# In routing problems, when is it ever necessary to include both 1) subtour elimination constraints, AND 2) elementary paths constraint?

In many routing problems, it is fairly common to include a constraint that ensures all vehicles follow an elementary path, meaning that no vertices are repeated.

However, when an elementary path is NOT required, then it is fairly common to include a constraint that ensures there are no subtours meaning that there are no isolated tours that do not start at, and return to, the specified depot.

From my understanding, it would never be necessary to include both of these constraints. However, in some cases I have seen both of these constraints, for example here: https://par.nsf.gov/servlets/purl/10074741

In routing problems, when is it ever necessary to include both 1) subtour elimination constraints, AND 2) elementary paths constraint?

EDIT I've found another resource which highlights my intuition: In this book it describes a similar problem, the vehicle routing problem with time windows. And, in its MILP formulation (shown below), it does not have subtour elimination constraints, and explicitly mentions the classical VRP subtour elimination constraints become redundant. Now, I would have expected this to be the same reason why the image 2 would not need the subtour elimination constraints, since in both cases, the direction of traversal is indicated by denoting $$(i,j)$$.

The difference between these two MILP formulations and problem statements might help me understand what you guys are getting at. Is the only difference that one of them imposes a unique direction?

• In the cited example, what are you interpreting as an elementary path constraint? If it is (4), keep in mind that (4) ensures (a) that every node is visited at least once and (b) that no two vehicles visit the same node.
– prubin
Commented Sep 23, 2023 at 19:41
• I'm not seeing how (4) can be interpreted that way. There is a summation over all the vehicles $\Sigma_k$. So, to me it says "Each vertex can be visited once and by only one vehicle (except for the depot)". Furthermore in the description below it says "(4) ensure that each node is visited exactly once. Commented Sep 24, 2023 at 6:34
• The necessary part of (4) is the $\ge$ from the $=$ which is not in the subtour elimination constraint. Without it, the solution is just to visit nothing Commented Sep 24, 2023 at 7:07
• @fontanf To me, (4) says "exactly one incoming arc for each vertex in $V^+$. Is that right? I don't understand what you mean by $\ge$. Constraint (4) is an equality, and the subtour elimination constraint is $\le$. Do you mind being more explicit? Commented Sep 25, 2023 at 22:10
• I editted the question with another problem formulation I've seen where the subtour elimination constraints are redundant. Maybe that will help me see what you're talking about? Commented Sep 26, 2023 at 6:59