Given a fully connected graph $G$, suppose that we are searching for a simple path $P$ with a specific cost $c$.
Is answering to that problem yes or no equivalent to subset-sum problem? What would be the complexity of finding such path?
I have made a reduction from subset-sum problem:
If each number in a set $S$ is a vertex of $G$ and weight of $<i,j>$ is $|i-j|$, then answering the question above yes or no is the same as solving the sumbet-sum problem.
P.S. The initial vertex I have visited is added to the cost.
Edit: Edge weights