us now recall the optimal solution of the continuous relaxation of a standard KP, namely the follower’s model - without constraints and constraints replaced by inclusion in $[0, 1]$. Under the assumption $\sum\limits_{i=1}^{n} w_{i} > C_l$, this solution has the following structure. Consider the sorting of the items by non-increasing ratios of profits over weights:

$$\label{eq:SortByEff}
\frac{p_1}{w_1} \geq \frac{p_2}{w_2} \geq \dots \geq \frac{p_n}{w_n}.$$

According to this order, items $j = 1,2,\dots$ are inserted into the knapsack as long as $\sum\limits_{k=1}^{j} w_{k} \leq C_l$. The first item $s$ which cannot be fully packed is commonly denoted in the knapsack literature as the *split* item (or *break*/*critical* item). The optimal solution of the KP linear relaxation is given by setting $y_j = 1$ for $j= 1,\dots,s-1$, $y_j = 0$ for $j= s+1,\dots,n$ and $y_s = (C_l - \sum\limits_{j=1}^{s-1} w_j)/w_s$. The solution with items $1, \dots, (s-1)$ is a feasible solution for KP and is commonly denoted as the *split solution*. In the remainder of the paper, we assume the ordering of the items . We denote by $KP(x)$ the follower’s knapsack problem induced by a leader’s strategy encoded in vector $x$, i.e. a knapsack problem with item set $$S:=\{i: x_i = 0, 