In [1] a graph problem is proposed:
- Given an undirected network,
- Find all paths between two nodes,
- Such that no arc is used more than once,
- And the path does not contain more than \(M\) arcs
Preliminaries
---- 30 SET n nodes n1 , n2 , n3 , n4 , n5 , n6 , n7 , n8 , n9 , n10 ---- 30 SET a directed arcs n1 n2 n3 n4 n5 n6 n1 YES YES + n7 n8 n9 n10 n1 YES |
| Shortest Path Model |
|---|
| \[\begin{align}\min&\sum_{i,j|a(i,j)}\color{darkred}f_{i,j}\\ & \sum_{i|a(i,n)} \color{darkred}f_{i,n} + \color{darkblue}{\mathit{inflow}}_n = \sum_{j|a(n,j)}\color{darkred}f_{n,j} &&\forall n\\ & \color{darkred}f_{i,j} \in [0,1] \end{align}\] |
where \[ \color{darkblue}{\mathit{inflow}}_n = \begin{cases} 1 & \text{if $n=n_1$}\\ -1 & \text{if $n=n_{10}$} \\ 0 & \text{otherwise}\end{cases}\] When we run this model, we will see:
---- 82 VARIABLE count.L = 2.000 length of shortest path (hops) ---- 82 VARIABLE f.L flow n8 n10 n1 1.000 |
---- 127 PARAMETER fsol solution from solution pool INDEX 1 = soln_sp_p1 n8 n10 n1 1 INDEX 1 = soln_sp_p2 n4 n8 n10 n1 1 INDEX 1 = soln_sp_p3 n2 n8 n10 n1 1 INDEX 1 = soln_sp_p4 n4 n6 n10 n1 1 INDEX 1 = soln_sp_p5 n2 n6 n10 n1 1 INDEX 1 = soln_sp_p6 n3 n8 n10 n1 1 |
All-Paths Model V1
| SP + MTZ Model |
|---|
| \[\begin{align} \min\>&\color{darkred}z \\ &\color{darkred}z= \sum_{i,j|a(i,j)}\color{darkred}f_{i,j} \\ &\color{darkred}z \le \color{darkblue}M\\ & \sum_{i|a(i,n)} \color{darkred}f_{i,n} + \color{darkblue}{\mathit{inflow}}_n = \sum_{j|a(n,j)}\color{darkred}f_{n,j} &&\forall n \\ & \color{darkred}t_j \ge \color{darkred}t_i + 1 + (\color{darkblue}N-1)(\color{darkred}f_{i,j}-1) && \forall a_{i,j}, i\ne n_1,j \ne n_{10} \\ & \color{darkred}f_{i,j} \in \{0,1\} \\ & \color{darkred}t_n\ge 0 \end{align}\] |
---- 125 PARAMETER fsol solution from solution pool INDEX 1 = soln_sp2_p1 n8 n10 n1 1 INDEX 1 = soln_sp2_p2 n2 n6 n10 n1 1 INDEX 1 = soln_sp2_p3 n2 n8 n10 n1 1 INDEX 1 = soln_sp2_p4 n4 n6 n10 n1 1 |
All-Paths Model V2
- I added nodes: \(src \text{-} n_1\) or \(n_{10} \text{-} snk\).
- The arcs are indicating adjacency. I.e. we have an arc \(n_1 \text{-} n_2 \rightarrow n_2 \text{-} n_8\).
- I removed self-loops such as : \(n_4 \text{-} n_4 \rightarrow n_4 \text{-} n_4\). They will not be used. (I could have left them in).
- A picture of the new graph is added as an appending (it is relatively large).
- As you can see in the augmented graph each node corresponds to an original arc. Each arc is a particular case of an original node. For instance: an arc related to \(n_2\) occurs two times: \(n_1 \text{-} n_2 \rightarrow n_2 \text{-} n_6\) and \(n_1 \text{-} n_2 \rightarrow n_2 \text{-} n_8\) .
| SP + MTZ Model |
|---|
| \[\begin{align} \min\>&\color{darkred}z \\ &\color{darkred}z= \sum_{i,j,p,q|a(i,j,p,q)}\color{darkred}f_{i,j,p,q} \\ &\color{darkred}z \le \color{darkblue}M+1\\ & \sum_{i,j|a(i,j,p,q)} \color{darkred}f_{i,j,p,q} + \color{darkblue}{\mathit{inflow}}_{p,q} = \sum_{i,j|a(p,q,i,j)}\color{darkred}f_{p,q,i,j} &&\forall p,q \\ & \color{darkred}t_{p,q} \ge \color{darkred}t_{i,j} + 1 + (\color{darkblue}{N}-1)(\color{darkred}f_{i,j,p,q}-1) && \forall a_{i,j,p,q}, \text{except src and snk} \\ & \color{darkred}f_{i,j,p,q} \in \{0,1\} \\ & \color{darkred}t_{i,j}\ge 0 \end{align}\] |
---- 206 PARAMETER fsoln solution from solution pool INDEX 1 = soln_sp3_p1 n1.n8 n8.n10 n10.snk n1 .n8 1 INDEX 1 = soln_sp3_p2 n1.n2 n2.n8 n8.n10 n10.snk n1 .n2 1 INDEX 1 = soln_sp3_p3 n1.n4 n4.n6 n6.n10 n10.snk n1 .n4 1 INDEX 1 = soln_sp3_p4 n1.n2 n2.n6 n6.n10 n10.snk n1 .n2 1 INDEX 1 = soln_sp3_p5 n1.n8 n3.n8 n8.n3 n8.n10 n10.snk n1 .n8 1 INDEX 1 = soln_sp3_p6 n1.n8 n3.n9 n8.n3 n9.n10 n10.snk n1 .n8 1 INDEX 1 = soln_sp3_p7 n1.n4 n3.n9 n4.n3 n9.n10 n10.snk n1 .n4 1 INDEX 1 = soln_sp3_p8 n1.n4 n3.n8 n4.n3 n8.n10 n10.snk n1 .n4 1 INDEX 1 = soln_sp3_p9 n1.n4 n4.n4 n4.n6 n6.n10 n10.snk n1 .n4 1 |
Conclusion
References
- Given an undirected graph, what is the optimal way to find all paths between node A and node B given a maximum amount of arcs? (in python), https://stackoverflow.com/questions/70684721/given-an-undirected-graph-what-is-the-optimal-way-to-find-all-paths-between-nod
- Longest path problem, https://yetanothermathprogrammingconsultant.blogspot.com/2020/02/longest-path-problem.html
Appendix A: augmented graph
Appendix B: GAMS model
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