Let $G=(V,E)$ be a graph, where $|E|=m$, and suppose we formulate the shortest path problem on $G$ as follows: minimize ${}^t(1,\dots,1)x$ such that $Bx={}^t(1,-1,0,\dots,0), x\in \{0,1\}^m$, where $B \in M_{n\times m}(\mathbb Z)$ is the incidence matrix (whose rows correspond to vertices and whose columns to edges) of $G$.
Here I want to show that the optimal solution doesn't change if $x$ runs through $[0,1]^m$ instead of $\{0,1\}^m$. I've already shown that we may assume $x$ runs through $[0, \infty)$, i.e., the vertices of the polytope defined by the problem are integral. It remains to show that the optimum is attained by $x \le 1$.
How can I show that?