  In the min-knapsack problem one aims at choosing a set of objects with
minimum total cost and total profit above a given threshold. In this paper, we
study a class of valid inequalities for min-knapsack known as bounded pitch
inequalities, which generalize the well-known unweighted cover inequalities.
While separating over pitch-1 inequalities is NP-hard, we show that approximate
separation over the set of pitch-1 and pitch-2 inequalities can be done in
polynomial time. We also investigate integrality gaps of linear relaxations for
min-knapsack when these inequalities are added. Among other results, we show
that, for any fixed $t$, the $t$-th CG closure of the natural linear relaxation
has the unbounded integrality gap.
