On the Relationship between Disjunctive Relaxations and Minors in Packing and Covering Problems

In 2002, Aguilera et al. analyzed the performance of the disjunctive lift-and-project operator defined by Balas, Ceria and Cornuéjols on covering and packing polyhedra, in the context of blocking and antiblocking duality. Their results generalize Lovász’s Perfect Graph Theorem and a theorem of Lehman on ideal clutters. This study motivated many authors to work on the same ideas, providing alternative proofs and analyzing the behaviour of other lift-and-project operators in the same context. In this paper, we give a survey of the results in the subject and add some new results, showing that the key of the behaviour of the disjunctive operator on these particular classes of polyhedra is the strong relationship between disjunctive relaxations and original relaxations associated to some minors.