of items $i \in [c,n]: x_i=0$ from the split solution without exceeding the follower capacity.

Notice that, the state-of-the-art algorithms for KP, *Minknap* ([@Pis97]) and *Combo* ([@MarPisTot99]) consider that in general only few items with ratio $p_i/w_i$ close to that of the critical item change their values in an optimal solution with respect to the values taken in the split solution. These items constitute the so-called *core* of the knapsack. *Minknap* and *Combo* start with the computation of the split solution and an expanding core initialized with the critical item only. Then, the algorithms iteratively enlarge the core by evaluating both the removal of items from the split solution and the addition of items after the critical item. The empirical evidence illustrates that an optimal (or close to be optimal) KP solution is typically found after few iterations.

We cannot precisely characterize the features of these exact algorithms by a set of constraints within an ILP model, but we can mimic the same algorithmic reasoning by considering subsets of the items set $c-\delta,...,c+\delta$ including the critical item $c$ for any given core size $2\delta+1$. In each subset, the items $i: i \leq c-1$ are removed from the split solution, while 