Simpler derivation of bounded pitch inequalities for set covering, and minimum knapsack sets

A valid inequality \alpha^Tx \ge \alpha_0 for a set covering problem is said to have pitch <= k ( a positive integer) if the k smallest positive \alpha_j sum to at least alpha_0. This paper presents a new, simple derivation of a relaxation for set covering problems whose solutions satisfy all valid inequalities of pitch and is of polynomial size, for each fixed . We also consider the minimum knapsack problem, and show that for each fixed integer p > 0 and 0 < \epsilon < 1 one can separate, within additive tolerance \epsilon, from the relaxation defined by the valid inequalities with coefficients in {0, 1, . . . , p} in time polynomial in the number of variables and 1/\epsilon.