Cover and pack inequalities for (mixed) integer programming

We review strong inequalities for fundamental knapsack relaxations of (mixed) integer programs. These relaxations are the 0–1 knapsack set, the mixed 0–1 knapsack set, the integer knapsack set, and the mixed integer knapsack set. Our aim is to give a unified presentation of the inequalities based on covers and packs and highlight the connections among them. The focus of this paper is on recent research on the use of superadditive functions for the analysis of knapsack polyhedra. We also present some new results on integer knapsacks. In particular, we give an integer version of the cover inequalities and describe a necessary and sufficient facet condition for them. This condition generalizes the well-known facet condition of minimality of covers for 0–1 knapsacks.