Given the linear programming formulation of the shortest path problem: $$ \begin{align*} \min & \sum_{u,v \in A} c_{uv} x_{uv}\\ \text{s.t } & \sum_{v \in V^{+}(s)} x_{sv} - \sum_{v \in V^{-}(s)}x_{vs} = 1\\ & \sum_{v \in V^{+}(u)} x_{uv} - \sum_{v \in V^{-}(u)}x_{vu} = 0 \quad \text{ for each } u \in V\setminus\{s,t\}\\ & \sum_{v \in V^{+}(t)} x_{tv} - \sum_{v \in V^{-}(t)}x_{vt} = -1\\ & x_{uv} \ge 0 \end{align*} $$
I am asked to show that it provides an optimal integer solution by building a solution using the Dijkstra algorithm for both the primal and the dual with both solutions having the same value.
I have no troubles in finding the dual, however I am unclear about what exactly I have to do. In particular, how do I apply Dijkstra in finding a solution for the above program?