On lifted cover inequalities: A new lifting procedure with unusual properties

Abstract

Lifted cover inequalities are well-known cutting planes for 0-1 linear programs. We show how one of the earliest lifting procedures, due to Balas, can be significantly improved. The resulting procedure has some unusual properties. For example, (i) it can yield facet-defining inequalities even if the given cover is not minimal, (ii) it can yield facet-defining inequalities that cannot be obtained by standard lifting procedures, and (iii) the associated superadditive lifting function is integer-valued almost everywhere.

A cut-and-branch algorithm for the Quadratic Knapsack Problem

“…The convex hull of vectors x ∈ {0, 1} n satisfying the knapsack constraint (1) is called the knapsack polytope, and has been studied in depth (see, e.g., [1,15,18,20,28,29]). The most well-known valid inequalities for the knapsack polytope are the lifted cover (LC) inequalities.…”

Lifting the knapsack cover inequalities for the knapsack polytope

“…Wolsey [25] devised an elegant way to perform simultaneous lifting approximately, based on superadditive functions. This approach, sometimes called sequence-independent lifting, has been used to good effect in, e.g., [1,13,18]. However, the resulting inequality is not guaranteed to define a facet of P .…”

“…There are many papers on valid inequalities for knapsack polytopes. Most of these focus on lifted cover inequalities (e.g., [2,3,8,12,13,15,16,18,23,26]), but there are a few papers on other families of inequalities. These include weight inequalities [22], lifted pack inequalities [1,16], Chvátal-Gomory cuts [17], Fenchel cuts [4], and the inequalities in [5], which are (somewhat confusingly) called knapsack cover inequalities.…”

Multi-cover Inequalities for Totally-Ordered Multiple Knapsack Sets: Theory and Computation

“…, n} such that j∈C a j > b. The associated cover inequality (CI) j∈C x j ≤ |C| − 1 is valid for the knapsack polytope conv(K knap ) and is not satisfied by the incidence vector of C. There is a long and rich literature on (lifted) cover inequalities for the knapsack polytope [1,12,23,10,16], and the reader is directed to the recent survey [13] for a thorough introduction.…”