Given a complete, weighted and undirected graph $G$, complexity of finding a path with a specific cost

Question

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

Answer 1

Given an instance $\langle \{s_1,\ldots,s_n\}, T \rangle$ of subset sum, construct the following weighted graph. The vertices are $$ v_0,v_1,\ldots,v_n,u_1,\ldots,u_n. $$ Connect $v_{i-1}$ and $v_i$ with an edge of weight $s_i$. Connect $v_{i-1}$ to $v_i$ via $u_i$ (i.e., add the edges $\{v_{i-1},u_i\},\{u_i,v_i\}$) with zero weight edges. There is a simple path of total cost $T$ iff there is a subset of $\{s_1,\ldots,s_n\}$ summing to $T$. This shows that your problem is NP-hard, and in fact NP-complete, since it's clearly in NP.