New inequalities for finite and infinite group problems from approximate lifting

In this paper, we derive new families of facet-defining inequalities for the finite group problem and extreme inequalities for the infinite group problem using approximate lifting. The new valid inequalities for the finite group problem include two- and three-slope facet-defining inequalities as well as the first family of four-slope facet-defining inequalities. The new valid inequalities for the infinite group problem include families of two- and three-slope extreme inequalities. These new inequalities not only illustrate the diversity of strong inequalities for the finite and infinite group problems, but also provide a large variety of new cutting planes for solving integer and mixed-integer programming problems.