items within a CPU time limit of 60 seconds.

The paper is organized as follows. In Section \[sec:themodel\], the bilevel linear programming formulation of the problem is introduced. In Section \[sec:LB\], we discuss the lower bounds for BKP. We outline the proposed exact solution approach in Section \[sec:ExaApp\] and discuss the computational results in Section \[sec:ComRes\]. Section \[sec:Concl\] provides some concluding remarks.

Notation and problem formulation {#sec:themodel}
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In BKP a set of $n$ items and two knapsacks are given. Each item $i$ $(=1,\dots, n)$ has associated a profit $p_i > 0$ and a weight $w_i > 0$ for the follower’s knapsack and a weight $v_i > 0$ for the leader’s knapsack. Leader and follower have different knapsack capacities denoted by $C_u$ and $C_l$, respectively. Quantities $p_i$, $v_i$, $w_i$ $(i=1,\dots, n)$, $C_u$, $C_l$ are assumed to be integer, with $v_i < C_u$ and $w_i < C_l$ for all $i$. To avoid trivial instances, it is also assumed that $\sum\limits_{i=1}^{n} v_{i} > C_u$ and $\sum\limits_{i=1}^{n} w_{i} > C_l$. We introduce $0/1$ variables $x_i$ $(i=1, \dots,n)$ equal to one if the leader selects items $i$ and $0/1$ variables $y_i$ equal to one if item $i$ is chosen by the follower. BKP can 