x_i \in x \}.$$ We also denote by $KP^{LP}(x)$ the corresponding Linear Programming (LP) relaxation. If $\sum_{i \in S} w_i > C_l$, we define the *critical* item $c$ of $KP^{LP}(x)$ as the last item with a strictly positive value in its optimal solution. Thus, we have $y_c \in (0, 1]$ and a corresponding split solution with profit $$\label{eq:SplitSol}
\sum\limits_{i \in  S: i < c} p_i = \sum\limits_{i = 1}^{c-1} p_i(1 - x_i)$$ which constitutes a feasible solution for $KP(x)$. Notice that we denote by $z(M)$ the optimal solution value of any given mathematical model $M$.

Computing lower bounds on BKP {#sec:LB}
=============================

Consider the optimal solution vector $x^*$. In the induced follower’s knapsack problem $KP(x^*)$ with item set $S$, two cases can occur: either there is no critical item in $KP^{LP}(x^*)$, namely $\sum_{i \in S} w_i  \leq C_l$, or one critical item exists, namely $\sum_{i \in S} w_i > C_l$. The first case can be easily handled by considering that the follower will pack all items not interdicted by the leader. This case is discussed in Section \[NOcrit\].\
In the second case, we derive effective lower bounds on BKP that constitute the main ingredient of the exact approach presented in 