by applying the same argument of Proposition \[firstProp\].

A new exact approach for BKP {#sec:ExaApp}
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Overview
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We propose an exact algorithm for BKP that considers the possible existence of a critical item in $KP^{LP}(x^*)$ and exploits the bounds provided by model $CRIT_2(c)$. The approach involves two main steps. In the first step, the possible non-existence of a critical item is first evaluated. Then, the approach assumes the existence of a critical item and identifies a set of possible candidate items. For each candidate item $c$ and a parameter $\delta$ to identify the core size, model $CRIT_2(c)$ is built by considering several subsets of additional constraints (\[TupleCons\]). Then the linear relaxation $CRIT_2^{LP}(c)$ is solved, where the integrality constraints and are replaced by inclusion in $[0,1]$.

The feasible problems $CRIT_2^{LP}(c)$ are sorted by increasing optimal value so as to identify an order of the most promising subproblems to explore. A limited number of feasible BKP solutions is also computed in this step.

In the second step, each relevant subproblem is explored by constraint generation until the subproblem can be pruned. An optimal BKP solution is eventually returned. The approach takes as input five parameters $\alpha$, $\beta$, $\delta$, $\mu$, $\gamma$ and 