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

Abstract

A valid inequality α T x ≥ α0 for a set covering problem is said to have pitch ≤ π (π a positive integer) if the π smallest positive αj sum to at least α0. This paper presents a new, simple derivation of a relaxation for set covering problems whose solutions satisfy all valid inequalities of pitch ≤ π and is of polynomial size, for each fixed π. We also consider the minimum knapsack problem, and show that for each fixed integer p > 0 and 0 < ǫ < 1 one can separate, within additive tolerance ǫ, from the relaxation defined by the valid inequalities with coefficients in {0, 1, . . . , p} in time polynomial in the number of variables and 1/ǫ.

A very fast method for guaranteed generation of one facet for 0-1 knapsack polyhedron

“…Most of the work in this direction has been focused on characterization of facets arising from lifting of the so-called minimal cover inequalities. Properties of the formal lifting procedure presented in [1,3,6,[17][18][19][20] and its resulting facets have been studied in several of the aforementioned references. The lifting procedure is dependent on an initial minimal cover inequality, and in most cases, the sequence of the variables chosen for lifting.…”