of constraints in $\mathcal{F}(\pi, x, k)$, we propose a different strategy that greedily selects the subsets according to the procedure denoted as $ComputeTuples$ and sketched below. For a given value of $\delta$, we consider the interval of items $[a, b]$, with $a=\max\{1;c-\delta\}$ and $b=\min\{c+\delta;n\}$. Starting by the empty set, we enumerate at most $\alpha$ “backward” sets with items $(c-1), \dots, a$ in increasing order of size. Each set has a profit and weight equal to the sum of profits and weights of the included items. We also compute at most $\beta$ “forward” sets with items $c, \dots, b$ in increasing order of size and with a weight not superior to the maximum weight of a backward set. This in order to exclude forward sets having less chance to be combined with a backward set.

Then the backward (resp. forward) sets are ordered by increasing (resp. decreasing) profit. We combine each backward set with a forward set and generate a tuple $\tau$. If $p^{\tau} > 0$ and $w^{\tau} \leq w_c$, we add constraint to $\mathcal{F}(\pi, x, k)$. We continue adding constraints to $\mathcal{F}(\pi, x, k)$ until their number is superior to an input parameter $\mu$. If not previously included, we also 