Integer programming: enforce the constraint that a subgraph contains at most $k$ connected components?

Question

I'm considering integer programming on an variation of Steiner Forest Problem:

Given a graph $G=(V,E)$, a cost function: $c:E \rightarrow R^{+}$, a terminal set $T \subseteq V$, and a positive integer $k$, find a subgraph containing at most $k$ connected components that includes all the terminals with minimum total edge cost.

The key point of constructing the ingeger programming is to enforce the $k$ connected components constraint. The only way I came up with is using the generalized subtour elimination constraints over edge variables for each connected component. Is there any other ways to enforcing the $k$ connected components constraint? Any clue or suggestion would be appreciated.

Answer 1

Base on Komus's constraint, we add another constraint which ensures a Steiner Tree on $G^{'}=(V, E^{'})$, where $E'=\{(i,j): i,j \in V\}$: $$\sum_{e \in cut(U,V)}x_e \ge 1, \forall u,v \in T, \forall \text{ }u-v\text{ }cut \text{ }(U,V)$$where $cut(U,V)$ denotes the cut set of $(U,V)$.

Together with Komus's constraint, our model is obtained as follows:

$$min \sum_{e \in E}c_e x_e$$ subject to $$\sum_{e \in cut(U,V)}x_e \ge 1, \forall u,v \in T, \forall \text{ }u-v\text{ }cut \text{ }(U,V)$$ $$\sum_{e \notin E} x_e \le k-1$$

Base on Cao's comment, one more step is needed to prune the solution of our model: if $\sum_{e \notin E}x_e < k-1$, then we delete the largest $\Delta$ edges in the solution, where $\Delta=k-1-\sum_{e \notin E}x_e$. Then we get an optimal solution.