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Given a set $S$ which we need to travel following TSP rules.

I was wondering if this sub tour elimination method is good enough or not?

Let $b_{i,j}$ denote edge from $i$ to $j$ is taken or not and $d_{i,j} > 0$ denotes distance from $i$ to $j$.

\begin{align}\min&\quad\sum_{i,j \in S} d_{i,j} \cdot b_{i,j}\\\text{s.t.}&\quad\sum_{j \in S} b_{j,i} - \sum_{k \in S} b_{i,k} = 0\\&\quad\sum_{j \in S} b_{j,i} = 1\end{align}

Let $s_0$ be the start node. Now use a continuous variable $DS_i$ to store the distance at node $i$, with $DS_{s_0} = 0$.

$$ \forall j \in S \setminus \{s_0\} \quad DS_{j} = \sum_{i} b_{i,j} \cdot (DS_{i} + d_{i,j}) $$

The last constraint eliminates the sub-tour in the path.

My question is how efficient this sub tour elimination constraint is and how to calculate it.

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My understanding is that you effectively mimic the Miller, Tucker, Zemlin style of subtour elimination constraints by "pushing a counter along the tour", these are reportedly not the computationally strongest formulations.

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