How can I find the shortest path for all nodes in a graph from a source $s$?

Question

This is the shortest path problem. I've used a model where we can find the shortest path between the source and a specified destination.

The idea behind this model is that we assign a flow of 1 for the source and -1 for the destination and every other node has a flow of 0 because they're acting as transfer only.

However, I want to find the shortest path for every node in the graph from the source. It's similar to what the Dijkstra algorithm does however I want to use linear programming.

How can I adapt the model to give me the shortest path for every node in the graph from the source? Here's how the original model I used looks like.

[![graph][1]][1]

Where x12 is the arc from edge 1 to edge 2.

The basic idea would be to have all edges with a flow of -1 however when I try this it doesn't work. Any help would be appreciated. the graph used : [1]: https://i.stack.imgur.com/x8yuT.png

Answer 1

Dijkstra's algorithm finds a shortest path from $s$ to all other nodes in $N \setminus\{s\}$. The corresponding linear programming problem is to minimize $$\sum_{(i,j)\in A} c_{i,j} x_{i,j}$$ subject to $$\sum_{(i,j)\in A} x_{i,j} - \sum_{(j,i)\in A} x_{j,i} = \begin{cases} n-1 &\text{for $i=s$}\\ -1 &\text{for $i\in N \setminus \{s\}$} \end{cases}$$ and $x_{i,j}\ge 0$ for all $(i,j)\in A$.

That is, node $s$ has a supply of $n-1$, and every other node has a demand of $1$.