use any FHE.
Question 1: If we use "protocol 1" for PSI, would the computation complexity still be $O(c)$ or it may increase?
Question 2: In general, how do we compare the complexity of the two protocols?