Reductions from non decision problems

Question

I want to show a minimization problem $Y$ has no approximation factor of 1.36. To be more specific the problem $Y$ is the exemplar distance problem between two genomes. Could I reduce from the min vertex cover problem instead of the decision version of the vertex cover problem. The problem I am having with reducing from the decision version is that a vertex cover of size k maps to the $Y$ of size $ \leq ck$, where $c$ is a constant. A decision version for problem $Y$ for me makes no sense, as there will always be a brekpoint distance between two genomes. I tried to research on the internet but I always only find reductions from decision problems. Could we reduce from non-decision problems.

Also when doing reductions from the vertex cover problem. I can't assume the given instance $G,k$ is such that k is the size of the optimal vertex cover right? $k$ is just any size of a Vertex cover.

Answer 1

Yes. See the notion of an approximation-preserving reduction.