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Channel: Are there examples where introducing clusters of binary variables provides a benefit for solving? - Operations Research Stack Exchange
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# Are there examples where introducing clusters of binary variables provides a benefit for solving?

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I have a larger model with a large number of binary variables among many others. For the purpose of this question, consider the effect that the binary variables impose on the model to be similar to that of a Knapsack. The variable considers to take a certain item or not and in general taking an item provides a certain benefit. However, due to the rest of the model, the change is of course more complex. The overall size and complexity of the model makes it quite difficult to identify the right items to be taken, so I introduced some dependencies among the variables that may alter the optimal solution, but which led me to the idea of adding further variables and dependencies that do not affect the optimal solution. I was wondering on the effect of solving the model by introducing additional artificial binary variables that cluster the actual variables, by means of logical-or connections. More concretely, let's consider these two models

$$\max c^T \cdot x \\\text{s.t. } w^T \cdot x \leq W \\x_i \in \{0, 1\}$$

and

$$\max c^T \cdot x \\\text{s.t. } w^T \cdot x \leq W \\y_k \geq x_i \quad \forall \, k, i \in H_k \\y_k \leq \sum_{i \in H_k} x_i \\x_i \in \{0, 1\} \\y_k \in \{0, 1\}$$

Can there be a positive effect for solving the second model at all? And if yes, what would be the conditions (supposedly concerning $$H_k$$, but also the branching strategy of the solver) for such a positive effect? Or would presolve just eliminate all $$y_k$$ anyway?

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